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Composite structures under ballistic impact
Composite Structures 66 (2004) 579–590 www.elsevier.com/locate/compstruct
Composite structures under ballistic impact N.K. Naik *, P. Shrirao Aerospace Engineering Department, Indian Institute of Technology-Bombay Powai, Mumbai 400 076, India Available online 29 July 2004
Abstract In the present study, investigations on the ballistic impact behaviour of two-dimensional woven fabric composites has been presented. Ballistic impact behaviour of plain weave E-glass/epoxy and twill weave T300 carbon/epoxy composites has been compared. The analytical method presented is based on our earlier work. Different damage and energy absorbing mechanisms during ballistic impact have been identified. These are: cone formation on the back face of the target, tensile failure of primary yarns, deformation of secondary yarns, delamination, matrix cracking, shear plugging and friction during penetration. Analytical formulation has been presented for each energy absorbing mechanism. Energy absorbed during each time interval and the corresponding reduction in velocity of the projectile has been determined. The solution is based on the target material properties at high strain rate and the geometry and the projectile parameters. Using the analytical formulation, ballistic limit, contact duration at ballistic limit, surface radius of the cone formed and the radius of the damaged zone have been predicted for typical woven fabric composites. Ó 2004 Elsevier Ltd. All rights reserved. Keywords: Ballistic impact; Woven fabric composite; Energy absorbing mechanisms; E-glass/epoxy; T300 carbon/epoxy
1. Introduction Composite structures undergo various loading conditions during the service life. For the effective utilization of composites as protective structures, the critical requirement is the resistance to foreign object ballistic impact loading. For such applications, the condition for the penetration of the projectile into the composite target and the associated damage mechanisms should be clearly understood. Post ballistic impact residual strength is also an important consideration. Impact loads can be classified into three categories: low velocity impact, high velocity impact and hyper velocity impact. The reason for this classification is that energy transfer between projectile and target, energy dissipation and damage propagation mechanisms undergo drastic changes as the velocity of the projectile changes. One of the possible ways of enhancing the ballistic limit is to utilize textile composites. Kinetic energy of the projectile when impacted into the target is dissipated and absorbed in various ways by the target. The main energy absorbing mechanisms
*
Corresponding author. Fax: +91-22-2572-2602. E-mail address: [email protected] (N.K. Naik).
0263-8223/$ - see front matter Ó 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.compstruct.2004.05.006
during ballistic impact are: kinetic energy absorbed by the moving cone formed on the back face of the target, shear plugging of the projectile into the target, energy absorbed due to tensile failure of the primary yarns, energy absorbed due to elastic deformation of the secondary yarns, energy absorbed due to matrix cracking and delamination and frictional energy absorbed during penetration. An impact event is considered to be a low velocity impact if the contact period of impactor is longer than the time period of the lowest vibrational mode. In low velocity impact regime, the support conditions are crucial as the stress waves generated outward from the impact point have time to reach the edges of the structural element, causing its full-vibrational response. On the other hand, in high velocity or ballistic impact, the response of the structural element is governed by the ‘local’ behaviour of the material in the neighborhood of the impacted zone, the impact response of the element being generally independent of its support conditions. The contact period of the impactor is much smaller than the time period of lowest vibrational mode of the structure. Hyper velocity impact involves projectiles moving at extremely high velocities such that the local target materials behave like fluids and the stress induced by the impact is many times the material strength.
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Nomenclature a A Aql Adi b ce cp ct d da ; db dci EKEi ESPi EDi ETFi EDLi EMCi Emt EFi ETOTALi Fi GIIcd h hl KEpi MCi mp N
Yarn width cross-sectional area of fibre/yarn quasi-lemniscate area reduction factor damaged area at time ti stress wave transmission factor elastic wave velocity plastic wave velocity transverse wave velocity projectile diameter damage distances deceleration of the projectile during ith time interval kinetic energy of the moving cone at time ti energy absorbed by shear plugging till time ti energy absorbed by deformation of secondary yarns till time ti energy absorbed by tensile failure of primary yarns till time ti energy absorbed by delamination till time ti energy absorbed by matrix cracking till time ti energy absorbed by matrix cracking per unit volume energy absorbed by friction till time ti total energy absorbed by the target till time ti contact force during ith time interval critical dynamic strain energy release rate in mode II target thickness layer thickness kinetic energy of the projectile at time ti mass of the cone at time ti mass of the projectile number of layers being shear plugged in a time interval
The composite structural applications which involve survivability of personnel and equipments against penetration by high velocity projectiles are of much import. Thus, it is important to have an understanding of the penetration process of the composites by a projectile under ballistic velocities. Considering the importance of this subject there are many experimental studies on the ballistic impact of composites. But the experimental studies are specific to certain types of composite materials under certain ballistic impact conditions. Based on these experimental studies the ballistic impact behaviour of a class of composite materials cannot be characterized. For complete understanding of a class of composite materials, analytical studies are necessary based only on mechanical/fracture and geometrical properties of target and ballistic impact parameters. But the analytical studies are
Pd Pm rdi rpi rti rt1 rt2 SSP ta ; tb tc ti Vf Vi VBL ; V50 VI ; VO VR zi e ed ei ep epy esy e0 r rp rsy q Dt
percent delaminating layers percent matrix cracking radius of the damage zone at time ti distance covered by plastic wave till time ti surface radius of the cone at time ti , distance covered by transverse wave till time ti surface radius of the cone at time t1 surface radius of the cone at time t2 shear plugging strength time intervals contact duration of the projectile during ballistic impact event ith instant of time fibre volume fraction projectile velocity at time ti ballistic limit velocity incident ballistic impact velocity residual velocity height/depth of the cone at time ti , distance traveled by projectile at time tiþ strain damage threshold strain maximum tensile strain in primary yarns at time ti plastic strain strain in primary yarns strain in secondary yarns maximum strain in a yarn/fibre at any moment stress plastic stress stress in secondary yarns density of the target duration of time interval that tc is divided into
very limited. The ballistic impact performance of composite laminates is dependent on the mechanical characteristics of the reinforcement/matrix and the physical characteristics of the impacting projectile and the target. Mechanical characteristics of the target include the elastic modulus, tensile strength, fracture strain and laminate configuration. Projectile characteristics include mass and shape. Thickness and size are the target parameters that affect the ballistic impact performance. Ballistic impact is generally a low-mass high-velocity impact caused by a propelling source. Ballistic limit of a target is defined as the maximum velocity of a projectile at which the complete perforation takes place with zero exit velocity. Limited number of semi-analytical and semi-numerical models, which require input details from a few experiments, are available [1–11]. But in these studies all
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the energy absorbing mechanisms have not been considered. Also, complete details are not provided. The objective of the present work is to present an analytical model for the ballistic impact analysis based only on dynamic mechanical and fracture properties of the target material, geometrical parameters of the target and the projectile parameters, such as mass, velocity, shape and size. The studies have been presented for typical woven fabric E-glass/epoxy and T300 carbon/ epoxy composites impacted with flat ended cylindrical projectiles. The analytical method presented can be used for predicting the energy absorbed by different damage and energy absorbing mechanisms, ballistic limit, contact duration and the damage size. Ballistic impact behaviour of plain weave E-glass/epoxy and twill weave T300 carbon/epoxy composites has been compared. Energy absorbed by various mechanisms is calculated for a given mass and velocity of the projectile. By knowing the energy absorbed by various mechanisms, the total energy absorbed during impact event is calculated as the sum of the individual energies absorbed. The energy absorbed at each time interval is calculated. Based on this, the velocity at the end of the corresponding time interval is calculated. This procedure is continued for the entire contact duration. At ballistic limit the target absorbs the entire energy of the projectile. So the energy absorbed by the target is made equal to the initial kinetic energy of the projectile. From this the velocity of the projectile, which is the ballistic limit of the target, is calculated. The input to the present analytical model are: the dynamic mechanical properties of the target, target geometrical details and projectile shape, size, mass and incident ballistic impact velocity. The output that can be obtained are the ballistic limit of the target, contact duration and damage dimensions.
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back face deformation behaviour using high speed photography. Their study showed that a cone was formed on the back face of the composite during the ballistic impact event. Experimental studies by Zhu et al. [12] on woven Kevlar/polyester laminates of varying thickness also showed the cone formation on the back face of the composites during ballistic impact event. The cone formation on the back face can be explained on the basis of transverse wave propagation during ballistic impact. Figs. 1 and 2 show the deformation and cone formation on the back face of the composite during ballistic impact. Here, d is the diameter of the projectile, rti is the surface radius of the cone formed and zi is the distance traveled by the projectile. During the ballistic impact event, the distance traveled by the projectile and the depth of the cone formed are equal. Also, the velocities of the projectile and the cone would be equal. The yarns directly below the projectile are called primary yarns. These yarns provide the resistive force to the projectile penetration into the target. The remaining yarns within the conical region are known as secondary yarns. These yarns deform and cause some energy absorption. The conical portion has a finite velocity equal to that of the projectile. For the analysis of the ballistic impact event, the contact duration can be subdivided into certain number of time intervals of duration Dt. The damage mechanism and the energy absorption can be determined for each
2. Damage mechanisms For the complete understanding of the ballistic impact of composites, different damage and energy absorbing mechanisms should be clearly understood. Possible energy absorbing mechanisms are: cone formation on the back face of the target, deformation of secondary yarns, fracture of primary yarns/fibres, delamination, matrix cracking, shear plugging and friction between the projectile and the target. For different materials like carbon, glass or Kevlar, different mechanisms can dominate. Also, the reinforcement architecture can influence the energy absorbing mechanisms. 2.1. Cone formation on the back face of the target Morye et al. [9] carried out ballistic impact tests on woven Nylon prepreg composites. They monitored the
Fig. 1. Cone formation during ballistic impact on the back face of the composite target.
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primary yarns take the direct force and ultimately fail if the strain exceeds the maximum strain limit. As a result the complete target perforation takes place. All the yarns apart from the primary yarns, i.e., the secondary yarns, shown in Fig. 1 deform and absorb some energy. The energy absorbed by secondary yarns depends on the strain distribution within the secondary yarns. Strain in the secondary yarns near the tip of the cone, i.e., at point A is the same as that in the primary yarns and it decreases linearly to zero as we move out radially to point B. 2.3. Energy absorbed due to tensile failure of primary yarns
Fig. 2. Damage propagation in composite laminate during ballistic impact event.
time interval and this effect can be taken into account while proceeding with further time intervals. The surface radius of the cone formed can be calculated on the basis of transverse wave propagation. The transverse wave velocity depends on the stress-strain curve at high strain rates for the material. During the second time interval the surface radius of the cone increases from rt1 to rt2 . Surface radius rt2 is calculated based on how much additional transverse wave propagation has taken place during that time interval. As the transverse wave propagation continues, surface radius of the cone rti , cone depth zi , mass of the cone MCi and the strain in the fibres ei increase. As the strain within the fibre exceeds the permissible dynamic tensile strain, the corresponding fibre fails. Increase in rti , zi , MCi and ei continues until all the fibres are broken or the entire energy of the projectile is absorbed by the target. It may be noted that the strain within the primary yarns is not constant. At any moment, it is maximum at the point of impact and decreases as we move away from the point of impact. Failure of all the fibres indicates complete perforation of the projectile into the target. At this stage, if the projectile still has some kinetic energy, it would be exiting with certain velocity. If the exit velocity is zero, the corresponding incident ballistic impact velocity of the projectile is the ballistic limit. If the projectile velocity is zero during the contact event before all the fibres are broken, it indicates that only the partial penetration has occurred.
Primary yarns/fibres provide the force to resist the penetration of the projectile into the target. The strain is maximum in these yarns/fibres. These yarns fail under tension when the strain of these yarns exceeds the ultimate strain in tension at high strain rate. Failure of these yarns/fibres results in absorption of some energy. Fig. 1 indicates the strain variation in primary yarns within a layer and in through the thickness direction. During cone formation, the tensile strain would be maximum along the middle primary yarn, in each layer. Within the primary yarns in a layer strain reduces gradually, as we move away from the mid primary yarn. This is due to the fact that the middle yarn is pushed by complete projectile diameter, but other primary yarns face a projectile length which is lesser than projectile diameter. The top layers of the laminate when impacted get compressed apart from the cone formation, which causes the strain in the top layers to be more than that in the bottom layers. The extent of compression is based on the stress at the point of impact. This stress causes an extra straining of the top layers. The variation of strain in through the thickness direction is shown in Fig. 1. The strain in the top layers is more than the strain in the bottom layers. Each primary yarn has different strain. As and when the strain of a particular yarn reaches the dynamic tensile failure strain, failure occurs to that particular yarn. As a result, there will be a sequential failure of yarns starting with the yarns in the top layer, then proceeding toward the bottom layer. It may be noted that the strain within the primary yarn is not constant. At any moment it is maximum at the point of impact and decreases as we move away from the point of impact. 2.4. Energy absorbed due to shear plugging
2.2. Energy absorbed due to deformation of secondary yarns Possible cone formation on the back face of the target during ballistic impact event is shown in Fig. 1. The
Cantwell and Morton [13] observed frustum-shaped shear cut-out zone on impacting carbon fibre reinforced angle ply laminates with steel spherical bullets. Lee and Sun [14] and Ellis [15] reported shear plugging to be one
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target. Tensile failure of the yarns thus absorbs some energy of the projectile. During the ballistic impact event, delamination and matrix cracking take place in the laminate area which forms the cone. The total kinetic energy of the projectile that is lost during ballistic impact is the total energy that is absorbed by the target till that time interval and is given by ETOTALi ¼ EKEi þ ESPi þ EDi þ ETFi þ EDLi þ EMCi þ EFi Fig. 3. Schematic showing plug formation.
of the major damage modes on impact of angle ply graphite/bismaleimide by blunt/flat projectiles. Fig. 3 shows schematic arrangement of plug formation during ballistic impact. Brittle nature of carbon based composites was responsible for change in damage mechanism. Shear plug formation is not observed for glass reinforced composites which have high failure strain at high strain rates. 2.5. Energy absorbed by other mechanisms Delamination and matrix cracking would absorb some energy (Fig. 2). Additionally, friction between the projectile and the target would also absorb some energy.
3. Analytical formulation for woven fabric composites On striking a target, energy of the projectile is absorbed by various mechanisms like kinetic energy of moving cone EKE , shear plugging ESP , deformation of secondary yarns ED , tensile failure of primary yarns ETF , delamination EDL , matrix cracking EMC , friction energy EF along with other energy dissipation mechanisms. An analytical model is presented for the above-mentioned mechanisms, which dominate during ballistic impact of two-dimensional woven fabric composites [16,17]. 3.1. Analytical formulation A cone is observed on the back face of the target as shown in Fig. 1 when struck by a projectile [9,12]. Shear plugging on the front face and cone formation on the back face start taking place depending on the target material properties during the ballistic impact event. The velocity of the cone is the same as the velocity of the projectile. Initially, the cone has velocity equal to that of the projectile and has zero mass. As time progresses, the mass of the cone increases and velocity of the cone decreases. As the cone formation takes place, the yarns/ fibres deform and absorb some energy. The primary yarns which provide the resistive force to the projectile motion deform the most, thus leading to their failure. When all the primary yarns fail the projectile exits the
ð1Þ An analytical model to predict the ballistic limit is presented below. Firstly, modelling of a single yarn is presented. The model is further extended to the woven fabric composites. The analytical model is for the case when the projectile impacts the target normally. The following assumptions are made in the analytical model presented: (1) Projectile is perfectly rigid and remains un-deformed during the ballistic impact. (2) Projectile motion is uniform during penetration within each time interval. (3) Yarns/fibres in a layer act independently. (4) Energy absorption due to primary yarn/fibre breakage and deformation of the secondary yarns are treated independently. (5) Longitudinal and transverse wave velocities are the same in all the layers. 3.2. Modelling of a single yarn subjected to transverse ballistic impact When a yarn is impacted by a projectile transversely, longitudinal strain wavelets propagate outward along the filament [4,18,19]. The outermost wavelet, called the elastic wave propagates at velocity sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 dr ce ¼ ð2Þ q de e¼0 The innermost longitudinal wavelet called the plastic wave propagates at velocity sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 dr cp ¼ ð3Þ q de e¼ep Here, ep denotes the strain at which plastic region starts. As the strain wavelets pass a given point on the yarn, material of the filament flows inward toward the impact point. The material in the wake of the plastic wave front forms itself into a transverse wave, shaped like an inverted tent with the impact point at its vertex. In a fixed coordinate system, the base of the tent spreads outward with the transverse wave velocity. This is depicted in Fig. 4. The transverse wave velocity is given by
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critical strain energy release rates along warp/fill directions and the other directions. For simplicity, the cone formed can be considered to be of circular cross-section, rather than quasi-lemniscate. The base of the cone spreads with a transverse wave velocity sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ep ð1 þ ep Þrp 1 dr ct ¼ de ð7Þ q de q 0 and the plastic wave velocity is given by sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 dr cp ¼ q de e¼ep
Fig. 4. Configuration of a yarn/fibre before and after transverse impact: (a) before impact; (b)–(d) after impact.
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ep ð1 þ ep Þrp 1 dr ct ¼ de q de q 0
n¼i X
n¼i X
n¼i X
ctn Dt
ð9Þ
cpn Dt
ð10Þ
n¼0
ð4Þ rpi ¼
ctn Dt
ð5Þ
cpn Dt
ð6Þ
n¼0
rpi ¼
After time ti the transverse wave has traveled to a distance rti and the plastic wave has traveled to a distance of rpi . The projectile has moved through a distance zi . Radii rti and rpi after time ti ¼ iDt are given by rti ¼
If the complete impact event is divided into a number of small instants, then at ith instant, the time is given by ti . By that time the transverse wave has traveled to a distance rti and the plastic wave has traveled to a distance of rpi . The projectile has moved through a distance zi . Radii rti and rpi after time ti ¼ iDt is given by rti ¼
ð8Þ
n¼0
3.3. Modelling of woven fabric composite Woven fabric composites consist of warp and fill yarns, interlaced in a regular sequence. As the projectile impacts on to the woven fabric composite, there can be many yarns beneath the projectile. In the present analysis, all the warp and fill yarns are treated separately. Behaviour of each yarn is analysed as explained in the previous section. Overall behaviour of the woven fabric composite is presented considering the combined effect of all the individual yarns within a layer. 3.4. Strain and deceleration history: single yarn of woven fabric composite
n¼i X n¼0
Because of stress wave attenuation, the stress in the yarn and hence, the strain in the yarn varies with the distance from the point of impact. Stress and in turn strain is maximum at the point of impact and decreases as one moves away from the point of impact. At ith time interval, the strain at the point of impact is given by qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8 9 < ðd=2Þ þ ðrti ðd=2ÞÞ2 þ z2i þ rpi rti rpi = ei ¼ : ; bðrpi =aÞ 1 ln b ð11Þ a Once, the strain variation in a yarn/fibre is known, the energy absorbed by various mechanisms like kinetic energy of the moving cone, deformation energy absorbed by secondary yarns, tensile failure of primary yarns, delamination and matrix cracking can be calculated. The formulation for energy absorbed by various energy absorbing mechanisms is provided in the following sections. At the beginning of the first time interval of impact, entire energy is in the form of kinetic energy of the projectile. Later, this energy is divided into energy absorbed by various damage mechanisms and the kinetic energy of moving cone and projectile. Considering the energy balance at the end of ith time interval, we obtain KEp0 ¼ KEpi þ EKEi þ ESPði1Þ þ EDði1Þ þ ETFði1Þ
When a projectile impacts a woven fabric composite, a cone with quasi-lemniscate cross-section would be formed on the back face of the target. Quasi-lemniscate shape is because of difference in elastic properties and
þ EDLði1Þ þ EMCði1Þ þ EFði1Þ
ð12Þ
Rearranging the terms in the above equation, the following form is obtained:
N.K. Naik, P. Shrirao / Composite Structures 66 (2004) 579–590 1 m V2 2 p 0
Ei1 ¼ 12 mp þ MCi Vi 2
ð13Þ
3.5. Kinetic energy of cone formed The cone formed on the back face of the target absorbs some energy. By the end of ith time interval the radius of the cone formed is
where Ei1 ¼ ESPði1Þ þ EDði1Þ þ ETFði1Þ þ EDLði1Þ þ EMCði1Þ þ EFði1Þ
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ð14Þ
rti ¼
n¼i X
ð20Þ
ctn Dt
n¼0
Each of the contributing terms to Ei is explained in the succeeding sections. The terms on the right-hand side of Eq. (14) are known at (i 1)th instant of time. From this, the velocity of the projectile at the end of ith time interval can be obtained and it is given by sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 mp V 2 Ei1 Vi ¼ 21 0 mp þ MCi 2
ð15Þ
If the projectile velocity is known at the beginning and the end of ith time interval, then the deceleration of the projectile during that time interval can be found out. It is given by dci ¼
Vi1 Vi Dt
ð16Þ
Note that the velocity at the beginning of the ith time interval is the same as the velocity at the end of (i 1)th time interval. Distance traveled by the projectile zi up to ith time interval is given by zi ¼
n¼i X
Dzn
ð17Þ
n¼0
1 2 Dzi ¼ Vi1 Dt dci ðDtÞ 2
MCi ¼ prti2 hq
ð21Þ
The velocity of the cone formed is equal to Vi , the velocity of the projectile at the end of ith time interval. So the energy of the cone formed at the end of ith time interval is EKEi ¼ 12MCi Vi 2
ð22Þ
3.6. Energy absorbed due to shear plugging When the target material is impacted by the projectile, shear plugging stress in the material near projectile periphery rises. As and when the shear plugging stress exceeds shear plugging strength, shear plugging failure occurs. As a result, plug formation takes place. This phenomenon is observed for carbon/epoxy composites (Fig. 3). If at the beginning of the ith time interval, shear plugging stress exceeds shear plugging strength, then the energy absorbed by shear plugging during that time interval is given by the product of distance sheared, shear plugging strength and the area over which shear plugging stress is applied. It is given by DESPi ¼ Nhl SSP p dh
ð18Þ
Utilizing dci , the deceleration of the projectile during ith time interval, the force resisting the projectile motion can be calculated. It is given by Fi ¼ mp dci
Mass of the cone formed is
ð19Þ
The magnitude of the force on the projectile is the same as the magnitude of the force applied on the target, by the projectile. Hence, this force can be used to determine whether shear plugging takes place or not. The above process is repeated until all the primary yarns in the target fail, i.e., the complete perforation takes place. The velocity at the end of time interval, during which all the yarns are broken, is the residual velocity of the projectile. On the other hand, if the numerator in Eq. (15) becomes zero, then the projectile does not penetrate the target completely with the given initial velocity. Thus by repetition of above procedure with various velocities so as to get complete perforation with zero residual velocity, the ballistic limit of the target laminate can be obtained.
ð23Þ
where N indicates the number of layers shear plugged during ith time interval and SSP denotes shear plugging strength. The energy absorbed by shear plugging by the end of ith time interval is given by n¼i X ESPi ¼ DESPn ð24Þ n¼0
3.7. Energy absorbed due to deformation of secondary yarns The secondary yarns experience different strains depending on their position. The yarns which are close to the point of impact experience a strain equal to the strain in the outermost primary yarn, whereas those yarns which are away from the impact point experience less strain. The strain variation is assumed to be linear from A to B (Fig. 1). This imposes the following boundary conditions for the variation in strain at time ti : pffiffiffi esy ¼ epy at r ¼ d= 2; i:e:; at point A esy ¼ 0 at r ¼ rti ;
i:e:; at point B
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Here, epy is the strain in the outermost primary yarn in that layer. The energy absorbed in the deformation of all the secondary yarns can be obtained by the following integration: Z rti Z esyi EDi ¼ rsy esy desy h pffiffi d= 2
0
2pr 8r sin1 ðd=2rÞ dr
ð25Þ
3.8. Energy absorbed due to tensile failure of primary yarns The yarns directly below the projectile, known as the primary yarns, fail in direct tension. All the primary yarns within one layer do not fail at one instant of time. As and when the strain of a particular yarn reaches the dynamic failure strain in tension, the yarn fails. It may be noted that the length of yarns/fibres failing in tension is twice the distance covered by the longitudinal wave. Also, the complete length of a primary yarn is not strained to the same extent as explained earlier. The reason for this phenomenon is stress wave attenuation. When the strain in yarns/fibres exceeds failure strain, it fails and some energy is absorbed due to tensile failure. For a yarn/fibre of cross-section area A it is given by ! Z x Z e¼e0 bx=a ETF ¼ A rðeÞde dx ð26Þ 0
delamination may not occur at all the lamina interfaces. Towards the end of ballistic impact event, when only a few non-delaminated layers are left, these non-delaminated layers are more likely to bend rather than delaminate. The area undergoing delamination and matrix cracking in the conical region is of quasi-lemniscate shape, which is taken to be Aql percent of the corresponding circular area. During (i þ 1)th time interval the area of delamination and matrix cracking is given by 2 2 rdi ÞAql pðrdðiþ1Þ
ð27Þ
where rdi indicates the radius up to which the damage has propagated until ith time interval. So the respective energies absorbed by delamination and matrix cracking during this time interval are given by 2 2 DEDLi ¼ Pd pðrdðiþ1Þ rdi ÞAql GIIcd
ð28Þ
2 2 DEMCi ¼ Pm pðrdðiþ1Þ rdi ÞAql Emt h
ð29Þ
The factors Pd and Pm stand for percentage delamination and percentage matrix cracking. Energy absorbed till the ith time interval is given by EDLi ¼
n¼i X
DEDLn
ð30Þ
DEMCn
ð31Þ
n¼1
EMCi ¼
n¼i X n¼1
e¼0
where e0 is the ultimate strain limit. If during ith time interval N numbers of yarns/fibres are failing, then the right-hand side of the above expression is multiplied by N . 3.9. Energy absorbed due to delamination and matrix cracking Delamination and matrix cracking absorb some part of the initial kinetic energy of the projectile. A part of the conical area undergoes delamination and matrix cracking (Fig. 2). The extent to which composite has delaminated and the matrix has cracked till (i þ 1)th time instant can be calculated on the basis of strain profile in the composite at that time interval. From the results derived in the earlier section it can be observed that the strain at the impact point is the highest and it decreases as one moves away from the point of impact. The area in which strain is more than the damage initiation threshold strain ed , will undergo damage in the form of matrix cracking and delamination. However, complete matrix cracking may not take place. Evidence for this phenomenon is provided by the fact that after ballistic impact, matrix is still attached to the fibres and does not separate from the reinforcement completely. Due to matrix cracking, the interlaminar strength of the composite decreases. As a result, further loading and deformation causes delamination. This delamination is of mode II type. Again,
3.10. Calculation of contact duration The impact event starts when the projectile touches the front face of the target. End of the event is taken to be when all the primary yarns fail or when the velocity of the projectile becomes zero. Total time from the start of the impact event till the end of the event is called as contact duration. If the end of the event occurs during nth time interval, then the contact duration is obtained as tc ¼ nDt
ð32Þ
4. Results and discussion Ballistic impact behaviour of woven fabric composites when impacted by a flat ended projectile has been presented. This behaviour is based on the analytical method presented in the previous section. Specifically, ballistic limit, contact duration at ballistic limit and surface radius of the cone formed at the ballistic limit have been determined. These results have been compared with the experimental results. Behaviour of plain weave E-glass/epoxy and twill weave T300 carbon/epoxy composite structures under ballistic impact has been compared. Input data necessary for the analytical prediction of ballistic impact behaviour in the present study has been provided in Table 1.
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Table 1 Input parameters required for the analytical predictions of ballistic impact behaviour Projectile details (cylindrical) Mass Shape Diameter
2.8 g Flat ended 5 mm
2.8 g Flat ended 5 mm
Target details Material Vf Thickness Number of layers Density
Twill weave T300 carbon/epoxy 50% 2 mm 5 1400 kg/m3
Plain weave E-glass/epoxy 50% 2 mm 6 1750 kg/m3
Other details Stress–strain Time-step Quasi-lemniscate area reduction factor Stress wave transmission factor Delamination percent Matrix crack percent Shear plugging strength Mode II dynamic critical strain energy release rate Matrix cracking energy Fibre failure energy
Fig. 9 1 ls 0.9 0.95 100% 100% 90 MPa 800 J/m2 0.9 MJ/m3 6 MJ/m3
Fig. 9 1 ls 0.9 0.825 100% 100% – 1000 J/m2 0.9 MJ/m3 28 MJ/m3
4.1. Calculation of ballistic limit, contact duration and surface radius of the cone formed during ballistic impact Using the analytical method presented earlier ballistic impact behaviour of typical plain weave E-glass/epoxy and twill weave T300 carbon/epoxy composites has been evaluated. The results are presented in Table 2 and Figs. 5–8. For the case of E-glass/epoxy, complete perforation did not take place at VI ¼ 158 m/s whereas with VI ¼ 159 m/s, complete perforation took place with exit velocity of 54 m/s. The experimentally obtained ballistic limit is V50 ¼ 150 m/s. Analytically predicted ballistic limit is between 158 and 159 m/s. For the case of T300 carbon/ epoxy, complete perforation did not take place with VI ¼ 82 m/s whereas with VI ¼ 83 m/s, complete perforation took place with exit velocity of 18 m/s. The above results are with mp ¼ 2.8 g, d ¼ 5 mm and h ¼ 2 mm. For T300 carbon/epoxy the studies were also carried out with mp ¼ 1:8 g, h ¼ 2 mm and d ¼ 5 mm. For this case the experimentally obtained ballistic limit,
V50 ¼ 105 m/s. The analytically predicted ballistic limit VBL ¼ 99 m/s (Table 2). It is generally observed that there is a good co-relation between the experimental results and the predicted results for both E-glass/epoxy and T300 carbon/epoxy. From Figs. 5 and 7 it can be seen that the ballistic limit is much higher for E-glass/epoxy than for T300 carbon/epoxy. This is because of higher energy absorbed by E-glass/epoxy than by T300 carbon/epoxy (Fig. 9). Fig. 9 is for high strain rate loading. Also, shear plugging strength for E-glass/epoxy is much higher than that for T300 carbon/epoxy. For E-glass/epoxy, energy absorbed by secondary yarn deformation and tensile failure of yarns are the main energy absorbing mechanisms. Shear plugging does not take place in this case. On the other hand, for T300 carbon/epoxy, the ultimate failure is because of shear plugging. Energy absorbed by secondary yarn deformation and shear plugging are the main energy absorbing mechanisms (Figs. 6 and 8). In this case, tensile failure of the yarns does not take place.
Table 2 Ballistic impact test results for typical plain weave E-glass/epoxy and twill weave T300 carbon/epoxy composites, d ¼ 5 mm, h ¼ 2 mm Material
Plain weave E-glass/epoxy Twill weave T300 carbon/epoxy Twill weave T300 carbon/epoxy
Projectile mass, mp (g)
Predicted ballistic limit, VBL (m/s)
Experimental ballistic limit, V50 (m/s)
Predicted damage size, rd (mm)
Experimental damage size, rd (mm)
Predicted surface radius of the cone, rt (mm)
2.8
159
150
9.6
10
35
1.8
99
105
–
–
59
2.8
83
–
–
–
61
588
N.K. Naik, P. Shrirao / Composite Structures 66 (2004) 579–590 Projectile kinetic energy
(a)
Cone kinetic enery Shear plugging energy
Projectile kinetic energy
40
Cone kinetic energy Secodary yarn deformation energy
Secondary yarn deformation energy
10
(b)
35
Tensile failure energy
Delamination energy Matrix cracking energy
Delamination energy
30
8
Matrix cracking energy Energy,E(J)
Energy,E(J)
25 6
4
20 15 10
2
5 0
0
-5 0
100
200
300
400
500
600
0
700
20
40
Time,t (micro-sec)
60
80
100 120 140 160
Time,t (micro-sec)
Fig. 5. Energy absorbed by different mechanisms during ballistic impact event, mp ¼ 2:8 g, h ¼ 2 mm, d ¼ 5 mm: (a) twill weave T300 carbon/epoxy laminate, VI ¼ 82 m/s; (b) plain weave E-glass/epoxy laminate, VI ¼ 158 m/s.
90
1.0
Projectile velocity
80
Maximum strain in top layer
0.9
Maximum strain in bottom layer
0.8
(b)
Projectile velocity Maximum strain in to player Maximum strain in bottom layer
60
0.6
50
0.5 0.4
40
0.3
30
0.2
60
40
Contact force
5
7
140
6 4
120
20
5 3
100 80
2
60
Tensile strain (%)
0.7
160
Velocity, V(m/s)
Shear plugging stress
80 Shear plugging stress (MPa)
70
Velocity, V(m/s)
100
Tensile strain (%)
(a)
4 3 2
1
20
40
0.1
1 0
10
0.0
0
20
0
0
-0.1 0
200 400 Time,t (micro-sec)
0 -1 0
600
20
40
-1
60 80 100 120 140 160 Time,t (micro-sec)
Fig. 6. Projectile velocity, strain, contact force and shear plugging stress variation during ballistic impact event, mp ¼ 2:8 g, h ¼ 2 mm, d ¼ 5 mm: (a) twill weave T300 carbon/epoxy laminate, VI ¼ 82 m/s; (b) plain weave E-glass/epoxy laminate, VI ¼ 158 m/s.
(a)
(b) Projectile kinetic energy 10
40
Projectile kinetic energy
Cone kinetic energy
Secondary yarn deformation energy
8
Cone kinetic energy
35
Shear plugging energy
Secondary yarn deformation energy
Tensile failure energy
30
Delamination energy
Energy, E (J)
Energy, E (J)
Delamination energy
25
Matrix cracking energy 6
4
Matrix cracking energy
20 15 10
2
5 0
0
-5 -10
0
10
20
30 40 50 60 Time,t (micro-sec)
70
80
90
0
20
40
60
80 100 120 140 Time,t (micro-sec)
160
Fig. 7. Energy absorbed by different mechanisms during ballistic impact event, mp ¼ 2:8 g, h ¼ 2 mm, d ¼ 5 mm: (a) twill weave T300 carbon/epoxy laminate, VI ¼ 83 m/s; (b) plain weave E-glass/epoxy laminate, VI ¼ 159 m/s.
N.K. Naik, P. Shrirao / Composite Structures 66 (2004) 579–590
589
Projectile velocity Projectile velocity
(a)
Maximum strain in top layer
(b)
Maximum strain in top layer Maximum strain in bottom layer
Maximum strain in bottom layer Contact force
160
7
5
Shear plugging stress
160
140
0.5 50 0.4 40
0.3 0.2
30
0.1
100 80 60 40
120 Velocity,V (m/s)
0.6
60
Tensile strain (%)
Velocity,V (m/s)
120
0.7
Shear plugging stress (MPa)
0.8 70
6 4
140
5 3
100 80
2
60
Tensile strain (%)
0.9
80
0.0
10
40
0
10
20
30
40
50
60
70
80
2 1
20
0 0
0
-0.1 -10
3
1
20
20
4
Contact force, F (KN)
1.0
90
90
0
-1 0
20
Time,t (micro-sec)
40
-1
60 80 100 120 140 160 Time,t (micro-sec)
Fig. 8. Projectile velocity, strain, contact force and shear plugging stress variation during ballistic impact event, mp ¼ 2:8 g, h ¼ 2 mm, d ¼ 5 mm: (a) twill weave T300 carbon/epoxy laminate, VI ¼ 83 m/s; (b) plain weave E-glass/epoxy laminate, VI ¼ 159 m/s.
References
Fig. 9. Stress–strain curves under tensile loading at high strain rate: (a) E-glass/epoxy yarn; (b) T300 carbon/epoxy yarn.
It is generally observed that the energy absorbed by delamination and matrix cracking is of small magnitude.
5. Conclusions Ballistic impact behaviour of typical plain weave Eglass/epoxy and twill weave T300 carbon/epoxy composites has been studied. It has been observed that, for identical ballistic impact conditions, ballistic limit is higher for E-glass/epoxy than for T300 carbon/epoxy. For E-glass/epoxy, energy absorbed by secondary yarn deformation and tensile failure of primary yarn are the main energy absorbing mechanisms. For T300 carbon/ epoxy, the main energy absorbing mechanisms are the secondary yarn deformation and shear plugging.
[1] Zhu G, Goldsmith W, Dharan CK. Penetration of laminated Kevlar by projectiles––II. Analytical model. Int J Solids Struct 1992;29:421–36. [2] Jenq ST, Jing HS, Chung C. Predicting the ballistic limit for plain woven glass/epoxy composite laminate. Int J Impact Eng 1994;15: 451–64. [3] Goldsmith W, Dharan CK, Chang H. Quasi-static and ballistic perforation of carbon fibre laminates. Int J Solids Struct 1995;32: 89–103. [4] Parga-Landa B, Hernandez-Olivares F. Analytical model to predict behaviour of soft armours. Int J Impact Eng 1995;16: 455–66. [5] Sun CT, Potti SV. A simple model to predict residual velocities of thick composite laminates subjected to high velocity impact. Int J Impact Eng 1996;18:339–53. [6] Potti SV, Sun CT. Prediction of impact induced penetration and delamination in thick composite laminates. Int J Impact Eng 1997;19:31–48. [7] Jenq ST, Kuo JT, Sheu LT. Ballistic impact response of 3-D fourstep braided glass/epoxy composites. Key Eng Mater Part 1 1998;141–143:349–66. [8] Benloulo ISC, Sanchez-Galvez V. A new analytical model to simulate impact onto ceramic/composite armors. Int J Impact Eng 1998;21:461–71. [9] Morye SS, Hine PJ, Duckett RA, Carr DJ, Ward IM. Modelling of the energy absorption by polymer composites upon ballistic impact. Compos Sci Technonl 2000;60:2631–42. [10] Billon HH, Robinson DJ. Models for the ballistic impact of fabric armour. Int J Impact Eng 2001;25:411–22. [11] Wen HM. Penetration and perforation of thick FRP laminates. Compos Sci Technol 2001;61:1163–72. [12] Zhu G, Goldsmith W, Dharan CK. Penetration of laminated Kevlar by projectiles––I. Experimental investigation. Int J Solids Struct 1992;29:399–419. [13] Cantwell WJ, Morton J. Impact perforation of carbon fibre reinforced plastic. Compos Sci Technol 1990;38:119–41. [14] Lee S-WR, Sun CT. A quasistatic penetration model for composite laminates. J Compos Mater 1993;27:251–71. [15] Ellis RL. Ballistic impact resistance of graphite epoxy composites with shape memory alloy and extended chain polyethylene spectra hybrid components. Master of Science Thesis, Mechanical Engineering Department, Virginia Polytechnic Institute and State University, December 1996.
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[16] Reddy BCK, Shrirao P, Naik NK. Ballistic impact on brittle composites: shear-plugging phenomenon. In: Batra RC, Henneke EG, editors. Proceeding of the 14th US National Congress of Theoretical and Applied Mechanics. Contemporary Research in Theoretical and Applied Mechanics. Blacksburg: Virginia Tech; 2002. p. 363. [17] Naik NK, Reddy BCK, Shrirao P. Analytical modeling of ballistic impact behavior of woven fabric composites. In: Ko FK (Ed.),
Proceedings of TEXCOMP - 6, Drexel University, paper 6.08, 2002. [18] Smith JC, McCrackin FL, Schiefer HF. Stress–strain relationships in yarns subjected to rapid impact loading. Part V: Wave propagation long textile yarns impacted transversely. Textile Res J 1958;(April):288–302. [19] Roylance D, Wilde A, Tocci G. Ballistic impact of textile structures. Textile Res J 1973;(January):34–41.
Composite structures under ballistic impact N.K. Naik *, P. Shrirao Aerospace Engineering Department, Indian Institute of Technology-Bombay Powai, Mumbai 400 076, India Available online 29 July 2004
Abstract In the present study, investigations on the ballistic impact behaviour of two-dimensional woven fabric composites has been presented. Ballistic impact behaviour of plain weave E-glass/epoxy and twill weave T300 carbon/epoxy composites has been compared. The analytical method presented is based on our earlier work. Different damage and energy absorbing mechanisms during ballistic impact have been identified. These are: cone formation on the back face of the target, tensile failure of primary yarns, deformation of secondary yarns, delamination, matrix cracking, shear plugging and friction during penetration. Analytical formulation has been presented for each energy absorbing mechanism. Energy absorbed during each time interval and the corresponding reduction in velocity of the projectile has been determined. The solution is based on the target material properties at high strain rate and the geometry and the projectile parameters. Using the analytical formulation, ballistic limit, contact duration at ballistic limit, surface radius of the cone formed and the radius of the damaged zone have been predicted for typical woven fabric composites. Ó 2004 Elsevier Ltd. All rights reserved. Keywords: Ballistic impact; Woven fabric composite; Energy absorbing mechanisms; E-glass/epoxy; T300 carbon/epoxy
1. Introduction Composite structures undergo various loading conditions during the service life. For the effective utilization of composites as protective structures, the critical requirement is the resistance to foreign object ballistic impact loading. For such applications, the condition for the penetration of the projectile into the composite target and the associated damage mechanisms should be clearly understood. Post ballistic impact residual strength is also an important consideration. Impact loads can be classified into three categories: low velocity impact, high velocity impact and hyper velocity impact. The reason for this classification is that energy transfer between projectile and target, energy dissipation and damage propagation mechanisms undergo drastic changes as the velocity of the projectile changes. One of the possible ways of enhancing the ballistic limit is to utilize textile composites. Kinetic energy of the projectile when impacted into the target is dissipated and absorbed in various ways by the target. The main energy absorbing mechanisms
*
Corresponding author. Fax: +91-22-2572-2602. E-mail address: [email protected] (N.K. Naik).
0263-8223/$ - see front matter Ó 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.compstruct.2004.05.006
during ballistic impact are: kinetic energy absorbed by the moving cone formed on the back face of the target, shear plugging of the projectile into the target, energy absorbed due to tensile failure of the primary yarns, energy absorbed due to elastic deformation of the secondary yarns, energy absorbed due to matrix cracking and delamination and frictional energy absorbed during penetration. An impact event is considered to be a low velocity impact if the contact period of impactor is longer than the time period of the lowest vibrational mode. In low velocity impact regime, the support conditions are crucial as the stress waves generated outward from the impact point have time to reach the edges of the structural element, causing its full-vibrational response. On the other hand, in high velocity or ballistic impact, the response of the structural element is governed by the ‘local’ behaviour of the material in the neighborhood of the impacted zone, the impact response of the element being generally independent of its support conditions. The contact period of the impactor is much smaller than the time period of lowest vibrational mode of the structure. Hyper velocity impact involves projectiles moving at extremely high velocities such that the local target materials behave like fluids and the stress induced by the impact is many times the material strength.
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Nomenclature a A Aql Adi b ce cp ct d da ; db dci EKEi ESPi EDi ETFi EDLi EMCi Emt EFi ETOTALi Fi GIIcd h hl KEpi MCi mp N
Yarn width cross-sectional area of fibre/yarn quasi-lemniscate area reduction factor damaged area at time ti stress wave transmission factor elastic wave velocity plastic wave velocity transverse wave velocity projectile diameter damage distances deceleration of the projectile during ith time interval kinetic energy of the moving cone at time ti energy absorbed by shear plugging till time ti energy absorbed by deformation of secondary yarns till time ti energy absorbed by tensile failure of primary yarns till time ti energy absorbed by delamination till time ti energy absorbed by matrix cracking till time ti energy absorbed by matrix cracking per unit volume energy absorbed by friction till time ti total energy absorbed by the target till time ti contact force during ith time interval critical dynamic strain energy release rate in mode II target thickness layer thickness kinetic energy of the projectile at time ti mass of the cone at time ti mass of the projectile number of layers being shear plugged in a time interval
The composite structural applications which involve survivability of personnel and equipments against penetration by high velocity projectiles are of much import. Thus, it is important to have an understanding of the penetration process of the composites by a projectile under ballistic velocities. Considering the importance of this subject there are many experimental studies on the ballistic impact of composites. But the experimental studies are specific to certain types of composite materials under certain ballistic impact conditions. Based on these experimental studies the ballistic impact behaviour of a class of composite materials cannot be characterized. For complete understanding of a class of composite materials, analytical studies are necessary based only on mechanical/fracture and geometrical properties of target and ballistic impact parameters. But the analytical studies are
Pd Pm rdi rpi rti rt1 rt2 SSP ta ; tb tc ti Vf Vi VBL ; V50 VI ; VO VR zi e ed ei ep epy esy e0 r rp rsy q Dt
percent delaminating layers percent matrix cracking radius of the damage zone at time ti distance covered by plastic wave till time ti surface radius of the cone at time ti , distance covered by transverse wave till time ti surface radius of the cone at time t1 surface radius of the cone at time t2 shear plugging strength time intervals contact duration of the projectile during ballistic impact event ith instant of time fibre volume fraction projectile velocity at time ti ballistic limit velocity incident ballistic impact velocity residual velocity height/depth of the cone at time ti , distance traveled by projectile at time tiþ strain damage threshold strain maximum tensile strain in primary yarns at time ti plastic strain strain in primary yarns strain in secondary yarns maximum strain in a yarn/fibre at any moment stress plastic stress stress in secondary yarns density of the target duration of time interval that tc is divided into
very limited. The ballistic impact performance of composite laminates is dependent on the mechanical characteristics of the reinforcement/matrix and the physical characteristics of the impacting projectile and the target. Mechanical characteristics of the target include the elastic modulus, tensile strength, fracture strain and laminate configuration. Projectile characteristics include mass and shape. Thickness and size are the target parameters that affect the ballistic impact performance. Ballistic impact is generally a low-mass high-velocity impact caused by a propelling source. Ballistic limit of a target is defined as the maximum velocity of a projectile at which the complete perforation takes place with zero exit velocity. Limited number of semi-analytical and semi-numerical models, which require input details from a few experiments, are available [1–11]. But in these studies all
N.K. Naik, P. Shrirao / Composite Structures 66 (2004) 579–590
the energy absorbing mechanisms have not been considered. Also, complete details are not provided. The objective of the present work is to present an analytical model for the ballistic impact analysis based only on dynamic mechanical and fracture properties of the target material, geometrical parameters of the target and the projectile parameters, such as mass, velocity, shape and size. The studies have been presented for typical woven fabric E-glass/epoxy and T300 carbon/ epoxy composites impacted with flat ended cylindrical projectiles. The analytical method presented can be used for predicting the energy absorbed by different damage and energy absorbing mechanisms, ballistic limit, contact duration and the damage size. Ballistic impact behaviour of plain weave E-glass/epoxy and twill weave T300 carbon/epoxy composites has been compared. Energy absorbed by various mechanisms is calculated for a given mass and velocity of the projectile. By knowing the energy absorbed by various mechanisms, the total energy absorbed during impact event is calculated as the sum of the individual energies absorbed. The energy absorbed at each time interval is calculated. Based on this, the velocity at the end of the corresponding time interval is calculated. This procedure is continued for the entire contact duration. At ballistic limit the target absorbs the entire energy of the projectile. So the energy absorbed by the target is made equal to the initial kinetic energy of the projectile. From this the velocity of the projectile, which is the ballistic limit of the target, is calculated. The input to the present analytical model are: the dynamic mechanical properties of the target, target geometrical details and projectile shape, size, mass and incident ballistic impact velocity. The output that can be obtained are the ballistic limit of the target, contact duration and damage dimensions.
581
back face deformation behaviour using high speed photography. Their study showed that a cone was formed on the back face of the composite during the ballistic impact event. Experimental studies by Zhu et al. [12] on woven Kevlar/polyester laminates of varying thickness also showed the cone formation on the back face of the composites during ballistic impact event. The cone formation on the back face can be explained on the basis of transverse wave propagation during ballistic impact. Figs. 1 and 2 show the deformation and cone formation on the back face of the composite during ballistic impact. Here, d is the diameter of the projectile, rti is the surface radius of the cone formed and zi is the distance traveled by the projectile. During the ballistic impact event, the distance traveled by the projectile and the depth of the cone formed are equal. Also, the velocities of the projectile and the cone would be equal. The yarns directly below the projectile are called primary yarns. These yarns provide the resistive force to the projectile penetration into the target. The remaining yarns within the conical region are known as secondary yarns. These yarns deform and cause some energy absorption. The conical portion has a finite velocity equal to that of the projectile. For the analysis of the ballistic impact event, the contact duration can be subdivided into certain number of time intervals of duration Dt. The damage mechanism and the energy absorption can be determined for each
2. Damage mechanisms For the complete understanding of the ballistic impact of composites, different damage and energy absorbing mechanisms should be clearly understood. Possible energy absorbing mechanisms are: cone formation on the back face of the target, deformation of secondary yarns, fracture of primary yarns/fibres, delamination, matrix cracking, shear plugging and friction between the projectile and the target. For different materials like carbon, glass or Kevlar, different mechanisms can dominate. Also, the reinforcement architecture can influence the energy absorbing mechanisms. 2.1. Cone formation on the back face of the target Morye et al. [9] carried out ballistic impact tests on woven Nylon prepreg composites. They monitored the
Fig. 1. Cone formation during ballistic impact on the back face of the composite target.
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primary yarns take the direct force and ultimately fail if the strain exceeds the maximum strain limit. As a result the complete target perforation takes place. All the yarns apart from the primary yarns, i.e., the secondary yarns, shown in Fig. 1 deform and absorb some energy. The energy absorbed by secondary yarns depends on the strain distribution within the secondary yarns. Strain in the secondary yarns near the tip of the cone, i.e., at point A is the same as that in the primary yarns and it decreases linearly to zero as we move out radially to point B. 2.3. Energy absorbed due to tensile failure of primary yarns
Fig. 2. Damage propagation in composite laminate during ballistic impact event.
time interval and this effect can be taken into account while proceeding with further time intervals. The surface radius of the cone formed can be calculated on the basis of transverse wave propagation. The transverse wave velocity depends on the stress-strain curve at high strain rates for the material. During the second time interval the surface radius of the cone increases from rt1 to rt2 . Surface radius rt2 is calculated based on how much additional transverse wave propagation has taken place during that time interval. As the transverse wave propagation continues, surface radius of the cone rti , cone depth zi , mass of the cone MCi and the strain in the fibres ei increase. As the strain within the fibre exceeds the permissible dynamic tensile strain, the corresponding fibre fails. Increase in rti , zi , MCi and ei continues until all the fibres are broken or the entire energy of the projectile is absorbed by the target. It may be noted that the strain within the primary yarns is not constant. At any moment, it is maximum at the point of impact and decreases as we move away from the point of impact. Failure of all the fibres indicates complete perforation of the projectile into the target. At this stage, if the projectile still has some kinetic energy, it would be exiting with certain velocity. If the exit velocity is zero, the corresponding incident ballistic impact velocity of the projectile is the ballistic limit. If the projectile velocity is zero during the contact event before all the fibres are broken, it indicates that only the partial penetration has occurred.
Primary yarns/fibres provide the force to resist the penetration of the projectile into the target. The strain is maximum in these yarns/fibres. These yarns fail under tension when the strain of these yarns exceeds the ultimate strain in tension at high strain rate. Failure of these yarns/fibres results in absorption of some energy. Fig. 1 indicates the strain variation in primary yarns within a layer and in through the thickness direction. During cone formation, the tensile strain would be maximum along the middle primary yarn, in each layer. Within the primary yarns in a layer strain reduces gradually, as we move away from the mid primary yarn. This is due to the fact that the middle yarn is pushed by complete projectile diameter, but other primary yarns face a projectile length which is lesser than projectile diameter. The top layers of the laminate when impacted get compressed apart from the cone formation, which causes the strain in the top layers to be more than that in the bottom layers. The extent of compression is based on the stress at the point of impact. This stress causes an extra straining of the top layers. The variation of strain in through the thickness direction is shown in Fig. 1. The strain in the top layers is more than the strain in the bottom layers. Each primary yarn has different strain. As and when the strain of a particular yarn reaches the dynamic tensile failure strain, failure occurs to that particular yarn. As a result, there will be a sequential failure of yarns starting with the yarns in the top layer, then proceeding toward the bottom layer. It may be noted that the strain within the primary yarn is not constant. At any moment it is maximum at the point of impact and decreases as we move away from the point of impact. 2.4. Energy absorbed due to shear plugging
2.2. Energy absorbed due to deformation of secondary yarns Possible cone formation on the back face of the target during ballistic impact event is shown in Fig. 1. The
Cantwell and Morton [13] observed frustum-shaped shear cut-out zone on impacting carbon fibre reinforced angle ply laminates with steel spherical bullets. Lee and Sun [14] and Ellis [15] reported shear plugging to be one
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target. Tensile failure of the yarns thus absorbs some energy of the projectile. During the ballistic impact event, delamination and matrix cracking take place in the laminate area which forms the cone. The total kinetic energy of the projectile that is lost during ballistic impact is the total energy that is absorbed by the target till that time interval and is given by ETOTALi ¼ EKEi þ ESPi þ EDi þ ETFi þ EDLi þ EMCi þ EFi Fig. 3. Schematic showing plug formation.
of the major damage modes on impact of angle ply graphite/bismaleimide by blunt/flat projectiles. Fig. 3 shows schematic arrangement of plug formation during ballistic impact. Brittle nature of carbon based composites was responsible for change in damage mechanism. Shear plug formation is not observed for glass reinforced composites which have high failure strain at high strain rates. 2.5. Energy absorbed by other mechanisms Delamination and matrix cracking would absorb some energy (Fig. 2). Additionally, friction between the projectile and the target would also absorb some energy.
3. Analytical formulation for woven fabric composites On striking a target, energy of the projectile is absorbed by various mechanisms like kinetic energy of moving cone EKE , shear plugging ESP , deformation of secondary yarns ED , tensile failure of primary yarns ETF , delamination EDL , matrix cracking EMC , friction energy EF along with other energy dissipation mechanisms. An analytical model is presented for the above-mentioned mechanisms, which dominate during ballistic impact of two-dimensional woven fabric composites [16,17]. 3.1. Analytical formulation A cone is observed on the back face of the target as shown in Fig. 1 when struck by a projectile [9,12]. Shear plugging on the front face and cone formation on the back face start taking place depending on the target material properties during the ballistic impact event. The velocity of the cone is the same as the velocity of the projectile. Initially, the cone has velocity equal to that of the projectile and has zero mass. As time progresses, the mass of the cone increases and velocity of the cone decreases. As the cone formation takes place, the yarns/ fibres deform and absorb some energy. The primary yarns which provide the resistive force to the projectile motion deform the most, thus leading to their failure. When all the primary yarns fail the projectile exits the
ð1Þ An analytical model to predict the ballistic limit is presented below. Firstly, modelling of a single yarn is presented. The model is further extended to the woven fabric composites. The analytical model is for the case when the projectile impacts the target normally. The following assumptions are made in the analytical model presented: (1) Projectile is perfectly rigid and remains un-deformed during the ballistic impact. (2) Projectile motion is uniform during penetration within each time interval. (3) Yarns/fibres in a layer act independently. (4) Energy absorption due to primary yarn/fibre breakage and deformation of the secondary yarns are treated independently. (5) Longitudinal and transverse wave velocities are the same in all the layers. 3.2. Modelling of a single yarn subjected to transverse ballistic impact When a yarn is impacted by a projectile transversely, longitudinal strain wavelets propagate outward along the filament [4,18,19]. The outermost wavelet, called the elastic wave propagates at velocity sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 dr ce ¼ ð2Þ q de e¼0 The innermost longitudinal wavelet called the plastic wave propagates at velocity sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 dr cp ¼ ð3Þ q de e¼ep Here, ep denotes the strain at which plastic region starts. As the strain wavelets pass a given point on the yarn, material of the filament flows inward toward the impact point. The material in the wake of the plastic wave front forms itself into a transverse wave, shaped like an inverted tent with the impact point at its vertex. In a fixed coordinate system, the base of the tent spreads outward with the transverse wave velocity. This is depicted in Fig. 4. The transverse wave velocity is given by
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critical strain energy release rates along warp/fill directions and the other directions. For simplicity, the cone formed can be considered to be of circular cross-section, rather than quasi-lemniscate. The base of the cone spreads with a transverse wave velocity sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ep ð1 þ ep Þrp 1 dr ct ¼ de ð7Þ q de q 0 and the plastic wave velocity is given by sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 dr cp ¼ q de e¼ep
Fig. 4. Configuration of a yarn/fibre before and after transverse impact: (a) before impact; (b)–(d) after impact.
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ep ð1 þ ep Þrp 1 dr ct ¼ de q de q 0
n¼i X
n¼i X
n¼i X
ctn Dt
ð9Þ
cpn Dt
ð10Þ
n¼0
ð4Þ rpi ¼
ctn Dt
ð5Þ
cpn Dt
ð6Þ
n¼0
rpi ¼
After time ti the transverse wave has traveled to a distance rti and the plastic wave has traveled to a distance of rpi . The projectile has moved through a distance zi . Radii rti and rpi after time ti ¼ iDt are given by rti ¼
If the complete impact event is divided into a number of small instants, then at ith instant, the time is given by ti . By that time the transverse wave has traveled to a distance rti and the plastic wave has traveled to a distance of rpi . The projectile has moved through a distance zi . Radii rti and rpi after time ti ¼ iDt is given by rti ¼
ð8Þ
n¼0
3.3. Modelling of woven fabric composite Woven fabric composites consist of warp and fill yarns, interlaced in a regular sequence. As the projectile impacts on to the woven fabric composite, there can be many yarns beneath the projectile. In the present analysis, all the warp and fill yarns are treated separately. Behaviour of each yarn is analysed as explained in the previous section. Overall behaviour of the woven fabric composite is presented considering the combined effect of all the individual yarns within a layer. 3.4. Strain and deceleration history: single yarn of woven fabric composite
n¼i X n¼0
Because of stress wave attenuation, the stress in the yarn and hence, the strain in the yarn varies with the distance from the point of impact. Stress and in turn strain is maximum at the point of impact and decreases as one moves away from the point of impact. At ith time interval, the strain at the point of impact is given by qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8 9 < ðd=2Þ þ ðrti ðd=2ÞÞ2 þ z2i þ rpi rti rpi = ei ¼ : ; bðrpi =aÞ 1 ln b ð11Þ a Once, the strain variation in a yarn/fibre is known, the energy absorbed by various mechanisms like kinetic energy of the moving cone, deformation energy absorbed by secondary yarns, tensile failure of primary yarns, delamination and matrix cracking can be calculated. The formulation for energy absorbed by various energy absorbing mechanisms is provided in the following sections. At the beginning of the first time interval of impact, entire energy is in the form of kinetic energy of the projectile. Later, this energy is divided into energy absorbed by various damage mechanisms and the kinetic energy of moving cone and projectile. Considering the energy balance at the end of ith time interval, we obtain KEp0 ¼ KEpi þ EKEi þ ESPði1Þ þ EDði1Þ þ ETFði1Þ
When a projectile impacts a woven fabric composite, a cone with quasi-lemniscate cross-section would be formed on the back face of the target. Quasi-lemniscate shape is because of difference in elastic properties and
þ EDLði1Þ þ EMCði1Þ þ EFði1Þ
ð12Þ
Rearranging the terms in the above equation, the following form is obtained:
N.K. Naik, P. Shrirao / Composite Structures 66 (2004) 579–590 1 m V2 2 p 0
Ei1 ¼ 12 mp þ MCi Vi 2
ð13Þ
3.5. Kinetic energy of cone formed The cone formed on the back face of the target absorbs some energy. By the end of ith time interval the radius of the cone formed is
where Ei1 ¼ ESPði1Þ þ EDði1Þ þ ETFði1Þ þ EDLði1Þ þ EMCði1Þ þ EFði1Þ
585
ð14Þ
rti ¼
n¼i X
ð20Þ
ctn Dt
n¼0
Each of the contributing terms to Ei is explained in the succeeding sections. The terms on the right-hand side of Eq. (14) are known at (i 1)th instant of time. From this, the velocity of the projectile at the end of ith time interval can be obtained and it is given by sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 mp V 2 Ei1 Vi ¼ 21 0 mp þ MCi 2
ð15Þ
If the projectile velocity is known at the beginning and the end of ith time interval, then the deceleration of the projectile during that time interval can be found out. It is given by dci ¼
Vi1 Vi Dt
ð16Þ
Note that the velocity at the beginning of the ith time interval is the same as the velocity at the end of (i 1)th time interval. Distance traveled by the projectile zi up to ith time interval is given by zi ¼
n¼i X
Dzn
ð17Þ
n¼0
1 2 Dzi ¼ Vi1 Dt dci ðDtÞ 2
MCi ¼ prti2 hq
ð21Þ
The velocity of the cone formed is equal to Vi , the velocity of the projectile at the end of ith time interval. So the energy of the cone formed at the end of ith time interval is EKEi ¼ 12MCi Vi 2
ð22Þ
3.6. Energy absorbed due to shear plugging When the target material is impacted by the projectile, shear plugging stress in the material near projectile periphery rises. As and when the shear plugging stress exceeds shear plugging strength, shear plugging failure occurs. As a result, plug formation takes place. This phenomenon is observed for carbon/epoxy composites (Fig. 3). If at the beginning of the ith time interval, shear plugging stress exceeds shear plugging strength, then the energy absorbed by shear plugging during that time interval is given by the product of distance sheared, shear plugging strength and the area over which shear plugging stress is applied. It is given by DESPi ¼ Nhl SSP p dh
ð18Þ
Utilizing dci , the deceleration of the projectile during ith time interval, the force resisting the projectile motion can be calculated. It is given by Fi ¼ mp dci
Mass of the cone formed is
ð19Þ
The magnitude of the force on the projectile is the same as the magnitude of the force applied on the target, by the projectile. Hence, this force can be used to determine whether shear plugging takes place or not. The above process is repeated until all the primary yarns in the target fail, i.e., the complete perforation takes place. The velocity at the end of time interval, during which all the yarns are broken, is the residual velocity of the projectile. On the other hand, if the numerator in Eq. (15) becomes zero, then the projectile does not penetrate the target completely with the given initial velocity. Thus by repetition of above procedure with various velocities so as to get complete perforation with zero residual velocity, the ballistic limit of the target laminate can be obtained.
ð23Þ
where N indicates the number of layers shear plugged during ith time interval and SSP denotes shear plugging strength. The energy absorbed by shear plugging by the end of ith time interval is given by n¼i X ESPi ¼ DESPn ð24Þ n¼0
3.7. Energy absorbed due to deformation of secondary yarns The secondary yarns experience different strains depending on their position. The yarns which are close to the point of impact experience a strain equal to the strain in the outermost primary yarn, whereas those yarns which are away from the impact point experience less strain. The strain variation is assumed to be linear from A to B (Fig. 1). This imposes the following boundary conditions for the variation in strain at time ti : pffiffiffi esy ¼ epy at r ¼ d= 2; i:e:; at point A esy ¼ 0 at r ¼ rti ;
i:e:; at point B
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Here, epy is the strain in the outermost primary yarn in that layer. The energy absorbed in the deformation of all the secondary yarns can be obtained by the following integration: Z rti Z esyi EDi ¼ rsy esy desy h pffiffi d= 2
0
2pr 8r sin1 ðd=2rÞ dr
ð25Þ
3.8. Energy absorbed due to tensile failure of primary yarns The yarns directly below the projectile, known as the primary yarns, fail in direct tension. All the primary yarns within one layer do not fail at one instant of time. As and when the strain of a particular yarn reaches the dynamic failure strain in tension, the yarn fails. It may be noted that the length of yarns/fibres failing in tension is twice the distance covered by the longitudinal wave. Also, the complete length of a primary yarn is not strained to the same extent as explained earlier. The reason for this phenomenon is stress wave attenuation. When the strain in yarns/fibres exceeds failure strain, it fails and some energy is absorbed due to tensile failure. For a yarn/fibre of cross-section area A it is given by ! Z x Z e¼e0 bx=a ETF ¼ A rðeÞde dx ð26Þ 0
delamination may not occur at all the lamina interfaces. Towards the end of ballistic impact event, when only a few non-delaminated layers are left, these non-delaminated layers are more likely to bend rather than delaminate. The area undergoing delamination and matrix cracking in the conical region is of quasi-lemniscate shape, which is taken to be Aql percent of the corresponding circular area. During (i þ 1)th time interval the area of delamination and matrix cracking is given by 2 2 rdi ÞAql pðrdðiþ1Þ
ð27Þ
where rdi indicates the radius up to which the damage has propagated until ith time interval. So the respective energies absorbed by delamination and matrix cracking during this time interval are given by 2 2 DEDLi ¼ Pd pðrdðiþ1Þ rdi ÞAql GIIcd
ð28Þ
2 2 DEMCi ¼ Pm pðrdðiþ1Þ rdi ÞAql Emt h
ð29Þ
The factors Pd and Pm stand for percentage delamination and percentage matrix cracking. Energy absorbed till the ith time interval is given by EDLi ¼
n¼i X
DEDLn
ð30Þ
DEMCn
ð31Þ
n¼1
EMCi ¼
n¼i X n¼1
e¼0
where e0 is the ultimate strain limit. If during ith time interval N numbers of yarns/fibres are failing, then the right-hand side of the above expression is multiplied by N . 3.9. Energy absorbed due to delamination and matrix cracking Delamination and matrix cracking absorb some part of the initial kinetic energy of the projectile. A part of the conical area undergoes delamination and matrix cracking (Fig. 2). The extent to which composite has delaminated and the matrix has cracked till (i þ 1)th time instant can be calculated on the basis of strain profile in the composite at that time interval. From the results derived in the earlier section it can be observed that the strain at the impact point is the highest and it decreases as one moves away from the point of impact. The area in which strain is more than the damage initiation threshold strain ed , will undergo damage in the form of matrix cracking and delamination. However, complete matrix cracking may not take place. Evidence for this phenomenon is provided by the fact that after ballistic impact, matrix is still attached to the fibres and does not separate from the reinforcement completely. Due to matrix cracking, the interlaminar strength of the composite decreases. As a result, further loading and deformation causes delamination. This delamination is of mode II type. Again,
3.10. Calculation of contact duration The impact event starts when the projectile touches the front face of the target. End of the event is taken to be when all the primary yarns fail or when the velocity of the projectile becomes zero. Total time from the start of the impact event till the end of the event is called as contact duration. If the end of the event occurs during nth time interval, then the contact duration is obtained as tc ¼ nDt
ð32Þ
4. Results and discussion Ballistic impact behaviour of woven fabric composites when impacted by a flat ended projectile has been presented. This behaviour is based on the analytical method presented in the previous section. Specifically, ballistic limit, contact duration at ballistic limit and surface radius of the cone formed at the ballistic limit have been determined. These results have been compared with the experimental results. Behaviour of plain weave E-glass/epoxy and twill weave T300 carbon/epoxy composite structures under ballistic impact has been compared. Input data necessary for the analytical prediction of ballistic impact behaviour in the present study has been provided in Table 1.
N.K. Naik, P. Shrirao / Composite Structures 66 (2004) 579–590
587
Table 1 Input parameters required for the analytical predictions of ballistic impact behaviour Projectile details (cylindrical) Mass Shape Diameter
2.8 g Flat ended 5 mm
2.8 g Flat ended 5 mm
Target details Material Vf Thickness Number of layers Density
Twill weave T300 carbon/epoxy 50% 2 mm 5 1400 kg/m3
Plain weave E-glass/epoxy 50% 2 mm 6 1750 kg/m3
Other details Stress–strain Time-step Quasi-lemniscate area reduction factor Stress wave transmission factor Delamination percent Matrix crack percent Shear plugging strength Mode II dynamic critical strain energy release rate Matrix cracking energy Fibre failure energy
Fig. 9 1 ls 0.9 0.95 100% 100% 90 MPa 800 J/m2 0.9 MJ/m3 6 MJ/m3
Fig. 9 1 ls 0.9 0.825 100% 100% – 1000 J/m2 0.9 MJ/m3 28 MJ/m3
4.1. Calculation of ballistic limit, contact duration and surface radius of the cone formed during ballistic impact Using the analytical method presented earlier ballistic impact behaviour of typical plain weave E-glass/epoxy and twill weave T300 carbon/epoxy composites has been evaluated. The results are presented in Table 2 and Figs. 5–8. For the case of E-glass/epoxy, complete perforation did not take place at VI ¼ 158 m/s whereas with VI ¼ 159 m/s, complete perforation took place with exit velocity of 54 m/s. The experimentally obtained ballistic limit is V50 ¼ 150 m/s. Analytically predicted ballistic limit is between 158 and 159 m/s. For the case of T300 carbon/ epoxy, complete perforation did not take place with VI ¼ 82 m/s whereas with VI ¼ 83 m/s, complete perforation took place with exit velocity of 18 m/s. The above results are with mp ¼ 2.8 g, d ¼ 5 mm and h ¼ 2 mm. For T300 carbon/epoxy the studies were also carried out with mp ¼ 1:8 g, h ¼ 2 mm and d ¼ 5 mm. For this case the experimentally obtained ballistic limit,
V50 ¼ 105 m/s. The analytically predicted ballistic limit VBL ¼ 99 m/s (Table 2). It is generally observed that there is a good co-relation between the experimental results and the predicted results for both E-glass/epoxy and T300 carbon/epoxy. From Figs. 5 and 7 it can be seen that the ballistic limit is much higher for E-glass/epoxy than for T300 carbon/epoxy. This is because of higher energy absorbed by E-glass/epoxy than by T300 carbon/epoxy (Fig. 9). Fig. 9 is for high strain rate loading. Also, shear plugging strength for E-glass/epoxy is much higher than that for T300 carbon/epoxy. For E-glass/epoxy, energy absorbed by secondary yarn deformation and tensile failure of yarns are the main energy absorbing mechanisms. Shear plugging does not take place in this case. On the other hand, for T300 carbon/epoxy, the ultimate failure is because of shear plugging. Energy absorbed by secondary yarn deformation and shear plugging are the main energy absorbing mechanisms (Figs. 6 and 8). In this case, tensile failure of the yarns does not take place.
Table 2 Ballistic impact test results for typical plain weave E-glass/epoxy and twill weave T300 carbon/epoxy composites, d ¼ 5 mm, h ¼ 2 mm Material
Plain weave E-glass/epoxy Twill weave T300 carbon/epoxy Twill weave T300 carbon/epoxy
Projectile mass, mp (g)
Predicted ballistic limit, VBL (m/s)
Experimental ballistic limit, V50 (m/s)
Predicted damage size, rd (mm)
Experimental damage size, rd (mm)
Predicted surface radius of the cone, rt (mm)
2.8
159
150
9.6
10
35
1.8
99
105
–
–
59
2.8
83
–
–
–
61
588
N.K. Naik, P. Shrirao / Composite Structures 66 (2004) 579–590 Projectile kinetic energy
(a)
Cone kinetic enery Shear plugging energy
Projectile kinetic energy
40
Cone kinetic energy Secodary yarn deformation energy
Secondary yarn deformation energy
10
(b)
35
Tensile failure energy
Delamination energy Matrix cracking energy
Delamination energy
30
8
Matrix cracking energy Energy,E(J)
Energy,E(J)
25 6
4
20 15 10
2
5 0
0
-5 0
100
200
300
400
500
600
0
700
20
40
Time,t (micro-sec)
60
80
100 120 140 160
Time,t (micro-sec)
Fig. 5. Energy absorbed by different mechanisms during ballistic impact event, mp ¼ 2:8 g, h ¼ 2 mm, d ¼ 5 mm: (a) twill weave T300 carbon/epoxy laminate, VI ¼ 82 m/s; (b) plain weave E-glass/epoxy laminate, VI ¼ 158 m/s.
90
1.0
Projectile velocity
80
Maximum strain in top layer
0.9
Maximum strain in bottom layer
0.8
(b)
Projectile velocity Maximum strain in to player Maximum strain in bottom layer
60
0.6
50
0.5 0.4
40
0.3
30
0.2
60
40
Contact force
5
7
140
6 4
120
20
5 3
100 80
2
60
Tensile strain (%)
0.7
160
Velocity, V(m/s)
Shear plugging stress
80 Shear plugging stress (MPa)
70
Velocity, V(m/s)
100
Tensile strain (%)
(a)
4 3 2
1
20
40
0.1
1 0
10
0.0
0
20
0
0
-0.1 0
200 400 Time,t (micro-sec)
0 -1 0
600
20
40
-1
60 80 100 120 140 160 Time,t (micro-sec)
Fig. 6. Projectile velocity, strain, contact force and shear plugging stress variation during ballistic impact event, mp ¼ 2:8 g, h ¼ 2 mm, d ¼ 5 mm: (a) twill weave T300 carbon/epoxy laminate, VI ¼ 82 m/s; (b) plain weave E-glass/epoxy laminate, VI ¼ 158 m/s.
(a)
(b) Projectile kinetic energy 10
40
Projectile kinetic energy
Cone kinetic energy
Secondary yarn deformation energy
8
Cone kinetic energy
35
Shear plugging energy
Secondary yarn deformation energy
Tensile failure energy
30
Delamination energy
Energy, E (J)
Energy, E (J)
Delamination energy
25
Matrix cracking energy 6
4
Matrix cracking energy
20 15 10
2
5 0
0
-5 -10
0
10
20
30 40 50 60 Time,t (micro-sec)
70
80
90
0
20
40
60
80 100 120 140 Time,t (micro-sec)
160
Fig. 7. Energy absorbed by different mechanisms during ballistic impact event, mp ¼ 2:8 g, h ¼ 2 mm, d ¼ 5 mm: (a) twill weave T300 carbon/epoxy laminate, VI ¼ 83 m/s; (b) plain weave E-glass/epoxy laminate, VI ¼ 159 m/s.
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589
Projectile velocity Projectile velocity
(a)
Maximum strain in top layer
(b)
Maximum strain in top layer Maximum strain in bottom layer
Maximum strain in bottom layer Contact force
160
7
5
Shear plugging stress
160
140
0.5 50 0.4 40
0.3 0.2
30
0.1
100 80 60 40
120 Velocity,V (m/s)
0.6
60
Tensile strain (%)
Velocity,V (m/s)
120
0.7
Shear plugging stress (MPa)
0.8 70
6 4
140
5 3
100 80
2
60
Tensile strain (%)
0.9
80
0.0
10
40
0
10
20
30
40
50
60
70
80
2 1
20
0 0
0
-0.1 -10
3
1
20
20
4
Contact force, F (KN)
1.0
90
90
0
-1 0
20
Time,t (micro-sec)
40
-1
60 80 100 120 140 160 Time,t (micro-sec)
Fig. 8. Projectile velocity, strain, contact force and shear plugging stress variation during ballistic impact event, mp ¼ 2:8 g, h ¼ 2 mm, d ¼ 5 mm: (a) twill weave T300 carbon/epoxy laminate, VI ¼ 83 m/s; (b) plain weave E-glass/epoxy laminate, VI ¼ 159 m/s.
References
Fig. 9. Stress–strain curves under tensile loading at high strain rate: (a) E-glass/epoxy yarn; (b) T300 carbon/epoxy yarn.
It is generally observed that the energy absorbed by delamination and matrix cracking is of small magnitude.
5. Conclusions Ballistic impact behaviour of typical plain weave Eglass/epoxy and twill weave T300 carbon/epoxy composites has been studied. It has been observed that, for identical ballistic impact conditions, ballistic limit is higher for E-glass/epoxy than for T300 carbon/epoxy. For E-glass/epoxy, energy absorbed by secondary yarn deformation and tensile failure of primary yarn are the main energy absorbing mechanisms. For T300 carbon/ epoxy, the main energy absorbing mechanisms are the secondary yarn deformation and shear plugging.
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