Impact on Ceramic Target

8/18/2019 Impact on Ceramic Target

1/20

Chapter 7

Impact on Ceramic Target

7.1 Overview

The unique low-density high strength properties of ceramics make them suitable forweight-saving armouring system. The use ranges from military protection (see figure

7.1) for armoured vehicles and body-armours on soldiers to civil protection of aircraft

turbine, spacecraft modules and nuclear facilities at particular threat level. To harness

the maximum potential, the characterization, in-depth understanding of constitutive and

failure behaviour are a pre-requisite. Incidentally the experimental procedures, primarily

dealing with destructive testing at high strain rate, are expensive to have meaningful

data extraction. Recent trend tries to fuse the performance prediction through numerical

simulation with the test procedures (Krishnan et al., 2010) enhancing the cost efficiency

and understanding of material behaviour although the experimental design is unlikely tobe replaced completely.

(a) Armour vest with ceramic insert (b) Add-on ceramic tiles on armoured

vehicle

Fig. 7.1 Application of ceramics in ballistic protection

In fact the design optimization of armouring assembly uses information exchange

among different approaches as shown in figure 7.2.

Under impact loading the ceramic material undergoes yielding in compression and

137


2/20

Ceramic Target

Fig. 7.2 Methodology of add-on armour design optimisation (Gálvez and Paradela,

2009)

eventually may be fractured, pulverized, and ejected depending on induced impulse in-

tensity. Another interesting feature is tensile spalling near the opposite surface owing to

reflected tensile wave. These phenomenological characteristics can be studied with in-

sightful observation if a suitable computational framework can accommodate the related

constitutive behaviour with reasonable accuracy. The pseudo-spring SPH framework is

explored in this regard in the following sections after adapting a established damage

model chosen from the varieties reported in literature.

7.2 Ceramic as an armouring material

Ceramics are one of the mostly used material other than metal and polymers in ballistic

shielding application. They can be crystalline or amorphous with inter-atomic bondingsranging from purely ionic to totally covalent one (but not chemical bonding as in metals)

(Walley, 2010). Due to these type of bondings, ceramics exhibits hard, yet brittle be-

haviour without significant plastic deformation. Further, the micro-texture and packing

of grains get altered after different processing and treatments during manufacturing.

7.2.1 Armouring efficiency of ceramic

Some ceramics with high K IC (fracture toughness) values like zirconia do not exhibit

good ballistic performance. Also it has been observed in many ceramic systems that

the second phases and reinforcements which often increase the fracture toughness by

crack bridging and branching typically lead to poorer armour performance (Rice, 2000).

It appears that an optimum balance between the hardness and the fracture toughness

must be maintained. Even any further increase in thickness over the optimum range is

not necessarily beneficial. The better performing armour ceramics typically have high

Young’s Moduli and indentation hardnesses, moderate to low densities (at low porosity),

and moderate to fine grain sizes (Rice, 2000). The ceramic materials used for ballistic

138


3/20

7.2 Ceramic as an armouring material

protection can be either monolithic structural ceramics or ceramic matrix composites.

Oxide ceramics like alumina ( Al2O3) in both single crystal and polycrystalline forms,

and non-oxide ceramics like silicon carbide (SiC ), boron carbide ( B4C ) and titanium

diboride (TiB2) are the most commonly used monolithic structural ceramics.

TiB2 and SiC have reasonably high hardness values, but the hardest one is B4C . This

hardness is useful in eroding the projectile tip and hence is preferred with a high value.

Regarding the elastic moduli, T iB2 with E = 550GPa offers better resistance than SiC

and B4C . But the low reference density of B4C makes it able to have higher rate of

energy dissipation with higher sonic velocity. But ultimate failure is direct function of

fracture toughness and hence the fracture behaviour is found to be superior in case of

TiB2 followed by that of Al2O3, SiC and then B4C .

7.2.2 Relative comparison of ballistic performances

Ceramic materials under dynamic loading are generally characterised by wave profiles

from flyer plate tests, stress time plots from bar impact experiments and depth of pene-

tration measurements (Lamberts et al., 2007). O’Donnell (1992) performed impact tests

on SiC , B4C , Al2O3, and TiB2 tiles bonded to 2024-T351 aluminum alloy plates with

conical nose shaped steel projectiles. The surface area of the fragments generated was

computed. It was found that boron carbide had the maximum fragment surface area

while SiC had the least. TiB2 caused the greatest erosion (∼ 59% of initial projectilemass) in the penetrator.

Regarding the relative thickness of layers/components in add-on armours with metal-

lic backing, the thin backing experiences greater dishing action and subsequent bulging,whereas the thicker backing demonstrates crater formation and ceramic ejection as shown

in figure 7.3. The communuted portion (crushed or pulverized ceramic but still with

residue load carrying capacity) or the cone angle of that zone in front facing ceramic

also changes for different mode of deformations.

(a) Thin Backing plate (b) Thick Backing plate

Fig. 7.3 Schematic representation of wave propagation (Woodward et al., 1994)

139


4/20

Ceramic Target

7.2.3 Micro-mechanics

Ceramics with strong ionic/covalent bonding, the microstructural parameters such as es-

pecially the grain size, shape, and orientation play a key role in determining its basic

mechanical properties like tensile and compressive strengths, hardness, toughness, and

wear properties. With similarity to Hall Petch relationship in metals, ceramic compres-

sive strength and hardness is inversely proportional to the square root of the grain size.

The tensile or flexural strength vary in a similar trend (non-linear decrease in the tensile

strength with increasing grain size). Whereas the fracture toughness is a strong function

of bonding and microstructure. And in addition to grain size, it is influenced by grain

shape, presence of second phases, grain orientation, and porosity. At high loading rate,

failure fronts propagate at speeds approaching to sound speed (more precisely, Rayleigh

wave speed) in the medium and tune according to grain size. But for heavy projectile,

the dependency is not so clearly understood.

Ceramic manufacturing and processing include pressing, sintering, hot isostatic press-

ing with or without additives. Each of the processing steps can tap flaws potentially and

hence the structural behaviour of the finished product are sensitive to synthesis and fab-

rication techniques with large heterogeneity and less reliability. Because of low tensile

and fracture toughness (0.5−5 MPa√ m), ceramics are used generally in front of armoursystem to primarily absorb the initial kinetic energy from the projectile by carrying

compressive loads predominantly. The damage pattern observed under such loadings

include - extensively fractured but still interlocked debris with high micro-crack den-

sity (commonly known as comminuted or mescall zone) in high pressure-high shear

zone, frontal surface with visible macro radial cracks due to profuse dislocations, lateral

cracking owing to material spallation at backside from reflected tensile pulse. Studies

(Ashby and Hallam, 1986; Vekinis et al., 1991) have shown that micro-structural defects

such as pores, cracks, inclusion, inhomogeneities and stress concentration regions like

secondary phases and elasticity mismatch at grain boundaries, triple points and even

sub-grain features like twin, stacking faults may induce tensile micro-crack nucleation

under external loadings by transforming global compression into a population of local

tensile zones. But the effects of such defects distribution on dynamic failure behaviour

are not established well yet and needs more careful investigation. Confining effect in-

hibits the nucleation and propagation of those micro-flaws - increasing the materials

load carrying capacity, and even may cause the brittle material nominally yield due to

plasticity (Lankford, 1977) when high confinement effect is present in materials with

low flaw density.

Microplasticity is the failure mechanism for the finer grain sizes. But for the larger

grain sizes, where the flaw dimensions are comparable to or greater than the grain sizes,

the most anticipated failure mechanism is Griffith flaw failure.

140


5/20

7.3 Ceramic material models


Sudden rupture after linear elastic deformation is the general failure mode for ceramics

under tension. Whereas, this changes to more gradual behaviour under compression, and

if under confinement, only loss in load carrying capacity is partial in place of complete

failure. Regarding estimation of damage state in ceramic materials, the broad group of

micromechanical models takes into account micro-cracks nucleation and growth under

multi-axial loading reproducing measurements obtained from impact experiments on

specimen reasonably well; whereas in the phenomenological models, an explicit func-

tion of time and effective stress is used to predict damage propagation, via calibrating

coefficients of the function with other measured quantities.

7.3.1 Microphysical models

Micro-mechanical models to capture loading rate dependent dynamic damage evolu-tion are generally based on sliding crack model (Heard and Cline, 1980) through the

dependence of dynamic stress intensity factor on crack growth speed. However, the mi-

cromechanical models must assume randomly distributed initial cracks and their sizes,

and they are computationally very expensive, generally not applicable to large-scale

problems.

Taylor et al. (1986) proposed statistical evaluation of micro-crack distribution (TCK

model), where individual micro-crack growth is obtained via fracture mechanics the-

ory. By using a scalar damage value for micro-crack distribution within the material,

the strain-rate-dependent inelastic behaviour was modelled when under tension. In thisTCK model gradual decrease in compressive strength with damage accumulation and

pressure, strain-rate dependency were not included.

The effect of the strain-rate and damage-dependent compressive strength were cou-

pled with TCK model by Rajendran and Kroupa (1989) primarily for tensile dominant

damages under extremely high pressure (> 10 GPa). Rajendran and Grove (1996) fur-

ther introduced micro-cracking and plasticity based another damage model for simu-

lating shock wave tests output. But the difficult and exhaustive procedure of parameter

identification via trial-and-error simulations to match several test results limits the scope

and applicability of this model.

7.3.2 Phenomenological models

Though phenomenological models are often only applicable to the specific experiments

they were based on. The calibrated coefficients becoming invalid if any configuration

changes such as geometry and boundary conditions are made. For example, uniaxial

strain tests and uniaxial stress tests were dominantly used to calibrate these models.

141


6/20

Ceramic Target

7.3.2.1 Simha model

Fahrenthold (1991) developed a continuum model based on the idea of complimentary

energy density that used a second-order tensor to represent anisotropic damage. Simha

et al. (2002) developed a damage model similar to Fahrenthold’s model, inferred from

bar and plate impact tests on AD−99.5 alumina. In their model, it was assumed thatceramics comminute at the Hugoniot Elastic Limit (HEL) and yield strength of material

is weighted sum of intact and damaged strength.

σ y = σ intact (1− D)+σ f ailed D+ 32

˙̄ε

γ (7.1)

where σ f ailed = min(α P,σ max) and γ = γ 0eγ 1(P−P HE L).

γ 0, γ 1, α , σ max and P HE L are material parameters. The term γ controls the contribu-

tion of the effective deviatoric strain rate ˙̄ε . This rate dependency is the phenomenolog-

ical contribution of micro-crack sliding, dislocation activity and grain boundary sliding.

7.3.2.2 John-Holmquist models

Pressure-dependent strength, damage and fracture based on accumulated plastic strain,

significant strength reduction and bulking after fracture, and strain rate effects are incor-

porated in models developed by Johnson and his co-workers. The first version (Johnson

and Holmquist, 1992), JH-1 model, employs piecewise linear strength envelope with

sudden strength reduction and pressure increases due to bulking just when damage vari-

able D reaches 1. Improvisation (JH-2) was made on this by using analytic smooth

function as the strength envelope to avoid sudden change in strength profile at juncture

of piecewise linear fits (Johnson and Holmquist, 1994). And gradual softening, bulking

due to incremental damage accumulation was employed here. But the actual behaviour

of brittle ceramics is perceived to be more realistically represented by sudden change

in strength after complete damage ( D = 1), particularly in plate impact scenario and

demonstrating dwell-penetration transition (Simha et al., 2002). Hence Johnson et al.

(2003) introduced another variety (JHB) by keeping sudden change of strength as per

JH-1 but approximating the strength profile by smoothed analytic function as in JH-2.

Additionally phase-change and hysteresis during unloading were also incorporated. The

pressure estimation independent of internal energy (except bulking pressure), accumu-

lation of damage and effect of strain rate were represented similarly in all three versions.

Pressure is computed by polynomial fit (similar to Hugoniot fit) equation of state

(EOS) with η = ρρ0 −1 as,

P =

K 1η +K 2η

2 +K 3η3 +∆P , if, η > 1

K 1η , otherwise(7.2)

142


7/20


If the prevalent strain-rate (ε̇ ) is more than the reference strain rate (ε̇ 0) at which

intact and damage strength (σ eq,0) parameters are obtained, the effect of strain-rate hard-

ening is represented by,

σ eq = σ eq,0

1+Cln

ε̇

ε̇ 0

(7.3)

Damage ( D) is perceived to be accumulating with the increments in effective plastic

strain (ε e f f

pl ) as,

D =∑∆ε

e f f pl

ε f pl

(7.4)

where ε f pl is the variable fracture strain computed differently in different versions.

Since present numerical investigation is based on JH model, distinct features of its

three versions are now discussed in brief in turn below.

7.3.2.3 JH-1 model

(a) Strength without Strain Rate Effects) (b) Fracture Strain

Fig. 7.4 JH-1 model (Johnson and Holmquist, 1992)

The intact and fractured strengths (the Mises effective stresses σ eq,i and σ eq, f respec-

tively) are estimated from current state of pressure as,

σ eq,i =σ i,1

P+T P

1−T

,

−T

≤P


8/20

Ceramic Target

tion 7.4, where the fracture strain is estimated as,

ε f pl =

P+T

Pmax +T ε f max (7.7)

As evident from the figure 7.4, only input state variable in above equation is pressure

P, all other parameters being material constants determined from combinations of test

procedures.

After complete damage ( D = 1), in addition to strength reduction, an increase in

terms of bulking (∆P) due to volume increase with increased free surfaces in damaged

ceramic is added to P. This pressure increment is determined from decrease in internal

elastic energy (U = σ 2ev

6G, σ ev and G being von-Mises effective stress and shear modulus

respectively) due to damage as,

∆P = −K 1η + (K 1η +∆P)2 +2β f K 1∆U (7.8)

where, ∆U = U i−U f is the difference in internal elastic energy before and after damageand β f is the fraction (0 ≤ β f ≤ 1) of the internal deviatoric energy loss converted topotential hydrostatic energy.

7.3.2.4 JH-2 model

(a) Strength without Strain Rate Effects (b) Bulking pressure

Fig. 7.5 JH-2 model (Johnson and Holmquist, 1994)

Here current strength at damage D is interpolated from intact and failed strengths as

σ eq = σ eq,i− D(σ eq,i−σ eq, f ) where each of the strength envelope is represented as,

σ eq,i = A

P

P HE L+

T

P HE L

N σ HE L (7.9)

σ eq, f = B

P

P HE L

M σ HE L (7.10)

144


9/20


Additional normalization constants P HE L and σ HE L are the pressure and stress at the

Hugoniot elastic limit (HEL). Once HEL is known for a particular material, η HE L is

iteratively compueted from

HE L = K 1

∗η HE L +K 2

∗η2 HE L +K 3

∗η3 HE L +

4

3

G η HE L

1+η HE L

(7.11)

Then P HE L = K 1 ∗η HE L +K 2 ∗η2 HE L +K 3 ∗η3 HE L and σ HE L = 32( HEL−P HE L)The fracture strain here is computed as,

ε f pl = D1

P

P HE L+

T

P HE L

D2(0≤ D ≤ 1.0) (7.12)

And similarly like gradual strength reduction, bulking pressure is added incrementally

as,

∆Pt +∆t = −K 1ηt +∆t + (K 1ηt +∆t +∆Pt )2 +2β f K 1(U | D(t )−U | D(t +∆t )) (7.13)

7.3.2.5 JHB model

(a) Strength without Strain Rate Effects (b) Fracture strain

Fig. 7.6 JHB model (Johnson et al., 2003)

In this model the intact and failure strengths are computed through the following

smooth function of pressure P,

σ eq,i =

σ i

P+T Pi−T ,−T ≤ P ≥ Pi

σ i +(σ i,max−σ i)

1− e− (P−Pi)σ i

(Pi+T )(σ i,max−σ i)

,P ≥ Pi(7.14)

145


10/20

Ceramic Target

and,

σ eq, f =

Pσ f P f

,0≤ P ≥ P f

σ f + (σ f ,max−σ f )

1− e− (P−P f )σ f

P f (σ f ,max−σ f )

,P ≥ P f

(7.15)

The sudden application of bulking pressure is quantified similar to the procedure adopted

in JH-1 version (equation 7.8), and the fracture strain is estimated as,

ε f pl = D1

P

σ i,max+

T

σ i,max

n≤ ε f max (7.16)

Now the additional features include incorporation of phase change and hysteresis

during unloading. At the second phase after threshold compressibility η2, the pressure

is computed with modified coefficients of EOS as,

P = K̄ 1(η−η0)+ K̄ 2(η−η0)2 + K̄ 3(η−η0)3 (7.17)

whereas, a linear transition between Phase 1 and 2 is considered. If bulking is coupled

with phase change, the increment in pressure is estimated through interpolation as,

∆P =−K e f f 1 ηe f f f + (K e f f 1 η

e f f f )

2 +2β f K e f f 1 ∆U (7.18)

where, K e f f 1 = K 1(1−λ )+ K̄ 1λ , and η e f f f = η f −λη0 with λ = ηmax−η1η2−η1 . The effects

of phase change and that of hysteresis during unloading is shown in the figure 7.7.

(a) Phase Change (b) Hysteresis during unloading

Fig. 7.7 Pressure curve in JHB model (Johnson et al., 2003)

For the purpose of investigating the intended behaviour, Johnson et al. (2003) demon-

strated the behaviour of a ceramic block under uni-axial compression and then release

as shown in figure 7.8. With few representative parameters, it was shown that after ini-

tial elastic build-up (segment 1-2), the ceramic undergoes yielding with intact strength

146


11/20


(segment 2-4). After sufficient accumulation of plastic strain and damage ( D = 1), the

strength envelope is shifted suddenly to fractured strength (segment 5-7). Now the un-

loading starts and the material unload elastically (segment 7-8 with parallel slope of

segment 1-2). At point 8, the effective deviatoric stress passes through zero during

transition from compression to tension with pressure remaining compressive still - rep-

resentating a pure hydro-static state of stress. After this, the tensile elastic unloading

of deviatoric stress happens until it reaches the failure envelope again (segment 8-9).

The strength envelope at this condition is dominated by the fractured state because of

earlier damage due to compression (segment 9-1) until it reaches the initial stress free

condition.

Fig. 7.8 Loading and unloading under uni-axial compression (Johnson et al., 2003)

7.3.3 Model selection

The ceramic behaviour is primarily dominated by pressure and strain-rate dependent be-

haviour under compression while being comparatively weak in tension (tensile pressure

capacity capped at −T ≪ compressive capacity). Hence most damage models were de-veloped with primary focus on this characteristics. But model parameter identification

being sensitive to reference test configurations, only uni-axial test scenario as in flyer

plate test or shock wave testing is not always preferable. Lankford (1977) tried the iden-

tification via multi-axial compression in split Hopkinson pressure bar (SHPB) test but

with expensive features.

Now among the mostly used models with larger share of test data available in lit-

erature, Johnson-Holmquist models (JH-1, JH-2 and JHB) hold potential of a realistic

representation. Among them the JH-2 version employs gradual strength reduction with

increments in damage and hence is not chosen to represent sudden brittle behaviour

in the test cases to follow (flyer plate spall test, distal boundary effect in deep target

and conoid formation when backed by metallic plate). JHB version uses an improved

147


12/20

Ceramic Target

analytic funtion in computing strength envelope compared to piecewise linear fits in

JH-1. But JHB model also requires additional material parameters due to considera-

tions of phase change and hysteresis effects during unloading. As mentioned earlier,

identification of these parameters requires special attention and even may require tuning

to represent the actual real life behaviour. Avoiding those complexities, here the JH-1

version is chosen to represent the particular test cases considered.

The material considered in all test cases is Silicon Carbide (SiC with reference

density ρ0 = 3215kg/m3, elastic modulus E = 449GPa, Poisson’s ratio ν = 0.16 and

HEL= 11.6GPa), parameters for which are taken from (Holmquist and Johnson, 2002)

and reproduced in table 7.1

Table 7.1 JH-1 model parameters for silicon carbide (SiC) (Holmquist and Johnson,

2002)

Equation of State Intact strength Failed Strength Tensile Strength Damage

K 1=220 GPa P1=2.5 GPa c f =0.40 T=0.75 GPa Pmax=99.75 GPa

K 2=361 GPa σ i,1=7.1 GPa σ f ,max=1.3 GPa ε f ,max=1.2

K 3=0 P2=10.0 Gpa

β f =1.0 σ i,2=12.2 GPa

7.4 Validation via flyer plate test

Spall failure owing to excessive tensile stress is an important phenomena to study Hugo-

niot elastic limit (HEL) and spall fracture criterion in ceramics. When one disc is im-

pacted upon another target disc, a compressive stress front starts propagating in both

materials after emanating from the contact surface. Now generally, ceramics are strong

in compression compared to its tensile behaviour. If the stress wave magnitude (of the

order of ρC l v p, with material density ρ , medium’s acoustic speed C l and particle veloc-

ity v p) does not exceed the compressive strength, the material won’t experience fracture

though partial loss of strength may happen due to accumulated plastic strength (when

ρC l v p > H EL).

Following the experimental design of flyer-plate test impedance matched (same ma-

terial) plates with target thickness (2w) approximately double of the hitting plate (w),

the compressive fronts will travels towards free surfaces opposite to the impact faces.

They will reflect from the free surface as the rarefaction release waves. The hitting

plate will be completely traversed by the reflected tensile wave in (w+w)/C l time just

when the compressive front reaches the free surface of the target plate. Now the tensile

front from the hitting plate will transmit in the target plate at the contact surface because

of impedance match, whereas the compressive front will reflect and starts propagating

towards the contact surface. So, these two release waves now comes at each other grad-

ually releasing the compressed zone traversed by them in the target plate. If the radial

148


13/20


boundary is sufficiently away from the middle of contact plane (keeping diameter to

thickness ratio sufficiently large), the reflection from radial boundary will not disturb

(see figure 7.9) this uni-axial propagation significantly.

Fig. 7.9 Wave propagation in flyer and plate due to the impact (Hiermaier, 2007)

Now, the two release front will meet at the middle of the target plate resulting in

amplification. Depending on the material and initial impulse this amplified release may

exceed the tensile strength. For most ceramics, the tensile limit is very less (⋍ 1GPa).

Hence even at velocity 150-200 m/s, the ceramic target plate will demonstrate a dis-

tinct spall plane perpendicular to the wave motion. The remaining shock and rarefaction

waves are reflected from that new spall plane and subsequently travel back and forth

within the spalled out material (see figure 7.10, as zoomed). This type of failure is com-

mon in brittle materials with small tensile to compressive strength ratio and generally

under dynamic loading like impulse on rocks.

Fig. 7.10 Deformed configuration with spall plane in target

If measurements are taken for free surface velocity, it will show sharp jump just

149


14/20

Ceramic Target

when the first compressive front reaches the free surface. The signal will then remain at

that amplitude (vmax) when the first release wave travels towards spall zone. During the

spallation, there will be characteristic drop (∆vsp ) in the signal (see figure 7.11) until it

rises again as the reflected wave from spall plane reaches this free surface. From this

velocity pullback ∆vsp , the spall stress is computed as 0.5ρ0C l∆vsp . Though, to correct

for pre-deformation during pre-spall wave propagation corrections was suggested by

Stepanov (1976).

0 0.5 1 1.5 2 2.5 3 3.50

50

100

150

Time (µs)

P a r t i

c l e V e l . ( m / s )

Dandekar and Bartowski (ARL-TR-2430)Holmquist and Johnson (JAP ’02)

Quan et al (IJIE ’06)

Simulation Output

Fig. 7.11 Free surface velocity measured for SiC with a flyer velocity of 148 m/s

The obtained spall signal (with average inter-particle spacing of 0.1955 mm) is com-

pared with that of experiment by Dandekar and Bartkowski (2001), prescribed profile

by Holmquist and Johnson (2002) and the numerical counterpart via AUTODYN by

Quan et al. (2006) in figure 7.11. The pseudo-spring SPH version successfully produces

the conformal characteristic drop (∆vs p) by adaptation of piecewise linear JH-1 model.

A point is to note that for the mollifying effect of the SPH parameter ‘h’ (smoothing

length) and for the used artificial viscosity, the jump in the signal is bit smoothened over

a finite small time-scale as compared to that of experimental evidence.

Apart from that, the SPH framework as a collocation method uses information from

a finite sub-domain bounded by the kernel support (here ‘2h’) and hence is bound to

yield a non-local effect. But brittle materials like ceramic is experimentally evidenced

to demonstrate very small fracture processing zone. A possible way-out is reducing the

h/∆ p ratio (where ∆ p: average particle spacing) within the acceptable range of robust

computation.

This type of fine tuning was required in this particular simulation despite reduced

sensitivity of h when using gradient correction (see figure 3.5). But this tuning does not

opposes the claim of pseudo-spring version, that material strength/damage behaviour

would solely be represented by damage algorithm at pseudo-spring level, irrespective of

150


15/20


basic SPH framework. To prove that, with same parameters (∆ p, h, artificial viscosity

parameters, ∆t ), the simulation was conducted with and without the particular JH-1

based damage algorithm at pseudo-springs.

The case without damage algorithm does not produces any particle-level instability

as the target plate just moves away from the impactor without any spall plane being

developed. Moreover the velocity signals (see figure 7.12) at impact face and at free

surface of the target show similar zig-zag patterns with same order of amplitude indi-

cating no spalling. Although the obvious phase lag due to time required for the first

compression wave to reach respective points of measurements are present. Besides the

average amplitude of those signals do not reach that obtained via JH-1 based pseudo-

spring framework.

Fig. 7.12 Velocity signal with and without pseudo-spring level damage

These observations imply that the classical SPH without any fracture algorithm can

capture the propagation of waves but can not simulate the failure behaviour even with

adaptive resolution. This proves once again the necessity of using explicit damage in

material-independent kernel based particle simulation. Although the tuning related to h

is believed to be replaceable with adjustment of the failure parameter as was identified

by Quan et al. (2006).

The minute difference between experimental and model output signal (figure 7.11)

is perceived to be due to difference of real-life test scenario and ideal surrounding of nu-

merical test. Free-free condition is hard to achieve completely in real-life experiments

but easily modelled in numerical test environment. Air pressure resistance in front of

the moving plate may be another factor contributing. Further the attenuation of the spall

signal may also happen after ceramic micro-structure changes as the stress wave passes

through it. But this air-resistance or micro-structural changes are not included in the

present simulation regime. When incorporating the macro-level ceramic material model

(Johnson-Holmquist model is taken here), the definition of phase changes and consid-

eration of hysteresis (as in Johnson-Holmquist-Biessel Model) may improve this close

151


16/20

Ceramic Target

agreement. However that is subjected to availability of suitable material parameters for

SiC and otherwise more physically-based material model for ceramic.

7.5 Distal boundary effect

Whereas the last validation test case is related to uni-axial loading, multi-axial load-

ing coupled with shear and dynamic fracturing via lateral and radial cracking is a far

more complex phenomena. Ceramics behave differently under tensile and compressive

loading subjected to kinetic energy projectile impact. Tensile spalling remains one of

the dominant mode in case of particular loading rate. Though under certain condition,

ceramic may even exhibit sudden and brittle failure mode in compression. Also, a tran-

sition may occur between brittle-to-ductile mode depending on the induces shear and

confinement effects as the shear capacity also increases with confinement. The precise

effects can be investigated with tri-axial compression test. Although, extensive uni-axial

experiments like bar and plate impact were done to understand high strain-rate behaviour

(Hiermaier, 2007), multi-axial test results are comparatively scarce.

(a) @ 5 µ s (b) @ 8 µ s

(c) @ 10 µ s (d) @ 20 µ s

Fig. 7.13 Discrete crack path in deep ceramic target

In general, the ceramics exhibit hydrostatic pressure dependent behaviour - prevent-

ing the inter-grain boundary slip. Here impact of deformable steel projectile (deforms

as per J-C model) on deep ceramic target is investigated to check discrete crack orien-

tation. During initial impulse transfer, high-intensity pressure and shear forms series

152


17/20

7.6 Conoid formation in multi-layer target

of dynamic cracking events - ring cracks, radial cracks, lateral cracks - either simulta-

neously or sequentially. Thus the ceramic becomes extensively cracked, but still inter-

locked due to confinement via un-deformed portion away from the impact zone. This

state is commonly known as comminuted state or a Mescall zone.

Now if for sufficient time, the ceramic domain can withstand the impulse based on its

higher resistance due to greater thickness, the compressive wave may reach the opposite

free surface and reflected tensile waves starts its motion towards impact surface. At this

point portions of the rear surface may spall off to form an impact crater in the deep target.

Figure 7.13 shows snapshots at different time-instants depicting this distal boundary

effects. This simulation (with average inter-particle spacing of 0.25 mm) shows the

potential of the current framework in investigating coupling mechanisms of dynamic

fracturing in details.

7.6 Conoid formation in multi-layer target

Advanced add-on armours are produced by a clever combination of ceramic tiles backed

by metal or composite plates. In general, the ceramics exhibit hydrostatic pressure de-

pendent behaviour - preventing the inter-grain boundary slip. Here impact of deformable

steel projectile (deforms as per J-C model) on bi-layer ceramic-metal system (metal back

plate exerts influence on whole deformation process) is investigated to demonstrate for-

mation of ceramic conoid.

After impact, generated compressive stress pulse travels along the thickness of the

target as well as the axial direction of projectile. Within the target assembly, the wave

upon reaching the ceramic-metal interface partially get transmitted in backing plate and

partially get reflected in the facing ceramic plate. For sufficiently localised impulse, high

shear creates a punching tendency. Heavy projectiles may cause out-of-plane bending

in the plate assembly. But this bending is obstructed by the backing plate and in most

cases, the tensile cracks starts interacting within the ceramic in dynamic fracturing be-

fore sufficient inertia transfer to cause differential fibre tension between front and rear

surfaces. Thus a cone configuration (see phase 1 in figure 7.14) is formed which then

acts as a part of loading material.

For most typical ceramics (here SiC), mechanical impedance is higher than that of

metals (here aluminium). So the reflected wave becomes tensile and initiates failure

within the ceramic - generally with lower tensile capacity than compressive stress car-

rying capacity. And the ceramic conoid gets further fragmented (see phase 2 in figure

7.14) when the overall bending is still resisted by the backing plate. The pulveriza-

tion/crushing of the ceramic conoid then allows the projectile to move forward and cre-

ating bulging in the backing plate. From simulation aspect identification of conoid and

ceramic fragments as a conglomerate of undamaged particle domain bounded by a ve-

153


18/20

Ceramic Target

(a) Phase 1

(b) Phase 2

Fig. 7.14 Penetration phase (Zaera and Sánchez-Gálvez, 1998)

neer of damaged particles is critical in investigating these two phases differently. Figure

7.15-7.17 demonstrates these simulated phases (with average inter-particle spacing of

0.333 mm at different time-instant with thick and thin backing aluminium. The inter-

layer contact was modelled here with springs only active in tension and strength guided

by that of weaker material.

D

1

0.8

0.6

0.4

0.2

0

D

1

0.8

0.6

0.4

0.2

0

Fig. 7.15 Conoid formation @ 2 µ s with thick and thin backing

D

1

0.8

0.6

0.4

0.2

0

D

1

0.8

0.6

0.4

0.2

0

Fig. 7.16 Crushing of conoid @ 12 µ s with thick and thin Backing

Now, the projectile deformation takes place until the damaged zone reaches the

front face after initiating from bottom of the conoid. This deformation stage during

non-penetration into the target is called ‘dwell’ (Chen et al., 2007). Short projectile im-

pacting on high-resistant target may completely deformed plastically or shattered during

154


19/20

7.7 Closure

D

1

0.8

0.6

0.4

0.2

0

D

1

0.8

0.6

0.4

0.2

0

Fig. 7.17 Bulging at back plate @ 22 µ s with thick and thin Backing

long damage development time within the ceramic. Without any penetration, this phe-

nomenon is called ‘interface defeat’. Shockey et al. (1990) found that confined ceramics

were much more efficient in defeating the penetrator this way redistributing the impact

load to a larger area on the surface of the backing plate. Also flow and abrasive prop-

erties of the finely fragmented material govern the penetration resistance of confined

ceramics.

And the thickness of backing plate prevents overall bending by providing higher

resistance with higher thickness. The time before significant bulging in the thicker back

plate also influences the conoid angle. Simultaneously, by dissipating larger share of

imparted energy in plastic deformation (figure 7.18a), the thicker back plate improves

the ballistic efficiency of the overall target assembly. The lower residual kinetic energy

in case of thicker backing (figure 7.18b) provides evidence of of that improved efficiency

in this particular configuration

Time (sec)

P W

( J )

0 5 E-0 6 1 E- 05 1 .5 E-0 5 2 E-0 5 2 .5 E-0 5 3 E- 05 3 .5 E-0 50

50000

100000

150000

200000

250000

300000

350000

400000

450000

Thick Backing

Thin Backing

(a) Plastic work done comparison

Time (sec)

K E ( J )

0 5 E-0 6 1 E- 05 1 .5 E- 05 2 E-0 5 2 .5 E-0 5 3 E- 05 3 .5 E- 05

250000

300000

350000

400000

Thick Backing

Thin Backing

(b) Kinetic energy comparison

Fig. 7.18 Effect of backing plate thickness

7.7 Closure

The localised fracture process zone and subsequent crack interaction in confined ceramic

is investigated with the help of pseudo-spring SPH simulation framework after adapting

the constitutive models by Johnson and Holmquist (1992) (JH-1). The damage model

155


20/20

Ceramic Target

uses pressure dependent intact and failure envelope. The failure strength only to be

followed after complete damage ( D = 1) accumulation with increments in plastic stain.

Strain rate effect and bulking after complete damage are also fused in the model.

The spall signal from a flyer plate test on SiC disc is investigated to validate the

implementation aspects of the simulation framework. The smoothening effect of SPH

kernel is found to be introducing a non-local effect in the output signal. Hence the

smoothing length (h) is fine tuned for better performance. But the explicit damage al-

gorithm through pseudo-spring analogy is indispensable as that tuned kernel support is

found to be incapable of producing a conformal signal with earlier experimental and

numerical evidences from literature. Moreover the spall plane remained dormant. An-

other perspective proved that the tuning does not polluted the simulation with un-desired

numerical instabilities.

After successful reproduction of stress wave propagation in spall test scenario, the

distal boundary efffect in a deep ceramic target impacted upon by deformable cylindrical

projectile is investigated. Multi-axial interaction among discrete cracks and eventual

spalling from opposite face with delayed energy dissipation time in an deep target is

simulated with reasonable robustness.

Finally the conoid formation and fragment identification in a ceramic plate backed

by ductile metal plate is successfully simulated. The influence of relative thickness is

qualitatively studied to indicate influence on conoid angle deviation and energy dis-

sipation characteristics of the target assembly and the co-existing deformation in the

deformable steel projectile. This whole strategy has potential to create a virtual exper-

imental paradigm for efficient designing of target system harnessing the advantages of

both brittle and ductile materials by a parametric variation of design parameters and

converging towards the desired performance response.

The inter-layer contact was modelled with springs which are active only in tension

and the strength is guided by that of weaker material. More sophisticated interface defi-

nition and use of adhesive layers poses a scope of improvement in the current framework

other than the obvious requirement of accurate constitutive parameter characterization.

Optimum adhesive layer thickness for the best performance as was demonstrated by

López-Puente et al. (2005) to show various amount of spalling with different adhesive

layer thickness is to be investigated in near future.

Impact on Ceramic Target

Documents

Transcript of Impact on Ceramic Target