Computational Modelling Of High Impact Loading

Published Date: 02 Nov 2017

and Fragmentation of Concrete

Robert Da_ern

School of Engineering

Cardi_ University

Da_ernR@cardi_.ac.uk

April 10, 2013

Contents

1 Introduction 3

1.1 Importance of Studying High Impact Loading . . . . . . . . . . . . . . . 3

1.2 Aim of this Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 Response of Concrete to Impact Loading 5

2.1 Macroscopic Nature of Concrete . . . . . . . . . . . . . . . . . . . . . . . 5

2.1.1 General Behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.1.2 Response to High Velocity Impact . . . . . . . . . . . . . . . . . . 6

2.2 Dynamic Behaviour of Concrete . . . . . . . . . . . . . . . . . . . . . . . 8

2.2.1 Porosity of Concrete . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.2.2 Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2.3 Strain Rate Sensitivity . . . . . . . . . . . . . . . . . . . . . . . . 10

3 Material Modelling 12

3.1 Equation of State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3.1.1 Mie Gruneisen Form of Equation of State . . . . . . . . . . . . . . 13

3.1.2 Polynomial Equation of State . . . . . . . . . . . . . . . . . . . . 15

3.1.3 P-_ EOS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.2 Strength and Failure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.2.1 Drucker-Prager . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.2.2 RHT Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

4 Previous Work in the Literature 23

4.1 Tham (2004) - Inuence of Constitutive Model . . . . . . . . . . . . . . . 24

4.2 Clegg et al. (2002) - Implementation of a Tensile Crack-Softening Damage

Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

4.3 Beppu et al. (2008) - Damage evaluation of concrete plates by high-velocity

impact . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

5 Validation of Material Model 31

5.1 Experimental Con_guration . . . . . . . . . . . . . . . . . . . . . . . . . 32

5.2 Numerical Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

5.2.1 Model Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

5.2.2 Material Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . 35

5.3 Comparison of Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

6 Model Parameter Study 42

6.1 Projectile Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

6.2 Depth of Concrete . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

1

6.3 Mesh Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

6.4 Projectile Size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

6.5 Projectile Shape . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

6.6 Damage Process Examination . . . . . . . . . . . . . . . . . . . . . . . . 46

6.7 Material Model Parameter Study . . . . . . . . . . . . . . . . . . . . . . 48

7 Conclusions 49

8 References 50

2

Chapter 1

Introduction

1.1 Importance of Studying High Impact Loading

The ability to predict the behaviour of structures under high velocity conditions is of

critical importance for a range of applications. The rate of loading is far greater than that

which is usually considered and with it comes new phenomena and material behaviour.

For this reason, a better understanding of the intricate material properties is required

in order to successfully design against the loading. Being able to predict the damage

and failure of structures which are prone to impact loads is of major importance to

many industries. However, it goes further than this; in an age where bomb threats and

explosions are rife, The Ministry of Defence, for example, has obvious interest in the

study of impact loading in order to be able to mitigate threats arising from bombs and

other such high velocity impacts.

Although it is possible to determine the outcome of certain situations in this _eld

experimentally, it is both very di_cult and extremely expensive. Instead, it is more viable

to use simulation software. The modelling of impacts at velocities of this magnitude

requires specialist software and comprehensive theoretical models in order to accurately

predict the behaviour of the materials. For purposes of this paper, a general purpose hydrocode,

AUTODYN (Version 14.0), will be used. AUTODYN is a _nite element analysis

software which specialises in modelling high velocity, short time duration analyses.

3

1.2 Aim of this Thesis

The purpose of this study is to analyse the response of concrete plates to speci_c loading

conditions. The thesis aims _rstly to summarise the previous work surrounding the

area of high impact loading. The study will then explore the theory surrounding the

subject area and an introduction will be made to some relevant material models and the

most appropriate will be chosen. The dynamic behaviour of concrete is complex and its

response is largely dependent on factors such as strain rate hardening, pressure hardening

and strain hardening. In order to accurately predict the response of concrete to impact

loading, these properties of concrete must be implemented in the constitutive model. It

would be impossible to expect a material model to cover all responses to possible loading

conditions for all types of material. It is therefore imperative that an appropriate model

is selected for the problem at hand. The parts of the constitutive model which model

the speci_c material properties will be examined in detail. Validation of the constitutive

model used will be provided through comparisons with experimental data provided by

Hansson (2005). The areas of interest relating to validation are the penetration depth

and damage properties which include spalling on the front surface, scabbing on the back

surface and crack propagation patterns.

Having validated the employed constitutive model, various studies will be conducted

to determine the response of concrete to changing conditions. Parametric studies will

be performed on the given model setup to give insight into major inuences and speci_c

model sensitivities. Constitutive model dependencies and the resulting behaviour will

then be analysed and discussed with reference to the underlying physical properties.

4

Chapter 2

Response of Concrete to Impact

Loading

2.1 Macroscopic Nature of Concrete

2.1.1 General Behaviour

It is widely understood that concrete demonstrates a greater strength in compression

than it does in tension. This fact is used to advantage in columns and foundations in

buildings, as well as in structural defence. In quasi-static problems, which deal with very

low rates of loading, damage occurs in a global manner - i.e. the concrete su_ers exural

deformation, which, unlike local damage, may cause structural failure. This structural

failure in plain concrete will almost always occur in tension due to its reduced strength

when compared to that of in compression. This property of concrete is adjusted for by

use of reinforcements. These reinforcements are usually amde of steel which exhibits a

much higher strength in tension than concrete and fails plasticly - as opposed to concrete

which is a brittle material. The use of the reinforcements means that the concrete will

not fail suddenly when the tensile limit is reached - but instead the steel will fail in a

plastic fashion, causing cracking in the concrete before ultimate failure. Although this

5

fact is largely associated with that of static problems, reinforcements are used in almost

all concrete structures and thus an understanding of its implications is essential. This

study focuses on the impact of plain concrete, but many studies (Nystrom and Gylltoft

2011; Tham 2004) deal with impact loading on reinforced concrete. An extension to this

study would include the parameter e_ects and model geometry sensitivities applicable to

reinforced concrete.

2.1.2 Response to High Velocity Impact

The behaviour which concrete demonstrates under impact loading conditions is, however,

di_erent in many respects to that of quasi-static loading. The fact remains that the

behaviour of the concrete is largely dependent on the strength di_erence in compression

and in tension. The processes which cause the compression and tension in the concrete,

however, are very di_erent. Various dependencies and material properties dictate the

behaviour which concrete demonstrates under these conditions and are discussed below.

Failure Process

When a concrete plate is subjected to an impact from an object at high velocity, a

compressive stress wave is produced and expands through the thickness of the plate.

Once it reaches the back side of the plate, the wave becomes a tension wave and travels

back through the plate. Due to the lower strength in tension than in compression, this

tension wave causes cracks to form in the concrete which propagate through the material.

High velocity impacts cause localised damage as opposed to global damage. This local

damage is most often split into three modes; cratering, spalling and perforation - a

graphical representation of which can be seen in Figures 2.1 and 2.2. Throughout this

study, reference will be made to these modes of failure and the model parameters which

a_ect them. As well as these modes of failure, a major consideration will be a_orded

to the crack propagation patterns in concrete when subjected to high velocity impacts.

6

Tensile cracks occur in the material when the tensile stress exceeds that of the tensile

strength in the concrete. These cracks then expand through the material due to crack

softening. It has been shown (O^zbolt et al. 2011) that as the crack velocity reaches the

critical speed of propagation, crack branching starts to occur. A successful model must

be able to accurately replicate these failure modes with reference to experimental results

in order that future studies using the models can be appropriately relied upon.

Figure 2.1: Failure modes of local damage: (a) cratering;(b) spalling; and (c) perforation, _gure

reproduced from Beppu (2008)

Figure 2.2: Figure 2.3: Local impact e_ects on concrete target: (a) Penetration; (b) Cone

cracking; (c) Spalling; (d) Radial cracks on (i) impacted face and (ii) bottom face; (e) Scabbing;

(f) Perforation; (g) General impact response, _gure reproduced Li et al. (2005)

7

2.2 Dynamic Behaviour of Concrete

In order to successfully model the behaviour of concrete under impact loading rates, all

aspects which contribute to the process of damage and failure have to be understood in

detail. In reality, concrete is a non-homogenous material, constituting particles of di_ering

sizes and intermolecular forces. The micro-scale modelling of such a non-homogenous

material would be extremely costly and time consuming to implement. Therefore, for the

purposes of numerical modelling, concrete is assumed to be a homogenous material and

the behaviour being addressed is that of a macroscopic scale. However, the porous nature

of concrete is important and cannot be omitted and is discussed below. The numerical

simulation requires the formation of appropriate material constitutive relationships which

must be able to represent the macroscopic material properties which result from complex

micro-mechanical concrete behaviour. The behaviour which needs to be captured by the

material model includes phenomena such as pressure hardening, strain hardening, strain

rate hardening, and fracture damage.

2.2.1 Porosity of Concrete

Porous materials such as concrete are extremely e_ective in absorbing shocks and mitigating

impact pressures. The material compacts to its solid density (absent of pores) at

relatively low stress levels but, because the volume change is large, a large amount of energy

is absorbed by the material. This absorption causes a lengthening of the shockwave

in time and a reduction of its magnitude, thereby mitigating the impact pressure. The

energy lost by the incident shock is absorbed by the porous material during compaction

and this can lead to a temperature increase in the porous material. Adequate modelling

of the compaction of the material must take into account this energy rise to ensure accurate

solutions. The issue of modelling of this porous nature of concrete is discussed in

the next chapter.

8

2.2.2 Governing Equations

As many authors have demonstrated (Thorma et al. 1999; Riedel et al. 1999; Livingstone

et al. 2000), the non-linear response of materials to shock loading can be described

by solving the governing equations of the conservation of mass, momentum and energy

(Eqs 2.5) together with an additional relationship between the pressure, volume and

energy. This set of equations describe purely the hydrodynamic portions of the stress and

strains (all three normal stresses of the stress tensor are equal) and omit any resistance to

shear - that is, a description which only holds true for liquids and gasses. An extension to

solid behaviour must include the deviatoric strength. This will be explained in detail in

Chapter 3. In order to represent the material behaviour accurately, the material response

to both volumetric and deviatoric strains must be determined. The governing equations

for both bodies are given by the following:

Equation of Mass Conservation:

_V = _0 (2.1)

where V represents the relative volume, q denotes the current density, and q0 denotes the

reference density.

Equation of Momentum Conservation:

_ij;j + _fi = __i (2.2)

where _ij represents the Cauchy stress; fi represents the body force density, and _i

denotes the acceleration

Equation of Energy Conservation:

_E

= V Sij _ _ij ô€€€ (p + q)_ij (2.3)

where Sij denotes the deviatoric stresses and p denotes the hydrostatic pressures, as given

in

Sij = _ij + (p + q)_f ij (2.4)

9

where q denotes the bulk viscosity, _ij denotes the kronecker delta (_ij = 1, if i = j;

otherwise _ij = 0).

and when applied to a lagrangian referential (moving with the subgrid), the partial

di_erential equations can be derived as:

D_

Dt

+ _

@vi

@xi

= 0 (2.5a)

_

Dvi

Dt

= ô€€€

@p

@xi

+

@Sji

@xj

+ _fi (2.5b)

_

De

Dt

= _q_ +

@

@xi

_

k

@T

@xi

_

ô€€€ p

@vi

@xi

+ Sji

@vi

@xi

(2.5c)

These governing equations can then be solved together with the equation of state

to provide a complete solution for uids which demonstrate no shearing resistance, i.e.

Sij = 0 .

The volumetric stresses are calculated through an equation of state (EOS) and the

deviatoric stresses are calculated using a strength model. As well as these, a failure model

must also be included which describes the onset of damage in the material. These three

main areas are detailed in the following chapter. For the purposes of this study, two constitutive

models to describe the strength will be explored in detail. The Drucker-Prager

model and the RHT model (Riedel, Hiermaier and Thorma 1999) are both applicable to

the modelling of brittle materials at high loading rates. Justi_cations to aspects of the

material models are explained where applicable and are validated through the experimental

comparisons detailed in Chapter 5.

2.2.3 Strain Rate Sensitivity

Impact loading causes straining in a material to occur in a very short space of time - that

is, it causes a high strain rate in the material. Most materials, including concrete, are

10

strain rate dependent, meaning that an increase in strength is experienced at high strain

rates. In fact, concrete is a material which is very strain rate dependent and experiences

a large increase in stress at relatively low rates of strain. Furthermore, according to

Bischo_ (1991) and Ross et al. (1996), the ultimate uni-axial compressive strength can

almost double and the ultimate uni-axial tensile strength can increase by a factor of 5-7

for high rates of strain. For this reason, it is imperative that a constitutive model of

concrete includes this property of concrete in order to obtain an accurate representation

of the concrete behaviour under this type of dynamic loading. Typical strain rates for

di_erent loading scenarios is shown schematically in Figure 2.3. As can be seen from

the _gure, strain rates in the order of 100=s - 103=s are most applicable to this study.

The implementation of this material property is discussed in further detail in the next

chapter.

Figure 2.3: Strain Rates of Di_erent Loading Scenarios, _gure reproduced from Bischo_ and

Perry (1991)

11

Chapter 3

Material Modelling

In order to successfully model engineering situations, a comprehensive understanding of

the physics which determines the response of materials to loading is required. There is

obviously a wide range of types of materials and the behaviour which they demonstrate,

whether it be brittle, ductile, week, strong, hard, soft or a combination of them. These

types of materials may also behave di_erently to di_erent loading conditions and therefore

it is essential that a material model describes the behaviour accurately. It would be

impossible for one material model to cover the behaviour of all types of materials to all

loading conditions. It is therefore necessary to determine the properties of the material in

question and develop a material model which is applicable to the given scenario. Below

are detailed explanations of the di_erent aspects of material models relevant to the topic

of brittle materials subject to impact loading. These models are available in AUTODYN

Version 14.0 and references will be made to the implementation of the models where

appropriate.

3.1 Equation of State

As discussed in the previous section, having determined the partial di_erential equations

governing unsteady material dynamic motion to express the local conservation of mass,

12

momentum and energy, it is then necessary to introduce a further relationship between

ow variables. A material model is used to relate stress to deformation and internal

energy (or temperature). The stress tensor can be split into a hydrostatic pressure (all

normal stresses being equal) and a stress deviatoric tensor adrressing the resistance of

the material to shear deformation. The relationship between the hydrostatic pressure,

the local density (speci_c volume) and the local speci_c energy (temperature) is known

as the equation of state. It can be said, however, that \typically, the inuence of internal

temperature uctuations are disregarded in material models for concrete."(Tai and Tang

2006).

For the purposes of this study, the equation of state employed will be built up from

a simple Hooke's Law form into an equation of state which is applicable for the scenario

in question.

It has been determined that the equation of state relates the pressure, p to the volume,

V and the speci_c energy, i.e. p = f(V; e). However, reiterating the exclusion of the

inuence of changes in entropy, the pressure can be assumed to be entirely dependent

on the density (or speci_c volume). Alternatively, an initial elastic behaviour can be

expressed by an approximation to Hooke's Law which can be written as:

p = K_ (3.1)

where _ = (_/_0)ô€€€1, and K is the material bulk modulus. This form of the equation of

state is only appropriate for modelling fairly small compressions.

3.1.1 Mie Gruneisen Form of Equation of State

An improved form of the equation of state is the Mie-Gruneisen equation of state and is a

special form of the Gruneisen model which describes the e_ect on vibrational properties

of a material due to a changing volume (Riedel et al. 2009). The Mie-Gruneisen form of

equation of state is described below:

13

If the pressure, in terms of energy and volume is expressed as

p = f(e; v) (3.2)

then a change in pressure, dp, can be written as

dp =

_

@p

@v

_

e

dv +

_

@p

@e

_

v

de (3.3)

Integrating Eq (3.3) allows the pressure to be expressed in terms of the volume, v and

energy, e relative to the pressure at a reference volume, v0 and reference energy, e0.

Za

p0

dp =

Zv;e

v0;e0

_

@p

@v

_

e

dv +

Ze;v

e0;v0

_

@p

@e

_

v

de (3.4)

Integrating Eq (3.4) _rstly at constant energy from v0 to v, and then at constant volume

from e0 to e, gives

p = p0 +

Zv;e0

v0;e0

_

@p

@v

_

e

dv +

Ze;v

e0;v

_

@p

@e

_

v

de (3.5)

The Gruneisen Gamma, ô€€€ is de_ned as

ô€€€ = v

_

@p

@e

_

v

(3.6)

and, as before, if it is assumed that ô€€€ is a function of volume (or density) only, then the

second integral of Eq (3.5) can be evaluated as

Z _

@p

@e

_

v

de =

ô€€€(v)

v

[e ô€€€ e0] (3.7)

The _rst integral in Eq (3.5) is a function only of volume and the reference energy, e0.

Therefore, the Mie Gruneisen form of the equation of state can be expressed by the

equation

p = pr(v) +

ô€€€(v)

v

[e ô€€€ er(v)] (3.8)

since

Zv;e0

v0;e0

_

@p

@v

_

e

dv = pr(v) ô€€€ p0 (3.9)

The functions pr(v) and er(v) are assumed to be known functions of v on some reference

curve of the phase diagram, e.g. the shock Hugoniot, as shown in Figure 3.1. The shock

Hugoniot is described in more detail below.

14

It is widely understood that when a material is subjected to impact, deformation or

energy input, it will su_er changes in its thermodynamic state throughout its volume. It

may at any time have regions that are solid, liquid or gaseous. An ideal equation of state

would cover all regions in the phase plane, but this is obviously an unrealistic expectation.

Instead, di_erent equations of state are used to focus on speci_c areas of the phase plane.

For the purposes of this report, the area around states achievable by shocking a material

from its initial state is of most interest. The area in the pressure volume plane of all

states achievable by shocking the material from an initial state (p0; v0) is known as the

shock Hugoniot. It is usually adequate to assume that the material equation of state

can be _t directly on this shock Hugoniot for the duration of an impact. However, an

improvement to this would be to take into account the release adiabat and the di_erence

in volume experienced when a material is allowed to expand after an initial compression.

It is recognised (Zukas 1990) that the release adiabat is not very well known for brittle

materials such as concrete. As of yet, there is no implementation in the AUTODYN

software but an extension to this study would be to implement this extension to the

reference curve via user subroutines.

3.1.2 Polynomial Equation of State

The polynomial equation of state is a general form of the Mie Gruneisen equation of

state discussed above and it has di_erent analytical forms for states of compression and

tension. The model is implemented into the AUTODYN software with various input

parameters. The EOS de_nes the pressure

for _ > 0 (compression) as:

p = A1_ + A2_2 + A3_3 + (B0 + B1_) _0e (3.10)

and for _ < 0 (tension) as:

p = T1_ + T2_2 + B0_0e (3.11)

15

Figure 3.1: Phase Diagram, Hugoniot and Adiabats, _gure reproduced from AUTODYN Theory

Manual (1997)

where _ =

_

_0

ô€€€ 1, _0 is the solid density at zero pressure, e is the internal energy per

unit mass and A1, A2, A3, B0, B1, T1 and T2 are material constants. These parameters

can then be used to de_ne the reference curve as, e.g. the shock Hugoniot.

3.1.3 P-_ EOS

Up to this point, the description of the equation of state has only applied to the solid

material - that is, a material void of pores. However, as described previously, concrete is

a porous material which can absorb shock impact relatively e_ectively. This impact mitigation

results from the porous material compacting to its solid state, causing a reduction

in volume and an absorption of energy from the impact. This property of concrete must

be taken into account in the material model in order to accurately represent the material.

The relationship between the hydrostatic pressure and density is non-linear due to

the porosity of the material and thus the conventional stress-strain curve is inadequate.

At _rst, an elastic behaviour is demonstrated whereby the compaction of pores in the

16

material is reversible. After this elastic limit is reached (known as the Hugoniot Elastic

Limit), the pores collapse and the material undergoes plastic compaction. Once all of

the pores have collapsed, the material becomes granular and the relationship between the

hydrostatic pressure and density becomes linear again. Figure 3.2 shows the relationship

between the pressure, p and the density _. In this _gure, _crush represents the Hugoniot

elastic limit, _0 the initial density of the porous solid, and _s the density of the fully

compacted material after unloading.

Figure 3.2: Loading and Unloading Behaviour for a Porous Material

The P-_ equation of state is a model derived by Herrmann (1960) which describes the

elastic portion of the pressure-density relationship of porous materials subject to shock

loading. The P-_ equation of state de_nes the starting point of the plastic collapse and

the Polynomial equation of state (described above) de_nes the compaction phase. The

principal assumption of the P-_ model is that the speci_c internal energy is the same

for a porous material as for the same material at solid density at identical conditions of

pressure and temperature. The parameter _ is then introduced denoting the porosity of

a material, de_ned by:

_ =

v

vs

(3.12)

where v is the speci_c volume of the porous material, and vs is the speci_c volume of

the solid material - that is, the same material void of pores - at the same pressure and

temperature. _ becomes equal to 1 when the material compacts to a solid. As discussed

17

previously, if the equation of state of the solid material is:

p = f (v; e) (3.13)

then the equation of state of the porous material is simply

p = f

_ v

_

; e

_

(3.14)

where f(x; y) is the same function in both equations (3.13) and (3.14). This function

can be any solid equation of state which describe the compressed material, e.g. the

Polynomial EOS described previously. To complete the description, the porosity _ must

be speci_ed as a function of the thermodynamic state of the material as, say:

_ = g (p; e) (3.15)

As we are dealing with areas in the region of the shock Hugoniot of the phase diagram as

shown in Figure 3.1, it can be assumed (Hermann 1960) that the pressure and internal

energy are related by the Rankine-Hugoniot conditions, so therefore Eq (3.15) can be

expressed as:

_ = g (v) (3.16)

where the variation of internal energy is implicitly assumed from the pressure.

This P-_ model is combined with the Polynomial equation of state to describe the

response of porous concrete to hydrostatic pressures for impact loading conditions. The

parameters for the implementation of these models can be seen later in the report.

3.2 Strength and Failure

As stated in the previous section, in order to model solids, a shearing resistance must be

included in the material model - that is, the deviatoric of the stress tensor. There are

many models available which are appropriate in di_erent circumstances, depending on

the nature of the material in question and the type of loading it is subject to. For brittle

materials under high rates of loading, such as concrete subjected to impact loading, a

18

model which takes into account e_ects of pressure hardening, strain rate hardening, strain

hardening, fracture strength and other such factors is required. As in the previous section,

the theory surrounding the model used will be built up from a more simple form. Initially,

the Drucker-Prager model will be considered, and its limitations to this application will

be highlighted. Then, the more appropriate RHT model will be discussed together with

the reasons which make it most suitable. A study on the e_ects of constitutive model

has been conducted by Tham (2004).

3.2.1 Drucker-Prager

The Drucker-Prager model is used to represent the behaviour of dry soils, rocks, concrete

and ceramics where the cohesion and compaction behaviour of the materials result in

an increasing resistance to shear up to a limiting value of yield strength as the loading

increases. The yield strength of these materials is highly dependent on pressure. In

the original form of the Drucker-Prager model, the relationship between yield stress and

pressure is linear, as can be seen in Figure 3.3a. A non-linear form of this relationship can

be seen in Figure 3.3b. This non-linear form accounts for the reduced strength in tension

compared to that of compression for materials such as concrete. When the material is in

tension (negative pressure), the yield stress drops rapidly to zero to give a realistic value

of the limited tensile strength. The non-linear Drucker-Prager yield condition takes the

form:

J2Y =

Y0

3

[kY0 + 3(k ô€€€ 1)p] (3.17)

where J2Y is the second invariant of the deviatoric stress yield; Y0 is the yield strength

in simple tension; k is the ratio between the yield strengths in compression and tension;

and p is the pressure

Although this model does account for the reduced strength in tension than in compression,

it does not include many other properties of concrete which are essential if

19

(a) Linear Drucker-Prager Yield Criterion (b) Non-Linear Drucker-Prager Yield Criterion

accurate modelling is to be achieved. That is not to say that the model cannot be used

to advantage, as some authors have demonstrated (Beppu et al. 2009; Itoh et al. 1998)

by also including separate factors into the constitutive model.

3.2.2 RHT Model

In order to successfully model the behaviour of concrete at impact velocities, the e_ects

of pressure hardening, strain hardening, strain rate hardening, and tensile meridians

must be taken in to account. The RHT model (Riedel, Hiermaier and Thorma 1999)

is considered by many (Leppanen 2002; Clegg et al. 1999; Nystrom 2011) to be an

appropriate selection for the constitutive strength model for this type of study. A short

description of the model is provided below. For a detailed description, see Riedel(2000)

or AUTODYN Theory Manual(1997). The model consists of three pressure-dependent

surfaces: an elastic limit surface, a failure surface, and a surface for residual strength, as

can be seen in Figure 3.3.

The failure surface is de_ned as follows

f (p; _eq; _; _ _) = _eq ô€€€ YTXC(p) _ FCAP(p) _ R3(_) _ FRATE( _ _) (3.18)

Further details of how the model represents the various aspects of the concrete behaviour

are presented below.

20

Figure 3.3: The RHT model used for concrete, _gure reproduced from Leppanen (2002)

Fracture Surface

The fracture surface is de_ned through the expression

YTXC = fc0

h

AFail

ô€€€

P_ ô€€€ P_

spall _ FRATE

_NFail

i

(3.19)

where AFail and NFail describe the form of the compressive meridian that is a function of

pressure for principal stress conditions, P_ is pressure normalized with respect to fc0 , P_

spall

is the normalized hydrostatic tensile limit, and FRATE is the rate dependent enhancement

factor (see below).

Tensile and Compressive Meridians

The RHT model can represent the di_erence between the compressive and tensile meridian

in terms of material strength using the third invariant dependence term of the deviatoric

stress (R3). Similar to the non-linear Drucker-Prager model, this can be utilized

to represent the observed reduction in strength of concrete in triaxial tension, compared

with triaxial compression. The exact form of this expression can be seen in Riedel (2008).

21

Strain Hardening

Shear Damage

Strain Rate Hardening

Tensile Failure

Tensile Crack Softening

22

Chapter 4

Previous Work in the Literature

The study of high impact loading and the behaviour of materials subject to it has been

of great interest in the recent decades. Many studies have been conducted to attempt

to successfully model the behaviour of brittle materials when subject to impact loading

in order to accurately predict the outcomes of such events as terrorist strikes or impacts

on structures. Research in the _eld of concrete subject to high impact loading seems

to have started with the experimental investigation conducted by Abrams (1917). Since

then progress has been made in many areas allowing accurate representation of concrete

behaviour to be made. Below is a summary of some of the areas of recent study and the

conclusions that have been made with particular attention being paid to areas inuencing

this study.

Many of the recent studies have addressed the issue of constitutive modelling and in

particular how they a_ect the simulation accuracy. Below are brief summaries of two

such studies by Tham (2004), Clegg et al. (2002) and Beppu et al. (2008) together with

their conclusions and a statement on their relevance to this study.

Other studies focus on di_erent applications of the modelling techniques, such as that

by Chen et al. (2004) modelling the e_ects of oblique impacts on concrete structures, and

a study by Gama and Gillespie (2011) modelling the impact of projectiles on composite

structures.

23

4.1 Tham (2004) - Inuence of Constitutive Model

Tham conducted a study on the inuence of various constitutive models on the resulting

response of concrete to impact loading. Simulations were performed using the following

models: Constant Yield Strength; Pressure-Dependent Yield Strength; Pressure-

Dependent Yield Strength with Fracture Damage and with Strain Rate Hardening. In

the simulations, steel ogive projectiles with a diameter of 25:4mm and a mass of 0:5kg

were _red against a 406:4 x 406:4 x 178mm reinforced concrete target with a striking

velocity ranging from 300m/s to 1000m/s. The results obtained from the simulation were

compared with experimental results obtained from Hanchak (1992). Results were obtained

of the projectile residual velocities after perforation of the concrete and combined

with experimental data.

For the case where the constitutive model uses a constant yield strength, the concrete

is assumed to have an elastic-perfectly plastic behaviour with the yield strength equal to

the compressive strength of the concrete target of 48 MPa. The case of the constitutive

model with a pressure dependent yield strength, the shear resistance of the concrete is

represented using the Drucker-Prager yield criteria. The parameters for this model were

derived from the tri-axial test data outlined in the experimental series conducted by

Hanchak (1992). In these two cases, the compression and compaction of the concrete are

represented using the Porous equation of state. In order to model the fact that concrete

is an order of magnitude weaker in tension than in compression, a hydronamic tensile

failure criterion is speci_ed. Failure occurs when the tensile pressure in the concrete

target exceeds 4 MPa. The _nal case includes the response of concrete to strain rate.

The hardening experienced by concrete under high strain rates is described in more detail

in Chapter 2. Tham uses the RHT model to describe the concrete response to strain rate.

Figure 4.1 shows the comparison of the simulated results of the di_erent constitutive

models with the results obtained from experiments. The results for the case of the consti-

24

tutive model with constant yield strength exhibit consistently higher residual velocities

than those observed in the experiment. The case of the constitutive model which includes

a pressure dependent yield strength, the results from the low velocity tests tend to exhibit

higher residual velocities. In the third case, where the e_ects of strain rate and fracture

damage are included, the simulated residual velocities are in closest agreement with the

experimental results. Although these results are for the response of reinforced concrete to

impact loading, the increased accuracy of the inclusion of strain rate e_ects and fracture

damage can be assumed to also hold true for plain concrete.

Figure 4.1: Comparison of Calculated Residual Velocities with Experimental Results, _gure

reproduced from Tham (2004)

4.2 Clegg et al. (2002) - Implementation of a Tensile

Crack-Softening Damage Model

Clegg (2002) conducted a study on the implementation of a tensile crack softening model

for improved modelling of brittle materials such as concrete in the tensile regime.

Constitutive models such as the Johnson-Holmquist and RHT model have a damage

law that is formulated in such a way that material damage can only result from inelastic

deviatoric straining of the material. This is most applicable in the high pressure region

which exists ahead of a projectile during the penetration process. In other regions of

25

the target the pressures are lower and principal tensile material stresses can be of the

same order as the deviatoric stresses. Clegg et al. proposed that under these conditions,

the maximum principal tensile stresses can lead to crack extension and damage growth.

Principal stress includes volumetric as well as deviatoric stress components, and the

model takes into account damage resulting from both these components. In the model,

the initiation of tensile failure is based on principal stress and the post-failure tensile

strength of the material is calculated based on a Rankine plasticity failure surface. The

failure surface is superimposed onto the traditional constitutive model, for example the

RHT model. The Rankine failure surface provides a limit for the maximum principal

tensile stress in the material. When the trial elastic stress state violates the Rankine

failure criteria, the stress is returned to the failure surface which induces a reduction

in the trial elastic stresses. The resulting loss in strain energy causes an increase in

the e_ective crack strain. This increase in e_ective crack strain energy is used together

with the initial principal tensile failure stress and an input parameter - the critical strain

energy release rate - to calculate the rate of crack growth and hence damage.The result

of using this extension to the traditional damage model proves to be very encouraging

especially in terms of crack propagation patterns. A comparison of a simulated result

by Clegg et al. to an experiment performed by Hazell (1998) shows a very close relation

to the experimental observation and can be seen in Figure 4.2. Having conducted this

report, this addition to the damage model has since been included in the release of the

commercial software AUTODYN. This crack-softening model will be included later in

this report with an analysis of improvements observed with its use.

26

(a) AUTODYN simulation results at 0.02ms

(b) Results from an Experimental _ring, _gure reproduced from Hazell

(1998)

Figure 4.2: 25mm Sintox-FA after impact at 1449m/s by a 6.35mm steel ball, _gure reproduced

from Clegg et al. (2002)

4.3 Beppu et al. (2008) - Damage evaluation of concrete

plates by high-velocity impact

Beppu et al. conducted a study into the damage patterns and behaviour of concrete subject

to high velocity impacts. An experimental series was performed whereby a mushroom

shaped projectile was _red at velocities between 100m/s and 500m/s on concrete plates

of thicknesses between 3cm and13 cm. Simulated results were compared to experimental

results to assess constitutive model performance and resulting concrete failure modes.

For the simulations, the concrete was modelled with a linear equation of state and

a Drucker-Prager strength model. This model was used both in the traditional linear

form and in the improved, non-linear form, as discussed in Chapter 3, for comparison

purposes. The inuence of strain rate on the behaviour of concrete was also included in

the constitutive model via a relationship proposed by Fujikake et al. (2001) as shown in

27

Eq (4.1):

f0

cd

f0

cs

=

_

_ _

_ _s

_0:006

h

Log

_

_ _

_ _s

_i1:05

(4.1)

where f0

cs is the static compressive strength of the concrete, f0

cd is the dynamic compressive

strength of the concrete, __s is the static strain rate and __ is the dynamic strain

rate.

The results from the dtudy shown good agreement with the experimental results

when using the non-linear Drucker-Prager yield criteria as shown in Figure 4.3. The

report concludes that the damage area is more localised with the increase in strain rate.

This is understood to be because the increase rate of tensile strength is larger than that

of compressive strength and thus tensile rate increases rapidly with strain rate.

Figure 4.3: Variation of damage with type of numerical model, _gure reproduced from Beppu

et al. (2008)

The macroscopic damage processes were also analysed in the report. Immediately

after impact, the concrete surrounding the impact zone becomes plastic, and this plastic

area spreads rapidly through the material. A compressive wave also propagates through

the material until it reaches the free boundary on the back surface where it changes to a

tensile wave. Due to the reduced strength in tension than compression, tensile fracturing

28

occurs. However, at this point, spalling does not occur on the back surface, despite the

appearance of several tensile cracks in the back surface. Further through the impact

process, diagonal cracks form and propagate towards the back surface. These diagonal

cracks then form the spalling surface on the back face of the concrete. The simulated

process of cratering and spalling can be seen in Figure 4.4

(a) Process of cratering, _gure reproduced from Beppu et al. (2008)

(b) Process of Spalling, _gure reproduced from Beppu et al. (2008)

Figure 4.4: Simulated damage processes, _gure reproduced from Beppu et al. 2008)

The paper by Beppu et al. (2008) further highlights the requirement of the inclusion

of the e_ects of strain rate in the material model. Although a Drucker-Prager consitutive

model was used here, it con_rms the non-lienarity of the pressure dependent yield criteria

is a requirement to accurately simulate the damage process. The linear Drucker-Prager

model did not accurately simulate the localised fashion of the concrete tensile fracture

experienced by the concrete.

Other studies such as that by Itoh(1998) propose extensions to the Drucker-Prager

29

model to include e_ects of strain rate. The model improvements were implemented into

AUTODYN using subroutines which are exposed to the user. The paper concludes that

the inclusion of these strain rate e_ects improves the simulation accuracy particularly in

reference to scabbing and cratering dimensions.

30

Chapter 5

Validation of Material Model

In order for material models to be used with con_dence to predict the behaviour of

concrete under conditions such as impact loading, validation with experimental data must

take place. As stated in previous chapters, the main modes of failure which are of interest

when determining model accuracy are: penetration depth (or exit velocity), spalling on

the front surface, scabbing on the back surface, and crack propagation patterns. If the

simulated results _t well with the experimental data, then it can be assumed that the

material model employed is acceptable under similar conditions. Below are the results

from an experimental series conducted by Hansson (2005) together with the simulated

results using the material model described in Chapter 3.

The experimental work in question addresses the penetration of normal strength concrete

at di_ering impact velocities and angles. The study considers the penetration in

unreinforced concrete by projectiles of diameter 50mm. The nominal impact of the projectiles

are 420msô€€€1 and 460msô€€€1, respectively. The length of the projectile was 450mm

giving the length to diameter ratio as 9. Tests where the projectile penetrated the target

to some distance and also where the projectile perforated the target with some exit

velocity were considered. Simulations were conducted and the comparison with the experimental

results can be seen below.

31

5.1 Experimental Con_guration

The exit velocities of the projectiles were determined with a high-speed camera. The

properties of the concrete and projectiles are shown in Tables 5.1 and 5.2, respectively.

The concrete tests were conducted after the test series at an age of 9.6 months. Figure 5.1

shows the stress-strain relationship for the steel type with HV 450 and at a nominal strain

rate of 400msô€€€1. The projectile used for the tests is shown in Figure 5.2 and consists of

the steel outer casing, and _lled with an inert material of relatively low density.

Table 5.1: Properties of the concrete tested after the test series at an age of 9.6 months, _gure

reproduced from Hansson (2005)

Table 5.2: Properties of the penetrators, _gure reproduced from Hansson (2005)

Figure 5.1: Stress-strain relationship for SS 142541 with HV 450 at the nominal strain rate of

400msô€€€1, _gure reproduced from Hansson (2005)

32

Figure 5.2: The used 50mm radius projectile, _gure reproduced from Hansson (2005)

5.2 Numerical Simulation

The commercial software AUTODYN was used for the simulation series together with

the RHT concrete model and a P-_ equation of state. The model was set up as a 2-

dimensional axisymmetric model to reduce computation time. A 3-dimensional model

would prove too time-consuming for the scope of this study but is noted as an area of

further exploration. A lagrangian discretization will be used whereby the material moves

with the mesh. An alternative meshless approach using small particle hydrodynamics

(SPH) has been utilised by Clegg et al. (2000). axisymmetric model can be seen in

Figure 5.3. A description of the P-_ EOS and the RHT model can be seen in Chapter 3.

Figure 5.3: Discretized 2-Dimensional Asymmetric Model

33

5.2.1 Model Geometry

The models were set up in accordance with the experimental con_guration outlined above.

Cases 1, 3 and 4 were simulated. The omission of Case 2 is due to the non-symmetrical

nature of the problem and thus the inadequacy of the 2-dimensional asymmetric simulation.

An extension to this report is to simulate in three dimensions which would make

this oblique impact simulation possible. Initially the elements in the target were square

with side lengths of 5mm. An analysis of the e_ect of the element size can be seen later

in the report. The projectile was modelled in two parts; the stem of the projectile as a

rectangle and the nose as a half ogive with radius 400mm. These parts were then joined

so as to become one entity. The projectile was made up of three materials, the locations

of which can be seen in Figure 5.4. The half projectile was made up of 90 elements; 65 in

the stem, and 25 in the nose. The outer two rows of elements in the stem are modelled

with the steel casing, the interior being modelled with the inert _ll. The hardened steel

cap is used for the nose to avoid local failure due to unfavourable material geometry. The

total mass of the projectile is 4.53kg. The use of a steel con_ning plate has been found

(Hansson 1998; Leppanen 2002) to have little e_ect on the penetration depth and crater

size. Therefore, target boundary conditions have been omitted from the model.

Figure 5.4: Projectile Model Material Locations

34

5.2.2 Material Modelling

Concrete Target

A P-_ equation of state was used to describe the porosity of the concrete together with

a polynomial EOS for the behaviour of the soild material to obtain the response of the

concrete to hydrostatic pressure. The RHT model has been used to model the deviatoric

strength of the concrete target. The values of the parameters used in the models can

be seen in Table 5.3 and were determined from results of compressive strength tests on

the experimental specimens and estimations based on similar concrete type simulations

(Riedel et al. 2009; Tham 2004; Leppanen 2002; Thorma et al. 1999).

These values were used for the purposes of validation but a study was performed on

the sensitivities of these parameters and can be seen later in the report. The uni-axial

compressive strength was chosen as 48 MPa and this value is considered a fair approximation

based on the fact that the tests on the specimens were conducted 9.6 months

after the experimental series took place. This meant that the strengths determined in

the tests were slightly exaggerated values. Again, this assumed value of the uni-axial

compressive strength is an initial value and a sensitivity study can be seen of this and

other parameters later in the report. Numerical erosion is necessary when using this Lagrangian

formulation but, as observed by Leppanen (2002) if the erosion criterion is set

too low, the behaviour becomes inaccurate. This is due ot the fact that the inability to

transmit strain energy causes reduced con_nement. This reduction in con_nement may

lead to a slight exaggeration in the simulated projectile penetration as compared to the

experimental data. This results from the fact that the material can still provide at least

some resistance when in an over-strained state, but the simulation simply removes the

elements from the model when this strain value is reached. This fact will be considered

in the analysis. Leppanen (2002) also noted that for criterion for erosion for erosion is set

below 2.0, penetration depth is reduced and radial stresses become unstable. Therefore,

35

Table 5.3: Concrete Material Data

Equation of State P alpha

Reference density 2.32400E+00 (g/cm3 )

Porous density 2.31400E+00 (g/cm3 )

Porous soundspeed 3.00000E+03 (m/s )

Initial compaction pressure 3.50000E+04 (kPa )

Solid compaction pressure 6.00000E+06 (kPa )

Compaction exponent 3.00000E+00

Solid EOS Polynomial

Bulk Modulus A1 3.52700E+07 (kPa )

Parameter A2 3.95800E+07 (kPa )

Parameter A3 9.04000E+06 (kPa )

Parameter B0 1.22000E+00

Parameter B1 1.22000E+00

Parameter T1 3.52700E+07 (kPa )

Parameter T2 0.00000E+00 (kPa )

Reference Temperature 3.00000E+02 (K )

Speci_c Heat 6.54000E+02 (J/kgK )

Thermal Conductivity 0.00000E+00 (J/mKs )

Compaction Curve Standard

Strength RHT Concrete

Shear Modulus 1.67000E+07 (kPa)

Compressive Strength (fc) 4.80000E+04 (kPa)

Tensile Strength (ft/fc) 8.30000E-02

Shear Strength (fs/fc) 1.80000E-01

Intact Failure Surface Constant A 1.60000E+00

Intact Failure Surface Exponent N 6.10000E-01

Tens./Comp. Meridian Ratio (Q) 6.805000E-01

Brittle to Ductile Transition 1.05000E-02

G (elas.)/(elas.-plas.) 2.00000E+00

Elastic Strength / ft 7.00000E-01

Elastic Strength / fc 5.30000E-01

Fractured Strength Constant B 1.60000E+00

Fractured Strength Exponent M 6.10000E-01

Compressive Strain Rate Exp. Alpha 3.20000E-02

Tensile Strain Rate Exp. Delta 3.60000E-02

Max. Fracture Strength Ratio 1.00000E+20

Use CAP on Elastic Surface? Yes

Failure RHT Concrete

Damage Constant, D1 4.00000E-02 (none )

Damage Constant, D2 1.00000E+00 (none )

Minimum Strain to Failure 1.00000E-02 (none )

Residual Shear Modulus Fraction 1.30000E-01 (none )

Tensile Failure Hydro (Pmin)

a geometric strain of 3.0 (300%) is used as a criterion for erosion and is applied instantaneously

when this degree of strain is met. It should be noted, however, that this reduces

con_nement in the vicinity of the projectile and may lead to a slight exaggeration in the

simulated projectile penetration as compared to the experimental data. This is due to the

fact that the material can still provide at least some resistance when in an over-strained

36

Table 5.4: Steel Casing Material Data

Equation of State Shock

Reference density 7.75000E+00 (g/cm3 )

Gruneisen coe_cient 2.17000E+00 (none )

Parameter C1 4.56900E+03 (m/s )

Parameter S1 1.49000E+00 (none )

Parameter Quadratic S2 0.00000E+00 (s/m )

Relative volume, VE/V0 0.00000E+00 (none )

Relative volume, VB/V0 0.00000E+00 (none )

Parameter C2 0.00000E+00 (m/s )

Parameter S2 0.00000E+00 (none )

Reference Temperature 3.00000E+02 (K )

Speci_c Heat 4.77000E+02 (J/kgK )

Thermal Conductivity 0.00000E+00 (J/mKs )

Strength Johnson Cook

Shear Modulus 8.18000E+07 (kPa )

Yield Stress 1.53900E+06 (kPa )

Hardening Constant 4.77000E+05 (kPa )

Hardening Exponent 1.80000E-01 (none )

Strain Rate Constant 1.22000E-02 (none )

Thermal Softening Exponent 1.00000E+00 (none )

Melting Temperature 1.76300E+03 (K )

Ref. Strain Rate (/s) 1.00000E+00 (none )

Strain Rate Correction 1st Order

state, but the simulation simply removes the elements from the model when this strain

value is reached. This fact will be considered in the analysis.

Projectile

The projectile is split into three materials, as shown in Figure 5.4. The steel casing is

modelled using a shock equation of state and a Johnson and Cook strength Model, the

parameters of which can be seen in Table 5.4. The same material data is used for the

steel nose cap, except for the yield stress, which is increased to 2:5 GPa. The inert _lling

is described by a linear EOS with a density of 2:4 g/cm3, a bulk modulus of 20 GPa, and

a Von Mises strength model with a shear modulus of 12 GPa and a yield stress of 100

MPa.

37

5.3 Comparison of Results

As stated previously, the validation of results with experimental data takes various forms.

Penetration depths for projectiles which don't perforate the target and residual velocities

for those which do; crater dimensions on the front surface; scabbing dimensions on the

back surface; and crack propagation patterns are all results which indicate an accurate

representation of the response of concrete to high velocity impact. Not all of these results

are available from the experimental data provided in the report by Hansson (2005), but

reasonable assumptions are made where appropriate. The most accessible validation

result is that of penetration depth or residual velocity which is examined later in the

section.

Figure 5.5 shows the experimental result of the front and rear faces of the target

after impact from the projectile for Case 1. The simulated damage plots for each of

the validation cases are shown in Figure 5.6. From this _gure, the penetration of the

projectiles can be seen graphically with a view of the damage patterns resulting from

the impact. A more detailed analysis of the damage process for Case 1 can be seen

below. Observing the experimental and simulated results for Case 1, it can be seen that

the cratering and spalling dimensions are represented fairly well in the simulations. It

is obviously a limited representation due to the non-symmetrical nature of the concrete

damage, and the 2-dimensional symmetric model used for the simulation.

The experimental results obtained by Hansson (2005) are shown in Table 5.5 whilst

the simulated penetration results for Cases 1, 3 and 4 are shown in Figure 5.7. The _gure

shows the velocity pro_le of the projectiles through the depth of the concrete plates. The

front face of the concrete block is at X = 0 in each case. For Cases 3 and 4, where the

projectile does not perforate the target, the results show very close agreement with the

experimental results. For Case 3, where the projectile is _red at 409msô€€€1 into a 900mm

concrete block, the experimental observation was that of a 64cm penetration and the

38

Figure 5.5: Experimental Result for Front (left) and Back (right) Faces of Concrete after Impact,

_gure reproduced from Hansson (2005)

Figure 5.6: Damage Plots for Validation Cases 1, 3, and 4

simulation resulted in a 62:7cm penetration. For case 4, with a 460msô€€€1 projectile velocity

and 1200mm thick concrete, the experimental penetration depth was observed to be 69cm

and the simulated result was 73:7cm. Both Cases 3 and 4 show very good agreement with

the observed experimental results, with errors of 2:03% and 6:81% respectively. For Case

1, where the projectile perforated the target, the simulated results were quite di_erent

from the observed experimental results.

Figure 5.8 shows the damage process for Case 1 and it can be seen that as the projectile

approaches the rear face of the target, more tensile cracks start to form, indicating the

39

Table 5.5: Experimental Results, table reproduced from Hansson (2005)

failure of the concrete and the reduction in strength provided by it. This is the reason

for the sloping o_ of the curve shown in Figure 5.7. However, the extent to which the

reduction in strength occurs seems to be to rapid, leading to an exaggerated value of the

residual velocity of the projectile after perforation.

40

Figure 5.7: Simulated Results of Projectile Velocity with Depth of Penetration

Figure 5.8: Validation Case 1 Damage Process

41

Chapter 6

Model Parameter Study

6.1 Projectile Velocity

For this study, a 900mm concrete plate was subjected to impact velocities ranging from

380msô€€€1 to 460msô€€€1. Figure 6.1 shows the penetration pro_les of the projectile for the

simulations. It can be seen that there is an obvious relationship between the impact

velocity and depth of penetration. It can also be seen that as the impact velocity approaches

440msô€€€1, perforation of the target starts to occur, and the projectile deceleration

reduces. The strength provided by the concrete reduces as it becomes damaged, and as

the impact velocity increases, and the depth of penetration increases, the concrete starts

to fail at the back surface of the plate. This failure causes a reduction in the strength

provided and thus the deceleration of the projectile reduces. As the impact velocity is

increased further, this fact is enhanced, with the reduction in strength occurring earlier

in the simulation. The case with an impact velocity of 440msô€€€1 demonstrates the transition

phase between penetration and perforation. Impact velocities below this magnitude

demonstrate penetration of the projectile whereby it comes to rest and all kinetic energy

of the projectile is lost. Impact velocities above this level show perforation through the

concrete material and a residual velocity in the projectiles.

Table 6.1 shows the complete set of results for the projectile velocity study. The

42

energy lost in the projectile can be seen to be 100% for the cases where the projectile only

penetrates the concrete target. For cases demonstrating perforation, residual velocities

can be seen together with the loss in kinetic energy.

Figure 6.1: Simulated Results of Projectile Velocity with Depth of Penetration

Table 6.1: Simulated Results for Projectile Velocity Study

Impact Velocity Residual Velocity Penetration Depth Energy Lost (Projectile)

380 - 543 100

390 - 577 100

400 - 600 100

410 - 627 100

420 - 641 100

430 - 663 100

440 - 100

450 98.7 - 95.2

460 94.7 - 95.8

43

6.2 Depth of Concrete

6.3 Mesh Density

This study examines the di_erence in simulated behaviour of the concrete due to mesh

density. The increased a