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Numerical study on crack propagation in high velocity perforation
Computers and Structures 83 (2005) 989–1004 www.elsevier.com/locate/compstruc
Numerical study on crack propagation in high velocity perforation X. Teng, T. Wierzbicki
*
Impact and Crashworthiness Lab, Room 5-218, Massachusetts Institute of Technology, Cambridge, MA 02139, USA Received 18 November 2003; accepted 10 December 2004
Abstract Perforation response of a target under high velocity projectile impact is controlled by a number of factors such as the thickness of a target, the mass and geometrical shape of a projectile, etc. In this paper, through-thickness crack propagation in perforation problems is studied numerically on the basis of a newly constructed fracture locus formulated in terms of the effective plastic strain and the stress triaxiality. Through a series of numerical simulations using ABAQUS/ Explicit, an expression relating crack length to indentation depth is developed. This expression is very general and applicable to both a beam and a circular plate over a wide range of the impact velocity, the mass ratio, and the thickness, provided that the target fails by shear plugging. It is found that the critical indentation depth for the formation and ejection of a plug is about 20–30% of the target thickness for 2024-T351 aluminum alloy, which can be used as an elementary fracture criterion. The average crack tip speed was found to be of order of 1000 m/s, which is one-third of the elastic shear wave speed of 2024-T351 aluminum alloy. 2005 Elsevier Ltd. All rights reserved. Keywords: Finite element analysis; Crack tip speed; Crack propagation; Shear plugging; Perforation
1. Introduction Failure response of a target struck by a projectile has long been studied in the context of design of effective armor plates. It has also become important in industrial applications. For example, the problem of the fuselage skin or the fuel tank of an aircraft perforated by uncontained turbine engine debris, which caused several catastrophes in the past 20 years [1], has been of concern to aircraft structural engineers. The September 11th attack
* Corresponding author. Tel.: +1 617 253 2104; fax: +1 617 253 1962. E-mail address: [email protected] (T. Wierzbicki).
illustrated an extreme example, in which the exterior columns of the Twin Towers were cut through by the aircraft wings, and the core structures were severely destroyed by the fuselage leading the collapse of the Twin Towers. Much work has been carried out experimentally, theoretically, and numerically on the perforation problem. Comprehensive reviews on perforation mechanisms and analytical solutions can be found in the journal articles by Anderson and Bodner [2], Corbett et al. [3], and Goldsmith [4]. The major objective of most of perforation experiments is to determine the ballistic limit of the target. In such tests, the initial impact velocity of projectiles is varied, while other parameters such as the thickness of targets, the mass of projectiles, etc. are kept
0045-7949/$ - see front matter 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.compstruc.2004.12.001
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Nomenclature fraction of plastic work converted into heat material constant in Eq. (4) transverse deflection of the target engineering shear strain reference strain rate fracture strain effective plastic strain plastic strain rate mass ratio of the impact zone to the projectile q mass density r0 plastic flow stress rh hydrostatic stress ri initial stress r equivalent stress A initial yield stress in the JC material model a crack length B hardening modulus in the JC material model C strain rate parameter in the JC material qffiffiffiffi model c ¼ rq0 transverse plastic stress wave speed
a b d c e_ 0 ef epl e_ pl l
constant. The failure mechanism of the target is also controlled by other parameters besides the impact velocity. Corran et al. [5] investigated experimentally effects of projectile masses, nose shapes, target thicknesses, and boundary conditions on ballistic limits. Recently, Børvik et al. [6] studied the ballistic limit of Weldox 460 E steel plates over a wide range of thicknesses both experimentally and numerically. As an alternative to experiments, a large scale numerical simulation has now become affordable. Finite element methods have reached a state of maturity. Many commercial codes such as ABAQUS, LS-DYNA3D, etc. are capable of simulating the perforation process, provided that they are equipped with a suitable fracture criterion. A number of numerical simulations have been published in the open literature, for example, Ambur et al. [7] and Guo et al. [8], but they lacked corroboration from experiments. The most systematic, comprehensive studies to date combining numerical simulations with experiments are due to Børvik and his colleagues [9,10]. They first performed a series of simple tensile tests to determine material parameters in the constitutive equation as well as the fracture criterion, and then conducted a number of perforation experiments with varying impact velocities. They also carried out parallel numerical simulations using LS-DYNA3D. Rather good agreement with experimental results gives a much confidence in the capability and accuracy of finite element codes to predict perforation response.
cv D d e h K M0 n q R T T0 Tm t u ucr V0 Vr v
specific heat damage projectile breadth/diameter shear hinge length beam thickness material constant in Eq. (4) projectile mass material hardening exponent temperature parameter in the JC material model radius of round corners of a projectile temperature room temperature melting temperature time indentation depth critical indentation depth at complete perforation initial impact velocity residual velocity crack tip speed
As compared to extensive literature on the ballistic limit, only few papers deal with through-thickness crack propagation, even though crack propagation is always involved in the perforation problem. Børvik et al. [11] observed from scan electronic micrographs that microvoids and micro-cracks nucleate, grow, and coalesce in front of the crack tip. It indicates that a metal target under high velocity projectile impact fails by ductile fracture. As Rosakis and Ravichandran [12] pointed out, dynamic ductile fracture mechanics remains virtually unexplored. Analogous to static loading cases, dynamic stress intensity factors and generalized J integrals were proposed to predict dynamic fracture initiation and growth [12]. In the field of impact engineering, another approach has recently emerged, in which the fracture criterion is assumed to depend on the effective plastic strain and the stress triaxiality. Such a formulation can predict sequential material failure in the perforation problem, for example, Børvik et al. [13]. The idea of material ductile failure dependent on the stress triaxiality is not new and goes back to the fundamental work of McClintock [14]. Johnson and Cook [15] introduced this concept combined with effects of strain rates and temperature to predict dynamic ductile failure. However, detailed information on how a through-thickness crack propagates was not reported in the relevant literature. Most of perforation experiments were performed on opaque plate specimens such that crack growth through the target thickness is difficult to track down using currently
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available techniques. By contrast, finite element procedures are capable of overcoming this difficulty and providing abundant details about crack initiation, crack tip speeds, and crack trajectories. In an earlier paper by the present authors [16], effects of three different types of ductile fracture criteria on high velocity impact have been addressed. In the present paper, parametric studies of a number of factors including the initial impact velocity and mass of the projectile, the thickness and breadth of the beam, pre-loading, etc. are performed using the Bao–WierzbickiÕs fracture locus implemented in ABAQUS/Explicit. Attention is focused on crack propagation through the thickness of targets. The major objective of this research is to verify an analytical expression controlling crack growth.
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Fig. 2. Schematic representation of a circular plate struck by a rigid cylindrical body.
2. Finite element models Two types of projectile–target systems are considered in the present paper: a long, solid beam of rectangular cross-section impacted by a rigid cubic projectile, see Fig. 1, and a large circular plate impacted by a flatnosed, rigid, cylindrical projectile, see Fig. 2. In finite element modeling, both the beam length (the diameter of the circular plate) and the breadth (the diameter) of the projectile were kept constant, while other geometrical and mechanical parameters such as the mass of the projectile M0, the thickness of the target h, the impact velocity V0, etc. were varied to investigate their effects on the perforation process. Since attention is focused on crack propagation through the thickness, 2-D finite element models were built instead of 3-D solid element models. A narrow beam can expand laterally with little constraints when it is indented by a projectile. Thus, plane stress elements were used for this type of beams, while plane strain elements have to be used for a wide beam. For a circular plate under high velocity impact by a blunt projectile, shear plugging is a favorable failure mode rather than petalling, and thus a 2-D model can also be built using axisymmetric elements. The deformation and possible failure of the projectile was not taken into account in
Fig. 1. Schematic representation of a long beam struck by a rigid mass.
Fig. 3. Finite element meshes near and in the impact area with 50 elements through the thickness.
the present formulation. The projectile is represented by a rigid surface in the finite element model. A thorough investigation of mushrooming deformation and possible fracture of a cylindrical projectile in the Taylor test is a subject of another publication [17]. Finite element meshes near and in the impact area of the target are shown in Fig. 3. Square elements with the size 0.2 mm · 0.2 mm were generated in the impact area. There is a total of 50 elements through the target thickness. To reduce computational cost, relatively coarse meshes were used in the region away from the impact area. Such a mesh model is able to provide correct results. A study on mesh size sensitivity was performed and results are presented in Section 5. The Johnson–Cook (JC) constitutive model implemented in ABAQUS/Explicit was used to describe the plastic flow properties of the target such as strain hardening, strain rate hardening, and thermal softening [18]: q h i e_ pl T T0 n ¼ A þ Bepl 1 þ C ln r 1 : ð1Þ Tm T0 e_ 0 The target is assumed to be made of 2024-T351 aluminum alloy. The material constants of Eq. (1) are listed in Table 1, and the true stress–strain curves under various combinations of temperature and strain rates are illustrated in Fig. 4. The isotropic J2 plasticity model
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Table 1 Material constants for 2024-T351 aluminum alloy E (GPa) 74.66
m 0.3
q (kg/m3) 2700
e_ 0 (s1)
C 4
3.33 · 10
0.0083
cv (J/kg K)
a
Tm (K)
T0 (K)
q
875
0.9
775
293
1.0
A (MPa)
B (MPa)
n
r0 (MPa)
c (m/s)
352
440
0.42
565.5
457.6
Fig. 5. The Bao–WierzbickiÕs fracture locus for 2024-T351 aluminum alloy [19].
Fig. 4. True stress versus true strain at various strain rates and temperature for 2024-T351 aluminum alloy.
was selected from the ABAQUS library of material models. It is assumed that a crack is formed when the accumulated plastic strain, modified by a function of the stress triaxiality, rh = r, reaches a critical value, D = 1, Z epl 1 depl : ð2Þ D¼ f rrh 0 The weighting function f ðrh = rÞ is not a monotonically r, as comdecreasing function of the stress triaxiality rh = monly believed, but consists of several branches, each dominated by a different micro-fracture mechanism [19,20]. For example, Bao and Wierzbicki found that for 2024-T351 aluminum alloy the function f ðrh = rÞ consists of four distinct branches [19], Fig. 5: 8 rh 1 6 13 ; > r > > > 0:46 r < 0:1225 rrh þ 13 13 < rrh 6 0; h ¼ f 2 > r > 1:9 rrh 0:18 rrh þ 0:21 0 < rrh < 0:4; > > : 0:4 < rrh : exp 1:944 rrh ð3Þ A unique feature of the Bao–WierzbickiÕs (BW) fracture criterion is the cut-off value for the negative stress tri-
axiality at rh = r ¼ 1=3. The physical meaning of the cut-off value is that a crack would never be generated in a material element subjected to hydrostatic compression, i.e. rh = r < 1=3. This newly constructed fracture locus was used to predict crack formation in this paper. When the cumulative damage D reaches unity at an integration point of an element, all the stress components are suddenly set to zero at this point. If all the integration points in an element fail, this element will lose load-carrying capability and be removed from the mesh to model crack initiation and growth. A more detailed discussion on the application of the JCÕs material model and the BWÕs fracture model to the perforation problem was presented in a companion paper by the present authors [16].
3. Indentation and crack propagation Fig. 6 shows a sequence of the perforation process for one specific loading case. Immediately upon impact, the part of the beam beneath the projectile is indented. Large plastic deformation and large compressive (negative) stresses develops in this area, but no elements fail throughout the whole process. At a certain time, two cracks initiate near the corners of the projectile, and rapidly grow through the beam thickness until complete perforation. Finally a plug is formed and ejected from the target. These are typical failure scenarios of shear plugging. During this failure process, plastic deformations occur at three distinct scales: very localized shear deformation through the target thickness, large compression of the target directly beneath the projectile, and global plastic bending deformation. This type of failure pattern
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authorsÕ earlier work [16] suggests the following relationship between these quantities: b a c u ¼ K ð1 þ lÞ ; ð4Þ h V0 h where K and b are two material constants; the mass ratio l of the impact area of the beam to the projectile is defined by l¼
qhbd ; M0
ð5Þ
where q is the mass density of the beam, and b is the beam breath. In Eq. (4), c is the transverse wave speed defined by rffiffiffiffiffi r0 ; ð6Þ c¼ q where r0 is the plastic flow stress under quasi-static conditions. Differentiating Eq. (4) with respect to time gives the crack tip speed v: b v c u b1 ¼ bK ð1 þ lÞ ; ð7Þ V V0 h
Fig. 6. Shear plugging process of the plane-stress beam of h = 10 mm impacted by the blunt projectile of l = 0.1 at V0 = 240 m/s.
where v = da/dt; and V = du/dt is the indentation rate. If the global deformation of the target can be neglected, V is just the velocity of the projectile. A derivation of the expressions for both instantaneous and average crack tip speeds will be presented in another publication [21]. It should be mentioned that Eq. (4) was obtained by curve-fitting crack growth for a specific case in which the initial impact velocity was varied and other parameters were kept constant. The development of Eq. (4) in an analytical way remains open. The immediate objective of the present paper is to show that Eq. (4) derived for a specific set of input parameters considered in Ref. [16] is also valid in a wide range of impact speeds, plate thicknesses, and mass ratios. This is accomplished through an extensive parametric study in which only one parameter was varied at a time. 4. Parametric studies 4.1. Impact velocities
Fig. 7. Schematic of crack propagation showing the crack length a and the indentation depth u.
can be characterized by two variables: the crack length a and the indentation depth u, as shown in Fig. 7. The
Consider a narrow beam of the thickness h = 10 mm under high velocity projectile impact of the mass ratio l = 0.1. The impact velocity was varied from 160 m/s to 300 m/s, while other parameters were kept constant. Studied here is the effect of the impact velocity. As discussed in Ref. [16], there are three possible failure modes for a beam struck by a rigid projectile: tensile tearing, shear plugging, and adiabatic shear banding in the ascending order of the impact velocity. Numerical
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results indicate that the present beam fails by tensile tearing at V0 = 160 m/s, and by shear plugging at higher impact velocities. Plots of the normalized crack length with the normalized indentation can be fitted well by Eq. (4) with K = 10 and b = 4, except for the case with V0 = 160 m/s, see Fig. 8. As the impact velocity increases further, i.e. V0 > 300 m/s, the failure mode of the beam would change from shear plugging into adiabatic shear banding. The present expression in Eq. (4) cannot be extended to the range of adiabatic shear banding. The latter problem is the subject of current research [22]. Plots of crack propagation versus time at various impact velocities are shown in Fig. 9. It appears that crack propagation in shear plugging consists of two phases. In the first phase, uniform indentation is dominant and
cracks slowly grow, while in the second phase, localized shear deformation is dominant and crack propagation is much faster. The time history of the crack length for the present cases can be fitted well by piecewise linear functions, which indicates that the crack tip speeds are almost constant in the respective phases. The crack growth in the second phase is more important, because most of the elements fail in this phase. Hereinafter, the crack tip speed is referred to the average value in the second phase. At the same time, one can see from Fig. 9 that the crack tip speed strongly depends on the impact velocity of the projectile. The average crack tip speeds are given by v = 1056, 1007, and 707 m/s, corresponding to the impact velocity V0 = 300, 240, and 200 m/s. These crack growth speeds are comparable to the values in Børvik et al.Õs examples [13]. Two speeds of 667 m/s and 1300 m/s were estimated from Figs. 13 and 14 in their paper. Note, that different materials were used in both studies. No relevant experiments are found in the open literature to verify the accuracy of the obtained crack tip speeds. Compared with the residual velocity of the projectile, the crack tip speed would probably be more sensitive to numerical modeling techniques and material micro-structures. 4.2. Projectile mass
Fig. 8. Crack propagation at various impact velocities for the plane-stress beam of h = 10 mm and l = 0.1.
Fig. 9. Time history of crack length at various impact velocities for the plane-stress beam of h = 10 mm and l = 0.1.
Another parameter controlling the perforation process is the weight of the projectile. Its effect on the transition of the failure mode and crack growth in the narrow beam is investigated in this section. The mass ratio l was varied from 0.05 to 0.45 among ten cases, while the initial impact velocity of V0 = 240 m/sec and the beam thickness of h = 10 mm were kept constant. Depending on the projectile mass, two types of failure modes: shear plugging and tensile tearing, were observed from the failure pattern. For l < 0.35, the beam fails due to shear plugging. When the mass ratio falls in the range of 0.35 6 l 6 0.45, tensile tearing is the dominant failure mode. As the mass ratio continuously increases, i.e. the mass of the projectile further decreases, l > 0.45, the beam of an infinite length will never fracture. The critical mass ratio is about 0.35 at the transition of the failure mode from shear plugging to tensile tearing. Clearly, the critical mass ratio depends on the initial impact velocity and other parameters such as the beam thickness. The transition of the failure mode can also be observed from the relationship between the residual velocity and the mass ratio, see Fig. 10. In shear plugging, the residual velocities decrease with the mass ratios in a hyperbolic way, which is similar to the relation between the ballistic limit and the projectile mass, as found by Corran et al. [5] from a series of experiments. The residual velocity suffers then a jump as the failure mode
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Fig. 10. Residual velocity vs. mass ratio for the plane-stress beam of h = 10 mm at V0 = 240 m/s.
Fig. 12. Average crack tip speeds vs. mass ratios for the planestress beam of h = 10 mm at V0 = 240 m/s.
changes from shear plugging to tensile tearing. In tensile tearing, most of the kinetic energy of the projectile is absorbed by the beam in the form of plastic bending and axial stretching, while, in shear plugging, only a small part of the kinetic energy is dissipated in the beam due to localized shear deformation and indentation in the impact area. Hence, the residual velocity in shear plugging is much higher than that in tensile tearing. Additionally, it can be seen from Fig. 10 that the simple momentum conservation approach gives rather good estimates of the residual velocities in the range of 0.0 6 l 6 0.35, compared with the numerical solution. Plots of instantaneous crack length vs. indentation depth for the cases with l < 0.35 are shown in Fig. 11. Rather good fitting quality is obtained using Eq. (4) with K = 10 and b = 4. The crack tip speeds are calculated
from plots of the crack length vs. the time and shown in Fig. 12. It transpires that the crack tip speed almost linearly decreases as the mass ratio increases, i.e. the heavier the projectile, the faster the crack growth.
Fig. 11. Crack propagation in the plane-stress beam of h = 10 mm at V0 = 240 m/s.
Fig. 13. Crack propagation in the plane-stress beam of various thicknesses at V0 = 240 m/s and l = 0.1.
4.3. Beam thickness A target is thought of to be thin/thick if its thickness is much smaller/larger than the projectile breadth. The ratio of the beam thickness to the projectile breadth influences the failure mode of the target. But either a thin or intermediately thick target would fail by shear plugging if the impact velocity of the projectile is sufficiently high. Effects of the plate thickness on the ballistic limit were extensively studied experimentally and numerically by Børvik et al. [6], and by Corran et al. [5]. They found
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Fig. 14. Average crack tip speed vs. beam thickness for the plane-stress beam at V0 = 240 m/s and l = 0.1.
that there is a kink in the curve of ballistic limits vs. thickness corresponding to the transition of the failure mode from tensile tearing to shear plugging. In the present paper, effect of the target thickness on crack propagation is investigated. Three cases with different aspect ratios h/d = 0.25, 0.50, and 1.0 were considered, respectively. The ratio of h/d = 0.25 represents the case of a thin beam, while h/d = 1.0 the case of a thick beam. In finite element modeling, the breadth of the projectile d = 20 mm, the mass ratio l = 0.1, and the impact velocity V0 = 240 m/s were kept constant as well as the element size near and in the impact area. As shown in Fig. 13, the relation between the crack length and the indentation depth can be fitted well by Eq. (4) with the same set of constants K = 10 and b = 4. The difference in the average crack tip speed among three cases is very small, see Fig. 14. It indicates that as long as the target fails by shear plugging, the aspect ratio does not have much influence on crack propagation. Note, that this dimensionless group h/d is not included in Eq. (4).
pressed by axial loads applied at both ends of the beam. Static response of the beam was calculated with ABAQUS/Standard. Deformed meshes, stresses, and strains were transferred to ABAQUS/Explicit as the initial conditions of the second phase in which the beam was struck by the rigid, blunt projectile. With ABAQUS/Explicit, the perforation process of the beam was simulated. Four pre-loading levels were considered. The stress magnitude ri = ±300 MPa represents an elastic case for the beam under working conditions, while ri = ±450 MPa represents a plastic case for the beam already under plastic deformation before impact. Both the initial impact velocity V0 = 240 m/s and the mass ratio l = 0.1 were defined in these cases. As shown in Fig. 15, the residual velocity of the projectile in the neutral case is higher than that in the precompression case, whereas it is lower than that in the pre-tension case. Hence, in general, the pre-tension expedites crack propagation, while pre-compression retards crack propagation. This conclusion is also verified by plots of the crack tip speed versus the initial stress, see Fig. 16. The earlier analysis in Ref. [16] indicates that shear response is dominant and is very localized in plugging. The axial pre-compression increases the stress level of the beam, and thus contributes to the increase in plastic strain. However, pre-compression lowers the level of the stress triaxiality, and thus retards damage accumulation. To some extent, both effects cancel out each other as far as the damage accumulation is concerned. But, for the pre-tension case, both effects contribute positively to damage accumulation. This finding is consistent with Chen and YuÕs conclusion [23]. Plots of the crack length versus the indentation depth are shown in Fig. 17. It transpires that pre-tension reduces the critical indentation depth.
4.4. Axial pre-loading In practice, structural members are stressed to a certain level by operational loads in addition to accidental loads. For example, the ill-fated exterior columns of the World Trade Center were supporting the top part of the building as the airplanes cut through them. The effect of pre-loads on the failure process of structures has not drawn much attention. Chen and Yu [23] studied theoretically the failure response of a plastic pre-stretched beam under impulsive loading. They concluded that the pre-tension makes the beam easier to shear off. In this section, global response and crack propagation of the pre-loaded beam under projectile impact was investigated. The beam was first stretched or com-
Fig. 15. Residual velocity vs. initial stress for the plane-stress beam of h = 10 mm at V0 = 240 m/s and l = 0.1.
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Fig. 16. Average crack tip speed vs. initial stress for the planestress beam of h = 10 mm at V0 = 240 m/s and l = 0.1.
4.5. Round-cornered projectiles Both the flat-nosed projectile with sharp corners and the round-nosed projectile are commonly used in perforation experiments. In reality, impactors have all kinds of shapes. In this section, impact response of the plane-stress beam struck by the projectile with round corners was investigated, see Fig. 18. The radius R of the round corner was varied from 0.0 to d/2 to investigate its effect on the failure pattern of the beam. R = 0.0 and R = d/2 represent two limiting cases: a flat-nosed projectile with sharp corners and a semi-circular round-nosed projectile. In finite element modeling, both the impact velocity V0 = 240 m/s and the mass ratio l = 0.1 were kept constant.
(a)
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Fig. 17. Crack propagation in the plane-stress beam of h = 10 mm under axial pre-loading at V0 = 240 m/s and l = 0.1.
The failure processes of the beam for two cases with 2R/d = 0.05 and 2R/d = 0.50 are shown in Figs. 6 and 19, respectively. The case with 2R/d = 0.05 represents a real blunt projectile with smooth corners. It can be seen from Fig. 6 that several elements around the corners of the projectile are severely distorted. The cracks initiate at these elements, and rapidly propagate through the beam thickness. This shear plugging process is almost the same as the case with sharp corners presented in Ref. [16]. By contrast, in the case of 2R/d = 0.50, no elements around the projectile are severely distorted. Two cracks start inside the beam instead of on the top surface, and then rapidly grow upwards and downwards simultaneously. Fig. 20 shows the time history of the stress triaxiality and effective plastic strain of a crack starter point in
(b)
Fig. 18. Schematic representation of a round-cornered projectile and a round-nosed projectile.
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Fig. 20. Time history of stress triaxiality and effective plastic strain at the crack starter point for the case with 2R/d = 0.50 at V0 = 240 m/s.
Fig. 21. Deflection profiles of the plane-stress beam of h = 10 mm for the projectile of l = 0.1 with various corner radii at V0 = 240 m/s. Fig. 19. Perforation process for the plane-stress beam of h = 10 mm impacted by the projectile of l = 0.1 and 2R/ d = 0.50 at V0 = 240 m/s.
the middle of the beam. It can be observed that initially this point is dominated by the compressive stress wave. Then compression gives way to shear and tension as the stress triaxiality gradually increases. At failure, axial stretching becomes dominated at this point with rh = r 1=3. Hence, crack formation in this case is due to the combined action of shear and tension. As the radius of the round corner increase, the contribution of axial stretching to the fracture of the beam becomes evident. This can be discerned from the maximum deflection profiles of the beam at failure, see Fig. 21. For the case with the round corner radius smaller than 2R/ d 6 0.20, the maximum deflection of the beam is much
smaller than the beam thickness, which indicates that the beam fails mainly by shear plugging. For the case with the round corner radius in the range of 0.25 6 2R/d 6 1.0, the maximum deflection of the beam is of the order of the beam thickness. At such magnitude, axial stretching and shear are equally important for the failure of the beam. The transition of the failure mechanism of the target due to the change in the radius of the round-nosed projectile was observed in experiments by Corran et al. [5]. The geometrical shape of their projectile is different from the present case, see Fig. 18. They found that the target fails by tensile tearing in the case with a small radius, while by shear plugging in the case with a large radius. Note, that a round-nosed projectile with infinitely large radius is equivalent to a blunt projectile. Clearly, the
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Fig. 22. Crack propagation of the plane-stress beam of h = 10 mm impacted by the projectile of l = 0.1 with the circular corners at V0 = 240 m/s.
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Fig. 24. Crack propagation in the plane-strain beam of h = 10 mm at V0 = 240 m/s.
transition of the failure mode is also dependent on the impact velocity. If V0 is sufficiently large, a beam struck by a round-nosed projectile would also fail by shear plugging. The relation between indentation depth and crack length is shown in Fig. 22 for three cases with various radii. It can be concluded that Eq. (4) is applicable in the range of 2R/d = 0.0–0.1. 4.6. Plane strain beams A wide beam is usually discretized using plane-strain elements instead of plane-stress elements, because of the constraint in the lateral direction. In this section, three plane-strain cases with various impact velocities are investigated. The failure pattern of a specific planestrain case is shown in Fig. 23 which is quite similar to that of the plane-stress case, except that local bending occurs at the impact area of the wide beam. Since localized shear deformation is dominant, through-thickness crack propagation is not much influenced by the lateral constraint, as shown in Fig. 24. However, there is large difference in the velocity history
Fig. 23. Failure pattern of the plane-strain beam of h = 10 mm at V0 = 240 m/s and l = 0.1.
Fig. 25. Velocity history of the projectile for the circular plate and beams of h = 10 mm at V0 = 240 m/s and l = 0.1.
of the projectile between the plane-stress and planestrain cases, specially in the indentation phase, see Fig. 25. For the plane-stress case, the beam can expand in the lateral direction without any constraints when indented. But it is not the case for the plane-strain beam. The projectile is much resisted by the plane-strain beam in the indentation phase, and thus its velocity decreases much rapidly. However, the residual velocities of the projectile are almost the same in both cases. This indicates that a large part of the kinetic energy is dissipated in the form of shear cracking rather than indentation. The plane-stress and plane-strain models can be thought of as two limiting cases of a real beam with a finite breadth. If of interest is just the ballistic limit of the beam, either a plane-stress model or a plane-strain model is effective in the simulation of the perforation process.
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4.7. Circular plates Most perforation experiments reported in the literature were conducted on circular plates instead of beams. Here, the perforation process of the circular plate of the diameter 1.0 m and the thickness h = 10 mm struck by a flat-nosed, rigid, cylindrical projectile was simulated, see Fig. 2. The diameter of the projectile is d = 2h = 20 mm. The mass ratio in this case is defined by l¼
qpd 2 h : 4M 0
ð8Þ
Under high velocity impact by a blunt projectile, a circular plate usually fails by shear plugging, which can be captured using 2-D axisymmetrical finite element models. The failure pattern of the circular plate is similar to that of the beam, see Fig. 26. The crack growth is almost the same as that in the beam except in the initial phase, as shown in Fig. 27. Since very localized shear deformation is dominant in shear plugging and such a deformation mode does not introduce additional axial or circumferential strains, crack propagation in the cir-
Fig. 26. Failure pattern of the circular plate of h = 10 mm struck by the cylindrical projectile with V0 = 240 m/s and l = 0.1.
Fig. 27. Crack propagation in the circular plate of h = 10 mm under various impact velocities.
cular plate is similar to that in the beam in the second phase. The velocity history of the projectile is shown in Fig. 25. It can be seen that the residual velocity of the projectile in the circular plate case is lower than for the beam cases, because shear resistance is much larger in a circular plate than in a beam. This conclusion has been theoretically verified by the present authors [21].
5. Mesh size effects Because high stress and strain gradients in the vicinity of the propagating crack tip, finite element solutions may be sensitive to element size. This section investigates mesh size effects on global response and localized crack growth based on three mesh models. Since perforation tests on 2024-T351 aluminum alloy are not available in the open literature, experiments on circular Weldox 460 E steel plates impacted by a hard cylindrical projectile performed by Børvik et al. [6] were simulated. The configuration of the projectile–target system is illustrated in Fig. 28. Three axisymmetric finite element models with different element size were built, Fig. 29. The minimum element size is 0.4 mm · 0.4 mm, 0.2 mm · 0.2 mm, and 0.1 mm · 0.1 mm, respectively, for the coarse, medium, and fine mesh models. Note, that the medium mesh model was used in the preceding calculation. The material coefficients in the JC constitutive model for Weldox 460 E steel were taken from Børvik et al. [24]. The fracture model taken from Ref. [24] was modified by introducing the cut-off value at rh = r ¼ 1=3, see Fig. 30. Fig. 31 shows comparison of residual velocities among experimental results and the present numerical solutions based on the three mesh models. It appears that there is a good correlation with each other, which verifies the correctness of the present numerical procedure. Mesh size effects on crack growth for the case at V0 = 277.5 m/s are illustrated in Fig. 32. It can be concluded that if the element size is sufficiently small, the calculated residual velocities and crack growth are not much sensitive to the mesh size, specially at the initial impact velocity much higher than the ballistic limit,
Fig. 28. Schematic representation of the projectile–target system in the Børvik et al.Õs experiments.
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Fig. 31. Comparison of residual velocities among experimental results and numerical solutions based on three mesh models.
Fig. 29. Three mesh models for a steel circular plate impacted by a rigid cylindrical projectile.
Fig. 32. Crack growth in the steel circular plate at V0 = 277.5 m/s based on three mesh models.
An additional model with as many as 200 throughthickness element was developed to study the convergence of the solution. It appears that there is a unique asymptotic value o the residual velocity. However, the error depends on the impact velocity and is the largest for the impact velocity close to the ballistic limit, see Fig. 33.
Fig. 30. Fracture locus for Weldox 460 E steel with the cut-off value for the negative stress triaxiality.
where shear plugging is the favorable failure mode for the target. At the same time, Figs. 31 and 32 indicate that the medium mesh model with 50 elements through the target thickness is capable of providing accurate numerical results.
6. Discussion As shown in the previous sections, the relation of the crack growth and the indentation can be fitted well by Eq. (4) with two material constants K = 10 and b = 4, i.e. 4 a c u ¼ 10 ð1 þ lÞ : ð9Þ h V0 h
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Fig. 33. Calculated residual velocity vs. element number through the thickness of the steel plate.
This expression is very general and applicable to the beam case as well as the circular plate case, provided that the target fails by shear plugging. Setting a = h in Eq. (9), the critical indentation ucr at complete perforation is given by ucr 0:56 V 0 ¼ : h 1þl c
ð10Þ
For V0 = 180–300 m/s and l = 0.0–0.3, the critical indentation depth ranges from 0.2 h to 0.3 h. This result is consistent with an earlier finding by Jones [25]. It can be used as an elementary fracture criterion in the shear failure mode, for example, Jones and Shen [26]. It is believed that other expressions such as a polynomial function with several terms would give better curve-fitting for the relation of the crack length and the indentation depth. To be capable of deriving a closedform solution for the shear-plugging problem, the power function as shown in Eq. (4) is used here. As an alternative, the crack propagation can also be well approximated by a piecewise linear function, see Fig. 34. Specifically for the plane strain beam case and the circular plate case, the piecewise linear function is given by ( 0 0 6 u 6 ui ; a h i ¼ ð11Þ c u 4:5 ð1 þ lÞ V 0 h 1:4 ui 6 u 6 ucr ; h where ui is the indentation depth at which the cracks initiate, which is given by, for 2024-T351 aluminum alloy, ui 0:31 V 0 ¼ : h 1þl c
ð12Þ
The physical meaning of the piecewise linear fitting is that the whole perforation process can be ideally separated into two phases. In the first phase, only indenta-
Fig. 34. Crack propagation fitted by piecewise linear functions.
tion is involved without crack growth. When the indentation depth reaches a certain level, the cracks initiate and propagate forwards at a constant velocity in the second phase. It is interesting to point out that the curve of crack length vs. indentation depth is analogous to the relationship of the damage and the plastic strain in damage mechanics. Actually, the crack length normalized by the target height is a damage indictor for the target in shear plugging, and the indentation can be related to the shear strain by introducing the concept of shear hinge length [27]. Hence, Eq. (4) describing the crack propagation can be transformed into a damage function: b c e D ¼ K ð1 þ lÞ c ; V0 h
ð13Þ
where D = a/h is the damage; c is the engineering shear strain, c = u/e; and e is the shear strain length, e/ h = 0.433 for the beam and e/h = 0.342 for the circular plate [27]. This damage function may be implemented in regular shell or beam element formulation to predict the failure process in high velocity impact problems. The crack tip speed is controlled by many factors such as the impact velocity, the projectile mass, the target thickness, and the pre-loads, etc. Note that material microstructures also have a influence on the crack tip speed. In the present paper, the average value of crack speed for 2024-T351 aluminum alloy is of an order of 1000 m/s, which is 31% of the shear wave speed, and is twice as large as the transverse plastic wave speed c. Shear wave speed or Rayleigh wave speed is usually thought of as the upper limit of the crack propagating speed for Mode II in the context of linear elastic continuum mechanics [28].
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Most of perforation experiments were run on plate specimens. Crack propagation through the target thickness cannot be tracked by optical, high-speed cameras due to metal opacity. The results in the previous section indicate that the crack propagation through the thickness inside a narrow beam would be almost in phase with that in its lateral surface. The crack growth on the lateral surface may be tracked by high-speed, highresolution cameras. Hence, perforation experiments on a narrow beam would be helpful to validate the present numerical study.
7. Conclusions A parametric study on the perforation process of the target struck by the rigid projectile moving at the high velocity is performed using ABAQUS/Explicit. The effects of the factors controlling the failure process such as the thickness and width of the beam, the mass and geometric shape of the rigid projectile, pre-loading, etc. on crack propagation have been investigated. An expression (Eq. (4)) relating the crack length to the indentation depth is proposed based on a series of numerical simulations. This expression is shown to be very general and can be applied to both the beam and circular plate cases, provided that the target fails by shear plugging. Based on this expression, an analytical solution for the perforation process has been derived, and will be released in a separate publication [21]. It is also found that the critical indentation depth is about 0.2 h–0.3 h, which can be used as a failure indicator in shear failure. It is found that the crack tip speed is affected by the impact velocity, the projectile mass, etc. In the present case studied, the crack tip speed is of the order of 1000 m/s, and is one-third of the shear wave speed.
References [1] Knight NR, Jaunky N, Lawson RE, Ambur DR. Penetration simulation for uncontained engine debris impact on fuselage-like panels using LS-DYNA. Finite Elem Anal Des 2000;36:99–133. [2] Anderson CE, Bodner SR. Ballistic impact: the status of analytical and numerical modelling. Int J Impact Eng 1988;7(1):9–35. [3] Corbett GG, Reid SR, Johnson W. Impact loading of plates and shells by free-flying projectiles: a review. Int J Impact Eng 1996;18(2):141–230. [4] Goldsmith W. Non-ideal projectile impact on targets. Int J Impact Eng 1999;22(2–3):95–395. [5] Corran RSJ, Shadbolt PJ, Ruiz C. Impact loading of plates—an experimental investigation. Int J Impact Eng 1983;1(1):3–22.
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[6] Børvik T, Hopperstad OS, Langseth M, Malo KA. Effects of target thickness in blunt projectile penetration of Weldox 460 E steel plates. Int J Impact Eng 2003; 28(4):413–64. [7] Ambur DR, Jaunky N, Lawson RE, Knight NR. Numerical simulations for high-energy impact of thin plates. Int J Impact Eng 2001;25:683–702. [8] Guo J, Shi G, Wang Y, Lu C. Efficient modeling of panellike structures in perforation simulations. Comput Struct 2003;81:1–8. [9] Børvik T, Langseth M, Hopperstad OS, Malo KA. Perforation of 12 mm thick steel plates by 20 mm diameter projectiles with flat, hemispherical and conical noses: part I: experimental study. Int J Impact Eng 2002;27(1): 19–35. [10] Børvik T, Hopperstad OS, Berstad T, Langseth M. Perforation of 12 mm thick steel plates by 20 mm diameter projectiles with flat, hemispherical and conical noses: part II: numerical simulations. Int J Impact Eng 2002;27(1): 37–64. [11] Børvik T, Langseth M, Hopperstad OS, Malo KA. Ballistic penetration of steel plates. Int J Impact Eng 1999;22(9–10):855–86. [12] Rosakis AJ, Ravichandran G. Dynamic failure mechanics. Int J Solids Struct 2000;37:331–48. [13] Børvik T, Hopperstad OS, Berstad T, Langseth M. Numerical simulation of plugging failure in ballistic penetration. Int J Solids Struct 2001;38(34–35): 6241–64. [14] McClintock FA. A criterion for ductile fracture by the growth of holes. J Appl Mech 1968;35:363–71. [15] Johnson GR, Cook WH. Fracture characteristics of three metals subjected to various strains, strain rates, temperatures and pressures. Eng Fract Mech 1985;21(1): 31–48. [16] Teng X, Wierzbicki T. Effects of fracture criteria on high velocity perforation of thin beams. Int J Comput Methods 2004;1(1):171–200. [17] Teng X, Wierzbicki T, Rohr I, Hiermaier S. Numerical prediction of the Taylor test with fracture. Int J Solids Struct 2005;42(9–10):2929–48. [18] 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, Hague, Netherlands, 1983. p. 541–7. [19] Bao Y, Wierzbicki T. On fracture locus in the equivalent strain and stress triaxiality space. Int J Mech Sci 2004; 46(1):81–98. [20] Kao AS, Kuhn HA, Richmond O, Spitzig WA. Tensile fracture and fractographic analysis of 1045 spheroidized steel under hydrostatic pressure. J Mater Res 1990;5(1): 83–91. [21] Teng X, Wierzbicki T. Dynamic shear plugging of beams and plates with an advancing crack. Int J Impact Eng 2005;31(6):667–98. [22] Teng X, Wierzbicki T, Couque H. On the transition from adiabatic shear banding to fracture. Mech Mater, in preparation. [23] Chen FL, Yu TX. Influence of axial pre-load on plastic failure of beams subjected to transverse dynamic load. Key Eng Mater 2000;177–180:255–60.
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[24] Børvik T, Hopperstad OS, Berstad T. On the influence of stress triaxiality and strain rate on the behaviour of a structural steel. Part II: numerical study. Eur J Mech A/ Solids 2003;22:15–23. [25] Jones N. Structural impact. Cambridge, UK: Cambridge University Press; 1997. [26] Jones N, Shen WQ. Criteria for inelastic rupture of metal beams subjected to large dynamic loads. In:
Jones T, Wierzbicki T, editors. Structural crash worthiness and failure. Elsevier Applied Science; 1993. p. 95– 130. [27] Li QM, Jones N. Formation of a shear localization in structural elements under transverse dynamic loads. Int J Solids Struct 2000;37:6683–704. [28] Freund LB. Dynamic fracture mechanics. New York: Cambridge University Press; 1990.
Numerical study on crack propagation in high velocity perforation X. Teng, T. Wierzbicki
*
Impact and Crashworthiness Lab, Room 5-218, Massachusetts Institute of Technology, Cambridge, MA 02139, USA Received 18 November 2003; accepted 10 December 2004
Abstract Perforation response of a target under high velocity projectile impact is controlled by a number of factors such as the thickness of a target, the mass and geometrical shape of a projectile, etc. In this paper, through-thickness crack propagation in perforation problems is studied numerically on the basis of a newly constructed fracture locus formulated in terms of the effective plastic strain and the stress triaxiality. Through a series of numerical simulations using ABAQUS/ Explicit, an expression relating crack length to indentation depth is developed. This expression is very general and applicable to both a beam and a circular plate over a wide range of the impact velocity, the mass ratio, and the thickness, provided that the target fails by shear plugging. It is found that the critical indentation depth for the formation and ejection of a plug is about 20–30% of the target thickness for 2024-T351 aluminum alloy, which can be used as an elementary fracture criterion. The average crack tip speed was found to be of order of 1000 m/s, which is one-third of the elastic shear wave speed of 2024-T351 aluminum alloy. 2005 Elsevier Ltd. All rights reserved. Keywords: Finite element analysis; Crack tip speed; Crack propagation; Shear plugging; Perforation
1. Introduction Failure response of a target struck by a projectile has long been studied in the context of design of effective armor plates. It has also become important in industrial applications. For example, the problem of the fuselage skin or the fuel tank of an aircraft perforated by uncontained turbine engine debris, which caused several catastrophes in the past 20 years [1], has been of concern to aircraft structural engineers. The September 11th attack
* Corresponding author. Tel.: +1 617 253 2104; fax: +1 617 253 1962. E-mail address: [email protected] (T. Wierzbicki).
illustrated an extreme example, in which the exterior columns of the Twin Towers were cut through by the aircraft wings, and the core structures were severely destroyed by the fuselage leading the collapse of the Twin Towers. Much work has been carried out experimentally, theoretically, and numerically on the perforation problem. Comprehensive reviews on perforation mechanisms and analytical solutions can be found in the journal articles by Anderson and Bodner [2], Corbett et al. [3], and Goldsmith [4]. The major objective of most of perforation experiments is to determine the ballistic limit of the target. In such tests, the initial impact velocity of projectiles is varied, while other parameters such as the thickness of targets, the mass of projectiles, etc. are kept
0045-7949/$ - see front matter 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.compstruc.2004.12.001
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Nomenclature fraction of plastic work converted into heat material constant in Eq. (4) transverse deflection of the target engineering shear strain reference strain rate fracture strain effective plastic strain plastic strain rate mass ratio of the impact zone to the projectile q mass density r0 plastic flow stress rh hydrostatic stress ri initial stress r equivalent stress A initial yield stress in the JC material model a crack length B hardening modulus in the JC material model C strain rate parameter in the JC material qffiffiffiffi model c ¼ rq0 transverse plastic stress wave speed
a b d c e_ 0 ef epl e_ pl l
constant. The failure mechanism of the target is also controlled by other parameters besides the impact velocity. Corran et al. [5] investigated experimentally effects of projectile masses, nose shapes, target thicknesses, and boundary conditions on ballistic limits. Recently, Børvik et al. [6] studied the ballistic limit of Weldox 460 E steel plates over a wide range of thicknesses both experimentally and numerically. As an alternative to experiments, a large scale numerical simulation has now become affordable. Finite element methods have reached a state of maturity. Many commercial codes such as ABAQUS, LS-DYNA3D, etc. are capable of simulating the perforation process, provided that they are equipped with a suitable fracture criterion. A number of numerical simulations have been published in the open literature, for example, Ambur et al. [7] and Guo et al. [8], but they lacked corroboration from experiments. The most systematic, comprehensive studies to date combining numerical simulations with experiments are due to Børvik and his colleagues [9,10]. They first performed a series of simple tensile tests to determine material parameters in the constitutive equation as well as the fracture criterion, and then conducted a number of perforation experiments with varying impact velocities. They also carried out parallel numerical simulations using LS-DYNA3D. Rather good agreement with experimental results gives a much confidence in the capability and accuracy of finite element codes to predict perforation response.
cv D d e h K M0 n q R T T0 Tm t u ucr V0 Vr v
specific heat damage projectile breadth/diameter shear hinge length beam thickness material constant in Eq. (4) projectile mass material hardening exponent temperature parameter in the JC material model radius of round corners of a projectile temperature room temperature melting temperature time indentation depth critical indentation depth at complete perforation initial impact velocity residual velocity crack tip speed
As compared to extensive literature on the ballistic limit, only few papers deal with through-thickness crack propagation, even though crack propagation is always involved in the perforation problem. Børvik et al. [11] observed from scan electronic micrographs that microvoids and micro-cracks nucleate, grow, and coalesce in front of the crack tip. It indicates that a metal target under high velocity projectile impact fails by ductile fracture. As Rosakis and Ravichandran [12] pointed out, dynamic ductile fracture mechanics remains virtually unexplored. Analogous to static loading cases, dynamic stress intensity factors and generalized J integrals were proposed to predict dynamic fracture initiation and growth [12]. In the field of impact engineering, another approach has recently emerged, in which the fracture criterion is assumed to depend on the effective plastic strain and the stress triaxiality. Such a formulation can predict sequential material failure in the perforation problem, for example, Børvik et al. [13]. The idea of material ductile failure dependent on the stress triaxiality is not new and goes back to the fundamental work of McClintock [14]. Johnson and Cook [15] introduced this concept combined with effects of strain rates and temperature to predict dynamic ductile failure. However, detailed information on how a through-thickness crack propagates was not reported in the relevant literature. Most of perforation experiments were performed on opaque plate specimens such that crack growth through the target thickness is difficult to track down using currently
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available techniques. By contrast, finite element procedures are capable of overcoming this difficulty and providing abundant details about crack initiation, crack tip speeds, and crack trajectories. In an earlier paper by the present authors [16], effects of three different types of ductile fracture criteria on high velocity impact have been addressed. In the present paper, parametric studies of a number of factors including the initial impact velocity and mass of the projectile, the thickness and breadth of the beam, pre-loading, etc. are performed using the Bao–WierzbickiÕs fracture locus implemented in ABAQUS/Explicit. Attention is focused on crack propagation through the thickness of targets. The major objective of this research is to verify an analytical expression controlling crack growth.
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Fig. 2. Schematic representation of a circular plate struck by a rigid cylindrical body.
2. Finite element models Two types of projectile–target systems are considered in the present paper: a long, solid beam of rectangular cross-section impacted by a rigid cubic projectile, see Fig. 1, and a large circular plate impacted by a flatnosed, rigid, cylindrical projectile, see Fig. 2. In finite element modeling, both the beam length (the diameter of the circular plate) and the breadth (the diameter) of the projectile were kept constant, while other geometrical and mechanical parameters such as the mass of the projectile M0, the thickness of the target h, the impact velocity V0, etc. were varied to investigate their effects on the perforation process. Since attention is focused on crack propagation through the thickness, 2-D finite element models were built instead of 3-D solid element models. A narrow beam can expand laterally with little constraints when it is indented by a projectile. Thus, plane stress elements were used for this type of beams, while plane strain elements have to be used for a wide beam. For a circular plate under high velocity impact by a blunt projectile, shear plugging is a favorable failure mode rather than petalling, and thus a 2-D model can also be built using axisymmetric elements. The deformation and possible failure of the projectile was not taken into account in
Fig. 1. Schematic representation of a long beam struck by a rigid mass.
Fig. 3. Finite element meshes near and in the impact area with 50 elements through the thickness.
the present formulation. The projectile is represented by a rigid surface in the finite element model. A thorough investigation of mushrooming deformation and possible fracture of a cylindrical projectile in the Taylor test is a subject of another publication [17]. Finite element meshes near and in the impact area of the target are shown in Fig. 3. Square elements with the size 0.2 mm · 0.2 mm were generated in the impact area. There is a total of 50 elements through the target thickness. To reduce computational cost, relatively coarse meshes were used in the region away from the impact area. Such a mesh model is able to provide correct results. A study on mesh size sensitivity was performed and results are presented in Section 5. The Johnson–Cook (JC) constitutive model implemented in ABAQUS/Explicit was used to describe the plastic flow properties of the target such as strain hardening, strain rate hardening, and thermal softening [18]: q h i e_ pl T T0 n ¼ A þ Bepl 1 þ C ln r 1 : ð1Þ Tm T0 e_ 0 The target is assumed to be made of 2024-T351 aluminum alloy. The material constants of Eq. (1) are listed in Table 1, and the true stress–strain curves under various combinations of temperature and strain rates are illustrated in Fig. 4. The isotropic J2 plasticity model
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Table 1 Material constants for 2024-T351 aluminum alloy E (GPa) 74.66
m 0.3
q (kg/m3) 2700
e_ 0 (s1)
C 4
3.33 · 10
0.0083
cv (J/kg K)
a
Tm (K)
T0 (K)
q
875
0.9
775
293
1.0
A (MPa)
B (MPa)
n
r0 (MPa)
c (m/s)
352
440
0.42
565.5
457.6
Fig. 5. The Bao–WierzbickiÕs fracture locus for 2024-T351 aluminum alloy [19].
Fig. 4. True stress versus true strain at various strain rates and temperature for 2024-T351 aluminum alloy.
was selected from the ABAQUS library of material models. It is assumed that a crack is formed when the accumulated plastic strain, modified by a function of the stress triaxiality, rh = r, reaches a critical value, D = 1, Z epl 1 depl : ð2Þ D¼ f rrh 0 The weighting function f ðrh = rÞ is not a monotonically r, as comdecreasing function of the stress triaxiality rh = monly believed, but consists of several branches, each dominated by a different micro-fracture mechanism [19,20]. For example, Bao and Wierzbicki found that for 2024-T351 aluminum alloy the function f ðrh = rÞ consists of four distinct branches [19], Fig. 5: 8 rh 1 6 13 ; > r > > > 0:46 r < 0:1225 rrh þ 13 13 < rrh 6 0; h ¼ f 2 > r > 1:9 rrh 0:18 rrh þ 0:21 0 < rrh < 0:4; > > : 0:4 < rrh : exp 1:944 rrh ð3Þ A unique feature of the Bao–WierzbickiÕs (BW) fracture criterion is the cut-off value for the negative stress tri-
axiality at rh = r ¼ 1=3. The physical meaning of the cut-off value is that a crack would never be generated in a material element subjected to hydrostatic compression, i.e. rh = r < 1=3. This newly constructed fracture locus was used to predict crack formation in this paper. When the cumulative damage D reaches unity at an integration point of an element, all the stress components are suddenly set to zero at this point. If all the integration points in an element fail, this element will lose load-carrying capability and be removed from the mesh to model crack initiation and growth. A more detailed discussion on the application of the JCÕs material model and the BWÕs fracture model to the perforation problem was presented in a companion paper by the present authors [16].
3. Indentation and crack propagation Fig. 6 shows a sequence of the perforation process for one specific loading case. Immediately upon impact, the part of the beam beneath the projectile is indented. Large plastic deformation and large compressive (negative) stresses develops in this area, but no elements fail throughout the whole process. At a certain time, two cracks initiate near the corners of the projectile, and rapidly grow through the beam thickness until complete perforation. Finally a plug is formed and ejected from the target. These are typical failure scenarios of shear plugging. During this failure process, plastic deformations occur at three distinct scales: very localized shear deformation through the target thickness, large compression of the target directly beneath the projectile, and global plastic bending deformation. This type of failure pattern
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authorsÕ earlier work [16] suggests the following relationship between these quantities: b a c u ¼ K ð1 þ lÞ ; ð4Þ h V0 h where K and b are two material constants; the mass ratio l of the impact area of the beam to the projectile is defined by l¼
qhbd ; M0
ð5Þ
where q is the mass density of the beam, and b is the beam breath. In Eq. (4), c is the transverse wave speed defined by rffiffiffiffiffi r0 ; ð6Þ c¼ q where r0 is the plastic flow stress under quasi-static conditions. Differentiating Eq. (4) with respect to time gives the crack tip speed v: b v c u b1 ¼ bK ð1 þ lÞ ; ð7Þ V V0 h
Fig. 6. Shear plugging process of the plane-stress beam of h = 10 mm impacted by the blunt projectile of l = 0.1 at V0 = 240 m/s.
where v = da/dt; and V = du/dt is the indentation rate. If the global deformation of the target can be neglected, V is just the velocity of the projectile. A derivation of the expressions for both instantaneous and average crack tip speeds will be presented in another publication [21]. It should be mentioned that Eq. (4) was obtained by curve-fitting crack growth for a specific case in which the initial impact velocity was varied and other parameters were kept constant. The development of Eq. (4) in an analytical way remains open. The immediate objective of the present paper is to show that Eq. (4) derived for a specific set of input parameters considered in Ref. [16] is also valid in a wide range of impact speeds, plate thicknesses, and mass ratios. This is accomplished through an extensive parametric study in which only one parameter was varied at a time. 4. Parametric studies 4.1. Impact velocities
Fig. 7. Schematic of crack propagation showing the crack length a and the indentation depth u.
can be characterized by two variables: the crack length a and the indentation depth u, as shown in Fig. 7. The
Consider a narrow beam of the thickness h = 10 mm under high velocity projectile impact of the mass ratio l = 0.1. The impact velocity was varied from 160 m/s to 300 m/s, while other parameters were kept constant. Studied here is the effect of the impact velocity. As discussed in Ref. [16], there are three possible failure modes for a beam struck by a rigid projectile: tensile tearing, shear plugging, and adiabatic shear banding in the ascending order of the impact velocity. Numerical
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results indicate that the present beam fails by tensile tearing at V0 = 160 m/s, and by shear plugging at higher impact velocities. Plots of the normalized crack length with the normalized indentation can be fitted well by Eq. (4) with K = 10 and b = 4, except for the case with V0 = 160 m/s, see Fig. 8. As the impact velocity increases further, i.e. V0 > 300 m/s, the failure mode of the beam would change from shear plugging into adiabatic shear banding. The present expression in Eq. (4) cannot be extended to the range of adiabatic shear banding. The latter problem is the subject of current research [22]. Plots of crack propagation versus time at various impact velocities are shown in Fig. 9. It appears that crack propagation in shear plugging consists of two phases. In the first phase, uniform indentation is dominant and
cracks slowly grow, while in the second phase, localized shear deformation is dominant and crack propagation is much faster. The time history of the crack length for the present cases can be fitted well by piecewise linear functions, which indicates that the crack tip speeds are almost constant in the respective phases. The crack growth in the second phase is more important, because most of the elements fail in this phase. Hereinafter, the crack tip speed is referred to the average value in the second phase. At the same time, one can see from Fig. 9 that the crack tip speed strongly depends on the impact velocity of the projectile. The average crack tip speeds are given by v = 1056, 1007, and 707 m/s, corresponding to the impact velocity V0 = 300, 240, and 200 m/s. These crack growth speeds are comparable to the values in Børvik et al.Õs examples [13]. Two speeds of 667 m/s and 1300 m/s were estimated from Figs. 13 and 14 in their paper. Note, that different materials were used in both studies. No relevant experiments are found in the open literature to verify the accuracy of the obtained crack tip speeds. Compared with the residual velocity of the projectile, the crack tip speed would probably be more sensitive to numerical modeling techniques and material micro-structures. 4.2. Projectile mass
Fig. 8. Crack propagation at various impact velocities for the plane-stress beam of h = 10 mm and l = 0.1.
Fig. 9. Time history of crack length at various impact velocities for the plane-stress beam of h = 10 mm and l = 0.1.
Another parameter controlling the perforation process is the weight of the projectile. Its effect on the transition of the failure mode and crack growth in the narrow beam is investigated in this section. The mass ratio l was varied from 0.05 to 0.45 among ten cases, while the initial impact velocity of V0 = 240 m/sec and the beam thickness of h = 10 mm were kept constant. Depending on the projectile mass, two types of failure modes: shear plugging and tensile tearing, were observed from the failure pattern. For l < 0.35, the beam fails due to shear plugging. When the mass ratio falls in the range of 0.35 6 l 6 0.45, tensile tearing is the dominant failure mode. As the mass ratio continuously increases, i.e. the mass of the projectile further decreases, l > 0.45, the beam of an infinite length will never fracture. The critical mass ratio is about 0.35 at the transition of the failure mode from shear plugging to tensile tearing. Clearly, the critical mass ratio depends on the initial impact velocity and other parameters such as the beam thickness. The transition of the failure mode can also be observed from the relationship between the residual velocity and the mass ratio, see Fig. 10. In shear plugging, the residual velocities decrease with the mass ratios in a hyperbolic way, which is similar to the relation between the ballistic limit and the projectile mass, as found by Corran et al. [5] from a series of experiments. The residual velocity suffers then a jump as the failure mode
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Fig. 10. Residual velocity vs. mass ratio for the plane-stress beam of h = 10 mm at V0 = 240 m/s.
Fig. 12. Average crack tip speeds vs. mass ratios for the planestress beam of h = 10 mm at V0 = 240 m/s.
changes from shear plugging to tensile tearing. In tensile tearing, most of the kinetic energy of the projectile is absorbed by the beam in the form of plastic bending and axial stretching, while, in shear plugging, only a small part of the kinetic energy is dissipated in the beam due to localized shear deformation and indentation in the impact area. Hence, the residual velocity in shear plugging is much higher than that in tensile tearing. Additionally, it can be seen from Fig. 10 that the simple momentum conservation approach gives rather good estimates of the residual velocities in the range of 0.0 6 l 6 0.35, compared with the numerical solution. Plots of instantaneous crack length vs. indentation depth for the cases with l < 0.35 are shown in Fig. 11. Rather good fitting quality is obtained using Eq. (4) with K = 10 and b = 4. The crack tip speeds are calculated
from plots of the crack length vs. the time and shown in Fig. 12. It transpires that the crack tip speed almost linearly decreases as the mass ratio increases, i.e. the heavier the projectile, the faster the crack growth.
Fig. 11. Crack propagation in the plane-stress beam of h = 10 mm at V0 = 240 m/s.
Fig. 13. Crack propagation in the plane-stress beam of various thicknesses at V0 = 240 m/s and l = 0.1.
4.3. Beam thickness A target is thought of to be thin/thick if its thickness is much smaller/larger than the projectile breadth. The ratio of the beam thickness to the projectile breadth influences the failure mode of the target. But either a thin or intermediately thick target would fail by shear plugging if the impact velocity of the projectile is sufficiently high. Effects of the plate thickness on the ballistic limit were extensively studied experimentally and numerically by Børvik et al. [6], and by Corran et al. [5]. They found
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Fig. 14. Average crack tip speed vs. beam thickness for the plane-stress beam at V0 = 240 m/s and l = 0.1.
that there is a kink in the curve of ballistic limits vs. thickness corresponding to the transition of the failure mode from tensile tearing to shear plugging. In the present paper, effect of the target thickness on crack propagation is investigated. Three cases with different aspect ratios h/d = 0.25, 0.50, and 1.0 were considered, respectively. The ratio of h/d = 0.25 represents the case of a thin beam, while h/d = 1.0 the case of a thick beam. In finite element modeling, the breadth of the projectile d = 20 mm, the mass ratio l = 0.1, and the impact velocity V0 = 240 m/s were kept constant as well as the element size near and in the impact area. As shown in Fig. 13, the relation between the crack length and the indentation depth can be fitted well by Eq. (4) with the same set of constants K = 10 and b = 4. The difference in the average crack tip speed among three cases is very small, see Fig. 14. It indicates that as long as the target fails by shear plugging, the aspect ratio does not have much influence on crack propagation. Note, that this dimensionless group h/d is not included in Eq. (4).
pressed by axial loads applied at both ends of the beam. Static response of the beam was calculated with ABAQUS/Standard. Deformed meshes, stresses, and strains were transferred to ABAQUS/Explicit as the initial conditions of the second phase in which the beam was struck by the rigid, blunt projectile. With ABAQUS/Explicit, the perforation process of the beam was simulated. Four pre-loading levels were considered. The stress magnitude ri = ±300 MPa represents an elastic case for the beam under working conditions, while ri = ±450 MPa represents a plastic case for the beam already under plastic deformation before impact. Both the initial impact velocity V0 = 240 m/s and the mass ratio l = 0.1 were defined in these cases. As shown in Fig. 15, the residual velocity of the projectile in the neutral case is higher than that in the precompression case, whereas it is lower than that in the pre-tension case. Hence, in general, the pre-tension expedites crack propagation, while pre-compression retards crack propagation. This conclusion is also verified by plots of the crack tip speed versus the initial stress, see Fig. 16. The earlier analysis in Ref. [16] indicates that shear response is dominant and is very localized in plugging. The axial pre-compression increases the stress level of the beam, and thus contributes to the increase in plastic strain. However, pre-compression lowers the level of the stress triaxiality, and thus retards damage accumulation. To some extent, both effects cancel out each other as far as the damage accumulation is concerned. But, for the pre-tension case, both effects contribute positively to damage accumulation. This finding is consistent with Chen and YuÕs conclusion [23]. Plots of the crack length versus the indentation depth are shown in Fig. 17. It transpires that pre-tension reduces the critical indentation depth.
4.4. Axial pre-loading In practice, structural members are stressed to a certain level by operational loads in addition to accidental loads. For example, the ill-fated exterior columns of the World Trade Center were supporting the top part of the building as the airplanes cut through them. The effect of pre-loads on the failure process of structures has not drawn much attention. Chen and Yu [23] studied theoretically the failure response of a plastic pre-stretched beam under impulsive loading. They concluded that the pre-tension makes the beam easier to shear off. In this section, global response and crack propagation of the pre-loaded beam under projectile impact was investigated. The beam was first stretched or com-
Fig. 15. Residual velocity vs. initial stress for the plane-stress beam of h = 10 mm at V0 = 240 m/s and l = 0.1.
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Fig. 16. Average crack tip speed vs. initial stress for the planestress beam of h = 10 mm at V0 = 240 m/s and l = 0.1.
4.5. Round-cornered projectiles Both the flat-nosed projectile with sharp corners and the round-nosed projectile are commonly used in perforation experiments. In reality, impactors have all kinds of shapes. In this section, impact response of the plane-stress beam struck by the projectile with round corners was investigated, see Fig. 18. The radius R of the round corner was varied from 0.0 to d/2 to investigate its effect on the failure pattern of the beam. R = 0.0 and R = d/2 represent two limiting cases: a flat-nosed projectile with sharp corners and a semi-circular round-nosed projectile. In finite element modeling, both the impact velocity V0 = 240 m/s and the mass ratio l = 0.1 were kept constant.
(a)
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Fig. 17. Crack propagation in the plane-stress beam of h = 10 mm under axial pre-loading at V0 = 240 m/s and l = 0.1.
The failure processes of the beam for two cases with 2R/d = 0.05 and 2R/d = 0.50 are shown in Figs. 6 and 19, respectively. The case with 2R/d = 0.05 represents a real blunt projectile with smooth corners. It can be seen from Fig. 6 that several elements around the corners of the projectile are severely distorted. The cracks initiate at these elements, and rapidly propagate through the beam thickness. This shear plugging process is almost the same as the case with sharp corners presented in Ref. [16]. By contrast, in the case of 2R/d = 0.50, no elements around the projectile are severely distorted. Two cracks start inside the beam instead of on the top surface, and then rapidly grow upwards and downwards simultaneously. Fig. 20 shows the time history of the stress triaxiality and effective plastic strain of a crack starter point in
(b)
Fig. 18. Schematic representation of a round-cornered projectile and a round-nosed projectile.
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Fig. 20. Time history of stress triaxiality and effective plastic strain at the crack starter point for the case with 2R/d = 0.50 at V0 = 240 m/s.
Fig. 21. Deflection profiles of the plane-stress beam of h = 10 mm for the projectile of l = 0.1 with various corner radii at V0 = 240 m/s. Fig. 19. Perforation process for the plane-stress beam of h = 10 mm impacted by the projectile of l = 0.1 and 2R/ d = 0.50 at V0 = 240 m/s.
the middle of the beam. It can be observed that initially this point is dominated by the compressive stress wave. Then compression gives way to shear and tension as the stress triaxiality gradually increases. At failure, axial stretching becomes dominated at this point with rh = r 1=3. Hence, crack formation in this case is due to the combined action of shear and tension. As the radius of the round corner increase, the contribution of axial stretching to the fracture of the beam becomes evident. This can be discerned from the maximum deflection profiles of the beam at failure, see Fig. 21. For the case with the round corner radius smaller than 2R/ d 6 0.20, the maximum deflection of the beam is much
smaller than the beam thickness, which indicates that the beam fails mainly by shear plugging. For the case with the round corner radius in the range of 0.25 6 2R/d 6 1.0, the maximum deflection of the beam is of the order of the beam thickness. At such magnitude, axial stretching and shear are equally important for the failure of the beam. The transition of the failure mechanism of the target due to the change in the radius of the round-nosed projectile was observed in experiments by Corran et al. [5]. The geometrical shape of their projectile is different from the present case, see Fig. 18. They found that the target fails by tensile tearing in the case with a small radius, while by shear plugging in the case with a large radius. Note, that a round-nosed projectile with infinitely large radius is equivalent to a blunt projectile. Clearly, the
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Fig. 22. Crack propagation of the plane-stress beam of h = 10 mm impacted by the projectile of l = 0.1 with the circular corners at V0 = 240 m/s.
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Fig. 24. Crack propagation in the plane-strain beam of h = 10 mm at V0 = 240 m/s.
transition of the failure mode is also dependent on the impact velocity. If V0 is sufficiently large, a beam struck by a round-nosed projectile would also fail by shear plugging. The relation between indentation depth and crack length is shown in Fig. 22 for three cases with various radii. It can be concluded that Eq. (4) is applicable in the range of 2R/d = 0.0–0.1. 4.6. Plane strain beams A wide beam is usually discretized using plane-strain elements instead of plane-stress elements, because of the constraint in the lateral direction. In this section, three plane-strain cases with various impact velocities are investigated. The failure pattern of a specific planestrain case is shown in Fig. 23 which is quite similar to that of the plane-stress case, except that local bending occurs at the impact area of the wide beam. Since localized shear deformation is dominant, through-thickness crack propagation is not much influenced by the lateral constraint, as shown in Fig. 24. However, there is large difference in the velocity history
Fig. 23. Failure pattern of the plane-strain beam of h = 10 mm at V0 = 240 m/s and l = 0.1.
Fig. 25. Velocity history of the projectile for the circular plate and beams of h = 10 mm at V0 = 240 m/s and l = 0.1.
of the projectile between the plane-stress and planestrain cases, specially in the indentation phase, see Fig. 25. For the plane-stress case, the beam can expand in the lateral direction without any constraints when indented. But it is not the case for the plane-strain beam. The projectile is much resisted by the plane-strain beam in the indentation phase, and thus its velocity decreases much rapidly. However, the residual velocities of the projectile are almost the same in both cases. This indicates that a large part of the kinetic energy is dissipated in the form of shear cracking rather than indentation. The plane-stress and plane-strain models can be thought of as two limiting cases of a real beam with a finite breadth. If of interest is just the ballistic limit of the beam, either a plane-stress model or a plane-strain model is effective in the simulation of the perforation process.
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4.7. Circular plates Most perforation experiments reported in the literature were conducted on circular plates instead of beams. Here, the perforation process of the circular plate of the diameter 1.0 m and the thickness h = 10 mm struck by a flat-nosed, rigid, cylindrical projectile was simulated, see Fig. 2. The diameter of the projectile is d = 2h = 20 mm. The mass ratio in this case is defined by l¼
qpd 2 h : 4M 0
ð8Þ
Under high velocity impact by a blunt projectile, a circular plate usually fails by shear plugging, which can be captured using 2-D axisymmetrical finite element models. The failure pattern of the circular plate is similar to that of the beam, see Fig. 26. The crack growth is almost the same as that in the beam except in the initial phase, as shown in Fig. 27. Since very localized shear deformation is dominant in shear plugging and such a deformation mode does not introduce additional axial or circumferential strains, crack propagation in the cir-
Fig. 26. Failure pattern of the circular plate of h = 10 mm struck by the cylindrical projectile with V0 = 240 m/s and l = 0.1.
Fig. 27. Crack propagation in the circular plate of h = 10 mm under various impact velocities.
cular plate is similar to that in the beam in the second phase. The velocity history of the projectile is shown in Fig. 25. It can be seen that the residual velocity of the projectile in the circular plate case is lower than for the beam cases, because shear resistance is much larger in a circular plate than in a beam. This conclusion has been theoretically verified by the present authors [21].
5. Mesh size effects Because high stress and strain gradients in the vicinity of the propagating crack tip, finite element solutions may be sensitive to element size. This section investigates mesh size effects on global response and localized crack growth based on three mesh models. Since perforation tests on 2024-T351 aluminum alloy are not available in the open literature, experiments on circular Weldox 460 E steel plates impacted by a hard cylindrical projectile performed by Børvik et al. [6] were simulated. The configuration of the projectile–target system is illustrated in Fig. 28. Three axisymmetric finite element models with different element size were built, Fig. 29. The minimum element size is 0.4 mm · 0.4 mm, 0.2 mm · 0.2 mm, and 0.1 mm · 0.1 mm, respectively, for the coarse, medium, and fine mesh models. Note, that the medium mesh model was used in the preceding calculation. The material coefficients in the JC constitutive model for Weldox 460 E steel were taken from Børvik et al. [24]. The fracture model taken from Ref. [24] was modified by introducing the cut-off value at rh = r ¼ 1=3, see Fig. 30. Fig. 31 shows comparison of residual velocities among experimental results and the present numerical solutions based on the three mesh models. It appears that there is a good correlation with each other, which verifies the correctness of the present numerical procedure. Mesh size effects on crack growth for the case at V0 = 277.5 m/s are illustrated in Fig. 32. It can be concluded that if the element size is sufficiently small, the calculated residual velocities and crack growth are not much sensitive to the mesh size, specially at the initial impact velocity much higher than the ballistic limit,
Fig. 28. Schematic representation of the projectile–target system in the Børvik et al.Õs experiments.
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Fig. 31. Comparison of residual velocities among experimental results and numerical solutions based on three mesh models.
Fig. 29. Three mesh models for a steel circular plate impacted by a rigid cylindrical projectile.
Fig. 32. Crack growth in the steel circular plate at V0 = 277.5 m/s based on three mesh models.
An additional model with as many as 200 throughthickness element was developed to study the convergence of the solution. It appears that there is a unique asymptotic value o the residual velocity. However, the error depends on the impact velocity and is the largest for the impact velocity close to the ballistic limit, see Fig. 33.
Fig. 30. Fracture locus for Weldox 460 E steel with the cut-off value for the negative stress triaxiality.
where shear plugging is the favorable failure mode for the target. At the same time, Figs. 31 and 32 indicate that the medium mesh model with 50 elements through the target thickness is capable of providing accurate numerical results.
6. Discussion As shown in the previous sections, the relation of the crack growth and the indentation can be fitted well by Eq. (4) with two material constants K = 10 and b = 4, i.e. 4 a c u ¼ 10 ð1 þ lÞ : ð9Þ h V0 h
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Fig. 33. Calculated residual velocity vs. element number through the thickness of the steel plate.
This expression is very general and applicable to the beam case as well as the circular plate case, provided that the target fails by shear plugging. Setting a = h in Eq. (9), the critical indentation ucr at complete perforation is given by ucr 0:56 V 0 ¼ : h 1þl c
ð10Þ
For V0 = 180–300 m/s and l = 0.0–0.3, the critical indentation depth ranges from 0.2 h to 0.3 h. This result is consistent with an earlier finding by Jones [25]. It can be used as an elementary fracture criterion in the shear failure mode, for example, Jones and Shen [26]. It is believed that other expressions such as a polynomial function with several terms would give better curve-fitting for the relation of the crack length and the indentation depth. To be capable of deriving a closedform solution for the shear-plugging problem, the power function as shown in Eq. (4) is used here. As an alternative, the crack propagation can also be well approximated by a piecewise linear function, see Fig. 34. Specifically for the plane strain beam case and the circular plate case, the piecewise linear function is given by ( 0 0 6 u 6 ui ; a h i ¼ ð11Þ c u 4:5 ð1 þ lÞ V 0 h 1:4 ui 6 u 6 ucr ; h where ui is the indentation depth at which the cracks initiate, which is given by, for 2024-T351 aluminum alloy, ui 0:31 V 0 ¼ : h 1þl c
ð12Þ
The physical meaning of the piecewise linear fitting is that the whole perforation process can be ideally separated into two phases. In the first phase, only indenta-
Fig. 34. Crack propagation fitted by piecewise linear functions.
tion is involved without crack growth. When the indentation depth reaches a certain level, the cracks initiate and propagate forwards at a constant velocity in the second phase. It is interesting to point out that the curve of crack length vs. indentation depth is analogous to the relationship of the damage and the plastic strain in damage mechanics. Actually, the crack length normalized by the target height is a damage indictor for the target in shear plugging, and the indentation can be related to the shear strain by introducing the concept of shear hinge length [27]. Hence, Eq. (4) describing the crack propagation can be transformed into a damage function: b c e D ¼ K ð1 þ lÞ c ; V0 h
ð13Þ
where D = a/h is the damage; c is the engineering shear strain, c = u/e; and e is the shear strain length, e/ h = 0.433 for the beam and e/h = 0.342 for the circular plate [27]. This damage function may be implemented in regular shell or beam element formulation to predict the failure process in high velocity impact problems. The crack tip speed is controlled by many factors such as the impact velocity, the projectile mass, the target thickness, and the pre-loads, etc. Note that material microstructures also have a influence on the crack tip speed. In the present paper, the average value of crack speed for 2024-T351 aluminum alloy is of an order of 1000 m/s, which is 31% of the shear wave speed, and is twice as large as the transverse plastic wave speed c. Shear wave speed or Rayleigh wave speed is usually thought of as the upper limit of the crack propagating speed for Mode II in the context of linear elastic continuum mechanics [28].
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Most of perforation experiments were run on plate specimens. Crack propagation through the target thickness cannot be tracked by optical, high-speed cameras due to metal opacity. The results in the previous section indicate that the crack propagation through the thickness inside a narrow beam would be almost in phase with that in its lateral surface. The crack growth on the lateral surface may be tracked by high-speed, highresolution cameras. Hence, perforation experiments on a narrow beam would be helpful to validate the present numerical study.
7. Conclusions A parametric study on the perforation process of the target struck by the rigid projectile moving at the high velocity is performed using ABAQUS/Explicit. The effects of the factors controlling the failure process such as the thickness and width of the beam, the mass and geometric shape of the rigid projectile, pre-loading, etc. on crack propagation have been investigated. An expression (Eq. (4)) relating the crack length to the indentation depth is proposed based on a series of numerical simulations. This expression is shown to be very general and can be applied to both the beam and circular plate cases, provided that the target fails by shear plugging. Based on this expression, an analytical solution for the perforation process has been derived, and will be released in a separate publication [21]. It is also found that the critical indentation depth is about 0.2 h–0.3 h, which can be used as a failure indicator in shear failure. It is found that the crack tip speed is affected by the impact velocity, the projectile mass, etc. In the present case studied, the crack tip speed is of the order of 1000 m/s, and is one-third of the shear wave speed.
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