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Author’s Accepted Manuscript
Determination of Johnson cook material and failure
model constants and numerical modelling of
Charpy impact test of armour steel
A. Banerjee, S. Dhar, S. Acharyya, D. Datta, N.
To appear in: Materials Science & Engineering A
Received date: 13 December 2014
Revised date: 15 May 2015
Accepted date: 23 May 2015
Cite this article as: A. Banerjee, S. Dhar, S. Acharyya, D. Datta and N. Nayak,
Determination of Johnson cook material and failure model constants and
numerical modelling of Charpy impact test of armour steel, Materials Science &
Engineering A,
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Determination of Johnson Cook material and failure model constants and
numerical modelling of Charpy impact test of armour steel
A. Banerjeec*, S. Dhara, S. Acharyyaa, D. Dattab, N. Nayakd
Jadavpur University, Kolkata, India
Indian Institute of Engineering Science and Technology, Shibpur, India
PXE Chandipur, DRDO, Min of Defence, Govt. of India, presently under PhD programme of
Jadavpur University, Kolkata
PXE Chandipur, DRDO, Min of Defence, Govt. of India
The behavior of typical armour steel material under large strains, high strain rates and elevated
temperatures needs to be investigated to analyse and reliably predict its response to various types
of dynamic loading like impact. An empirical constitutive relation developed by Johnson and
Cook (J – C) is widely used to capture strain rate sensitivity of the metals. A failure model
proposed by Johnson and Cook is used to model the damage evolution and predict failure in
many engineering materials. In this work, model constants of J – C constitutive relation and
damage parameters of J – C failure model for a typical armour steel material have been
determined experimentally from four types of uniaxial tensile test. Some modifications in the J –
C damage model have been suggested and Finite Element simulation of three different tensile
tests on armour steel specimens under a relatively high strain rate (10-1 s-1), high triaxiality and
elevated temperature respectively has been done in ABAQUS platform using the modified J – C
failure model as user material sub – routine. The simulation results are validated by the
experimental data. Thereafter, a moderately high strain rate event viz. Charpy impact test on
armour steel specimen has been simulated using J – C material and failure models with the same
* Corresponding author, E-mail address: [email protected]
material parameters. Reasonable agreement between the simulation and experimental results has
been achieved.
Keywords: Johnson – Cook material and failure models, Charpy test, armour steel , Finite Element
analysis, numerical simulation
Dynamic loads are encountered in a wide spectrum of phenomena viz. automotive applications,
high speed machinery and defence applications like high speed projectile impact on armour. The
behavior of materials under dynamic loading conditions differs significantly from quasi static
loading due to the effects of inertia, stress reflection and rate sensitivity of material [1]. Finite
Element (FE) formulation based on dynamic equilibrium equation is used in explicit model to
take care of inertia. In order to capture the effect of rate sensitivity, appropriate modeling of the
behavior of the material under high strain rates is an essential prerequisite. However, due to the
complexity of the problem, various material models have been proposed by several researchers
on a case to case basis rather than development of a universal model catering for a large variety
of materials under different loading conditions.
Johnson – Cook material model [2] is a popular constitutive relation for metals, widely used in
simulation of impact and penetration related problems [3-7]. The popularity is more due to the
simple form of the equation and the relative ease of determination of the model constants. A few
tensile tests under various loading conditions are generally sufficient to determine the five
Impact testing techniques were established in order to ascertain the fracture characteristics of
materials and thereby prevent sudden or brittle failure of engineering materials when subjected to
dynamic loads. One of the most prominent impact tests is the Charpy impact test that gives
information regarding the behavior of material under impact load [8, 9, 10]. Charpy test is a low-
cost and reliable technique that measures the energy consumed in breaking a notched specimen
simply supported at both ends when hammered by a swinging pendulum. The presence of notch
simulates the pre-existing cracks found in large structures that increase the probability of brittle
fracture [11].
Armour steel is a low-alloy medium carbon steel suitably heat treated to produce a tempered
martensite microstructure that gives an excellent combination of strength and ductility. Steel
armour is used as the protective cover of Battle Tanks and armoured vehicles that are subjected
to ballistic impact by fast moving military projectiles. A high degree of hardness and toughness
is an essential requirement for armour material. The response of the armour steel material to the
high loads occurring over small intervals of time needs to be understood to analyse the ballistic
impact phenomena and make reliable predictions.
Zejian Xu et al [12] have experimentally investigated the plastic behavior of 603 armour steel at
strain rates ranging from 10-3 s-1 to 4500 s-1 and temperatures from 288 K to 873 K. Compression
tests were conducted on an MTS hydraulic testing machine for 10-3 s-1 ≤  ≤ 1 s-1 and on SHPB
for the higher strain rates. Elevated temperatures were attained with electro – thermal cells.
Thermal softening was found to play a leading role in its competition with the effects of strain
rate and work hardening. The five material constants of Johnson – Cook and six of KHL material
models determined from the experimental results have been used to model a dynamic
phenomenon viz. strain rate jump test. The predicted data were found to be satisfactorily close to
the experimental results, with the KHL model exhibiting superior applicability for the material.
Whittington et al [13] investigated the mechanical response and damage evolution of Rolled
Homogenous Armour (RHA) steel, processed to Mil-A-12560H specification, through low and
high strain rate tension, compression and torsion tests at 20oC and 300oC . Fractography was
performed to quantify the number density of nucleated voids and size distribution of voids. An
Internal State Variable (ISV) plasticity / damage model was used to capture the varying effects of
temperature, strain rate and stress state for the RHA steel. Reasonable accuracy of the model has
been demonstrated.
However, a detailed experimental work for the determination of J – C material and J – C failure
model constants for armour steel material, in general, is not readily available in the open
literature. In this work, the behavior of a low alloy, medium carbon steel material typically used
in fabrication of armoured vehicles has been studied through extensive experimentation under
various loading conditions. Material constants for J – C material and failure models have been
determined from an analysis of the experimental data. A modified damage growth law has also
been proposed. The model constants have been validated through numerical modeling of tensile
tests on armour steel specimens conducted under a relatively high strain rate (10-1 s-1), high
triaxiality level and elevated temperature (500oC) followed by comparison of the simulation
results with experimental data. Thereafter, an attempt has been made to simulate a moderately
high strain rate phenomenon like Charpy impact test on the same material using J – C material
and failure models and the experimentally determined material constants in a commercial FE
code, ABAQUS - Explicit. The predicted behavior has then been compared with experimental
2.1 Johnson – Cook constitutive relation
In the framework of von Mises incompressible plasticity, the yield function is given as
  Sij Sij - σc = 0
Here, Sij stands for deviatoric stress tensor and σc is the current yield stress.
The flow rule is the normality rule suitable for von Mises yield function. Therefore,
dεppq =
(dSij nij) npq
The unit normal, nij is given below as
nij =
 ij
 ij
and h stands for plastic modulus.
The constitutive relation or the hardening rule is taken as Johnson – Cook type that expresses the
equivalent stress, σeq as a function of plastic strain, strain rate and temperature given by the
σeq = [A+Bεpn][1+c ln(  *)][1-T*m]
where A, B, n, c and m are the model constants, εp is the accumulated plastic strain,  * = (  p /
 0) is a dimensionless strain rate,  0 is the reference strain rate and T* = (T – T0)/(Tm – T0); T,
T0 and Tm being the working temperature, room temperature and melting temperature
From the consistency relation,
dФ =
dσij +
dσc = 0
 ij
 c
Using equations (1) to (5), the plastic modulus, h =
2.2 Johnson – Cook failure model
d c
is obtained.
d p
The fracture criterion proposed by Hancock and Mackenzie [14] has been extended by Johnson
and Cook [3] to make the failure strain sensitive to stress triaxiality, temperature, strain rate and
the strain path. The model assumes that damage accumulates in the material element during
plastic straining which accelerates immediately when the damage reaches a critical value. D is
defined as a damage variable which varies between 0 (material not damaged) and 1 (fully failed
material). The failure criterion is based on the value of equivalent plastic strain at element
integration points. Failure is assumed to occur when D reaches or exceeds the value of unity [15].
D is defined as
 pl
 p, f
where, Δεpl is an increment of the equivalent plastic strain, εp,f is the equivalent plastic strain at
failure and the summation is performed over all increments of deformation [16].
However, the critical value of damage variable, i.e. the value at which a macro crack occurs is
less than one. Hence, the failure criterion becomes
D = Dc ≤ 1.
Furthermore, experiments indicate that damage remains equal to zero during the build up of
dislocations generating micro cracks. There may exist a threshold of the accumulated plastic
strain at which damage starts to evolve. Based on these observations, a damage rule can be
proposed as
D= D
 pl ,
  f   p ,d
when,  pl
when,  pl
 p ,d
 p ,d
where Dc is the critical damage, εp,d the damage threshold and εf the fracture strain [17].
In this work, the damage growth law as mentioned above is modified. The damage is considered
to occur in two stages. In the first stage, the damage growth rate is very slow. This is due to the
fact that at the initial stage, the damage is due to void nucleation only. The damage growth law is
a linear function of equivalent plastic strain, given as
 p0
 p,eq
Here, εp0 is the plastic strain and Dc1 is the damage at the ultimate point (initiation of necking).
Dc1 is assumed to take a low value of 2%.
After the ultimate point, during necking process, the damage growth is accelerated due to void
growth and coalescence. The damage growth at this stage is increasing non – linearly and given
Dc 2  Dc1
D D 
 p,eq
D = c2
 p, f   p0
Here, Dc2 is the critical damage at fracture and taken as 0.8, εp,f is the strain at failure expressed
as equation (11) by Johnson and Cook [3].
The model by Johnson and Cook [3] proposes that εf depends on stress triaxiality, strain rate and
temperature and can be expressed as
εf = [D1 + D2 exp (D3σ*)] [1 + D4 ln (  p*)] [1 + D5T*]
where D1 to D5 are material constants, σ* = σm / σeq is the stress triaxiality ratio and σm is the
mean stress or hydrostatic stress.
Also, since damage degrades the material strength during deformation, the constitutive equation
for the damaged material can be written as
σD = (1 – D) σeq
where, σD is the stress at damaged state and σeq is obtained from Johnson – Cook constitutive
relation for the original (undamaged) material as given in (4).
The constitutive model and failure model defined in equations (1) through (12) are implemented
in the ABAQUS Explicit code through VUMAT sub – routine. When the failure criterion is met,
the stress components are set to zero and they remain zero during the remaining part of the
analysis. An element – kill algorithm implemented in the FE code removes the failed elements
from the mesh.
3.1 Armour steel material
The armour material studied in this work is a medium carbon, low alloy steel with chemical
composition as given in Table 1. The material is obtained from a 50mm thick armour plate. The
plate is hot rolled and made to undergo heat treatment schedule as given below to attain a
tempered martensite structure to ensure excellent combination of high strength as well as
i). Soft annealing by heating to 700oC @ 20 – 25oC per hour, soaking for 12 – 14 hours followed
by furnace cooling down to 100oC and thereafter air cooling.
ii). Hardening by heating and soaking at 910oC for 100 minutes followed by oil quenching
iii). Tempering at a temperature of 600oC for 200 minutes followed by air cooling
Specimens for all the tensile tests and Charpy Impact test are fabricated from the same heat
treated plate only in the rolling direction to ensure consistent microstructure and properties in all
the specimens.
Table 1. Chemical composition (weight %) of armour steel
3.2 Tensile tests for determination of Johnson – Cook material and failure model constants
Four types of tensile tests have been done to determine the Johnson – Cook material and failure
model constants for the armour steel material. Tensile tests of specimens as shown in Fig 1a were
conducted at low strain rates, between 10-4 s-1 (quasi – static) and 10-1 s-1 and room temperature
in an INSTRON make Universal Testing Machine (Model 8801) to determine the elastic
constants, the initial yield stress, A and the hardening parameters, B & n of the Johnson – Cook
constitutive relation. In the same machine, tensile tests on notched specimens having different
thickness and notch radii (Fig 1b) were done to determine the fracture strain at different
triaxiality ratios and evaluate therefrom the damage model parameters, D1, D2 and D3.
Tensile tests at strain rates between 100 to 1.5x102 s-1 were conducted on typically long, flat and
thin specimens (Fig 1c) in a servo hydraulic High Strain Rate Testing Machine, (Make:
INSTRON, model no. VHS 65/80-20). From the data obtained through tensile tests conducted at
strain rates between 10-4 s-1 and 1.5x102 s-1, the strain – rate dependent material model constant, c
and failure model constant, D4 were calculated.
Finally, tensile tests of cylindrical specimens (Fig 1d) at a constant strain rate (10-3 s-1) and
elevated temperatures between 200oC and 500oC were performed on INSTRON make universal
testing machine (Model 5582) for determination of the thermal softening constant, m and the
temperature dependent failure model constant, D5.
Fig 1. Geometry of specimens for tensile tests at (a) quasi static and low strain rates upto 10-1 s-1,
(b) various levels of triaxialities generated by different thickness and notch radii, (c) dynamic
strain rates, (d) elevated temperatures. All dimensions are in mm.
3.3 Results of tensile tests and extraction of J – C material and failure model constants
The true stress vs. true strain data generated from tensile test at quasi-static strain rate (10-4 s-1)
shows a distinct and high yield point and moderate strain hardening for the armour steel material.
The test results are used to determine the elastic constants as well as initial yield stress, A. Strain
hardening coefficient and exponent, B & n respectively are determined using a power law fit for
the true stress and plastic strain data taken after yield point.
The stress vs. strain curves obtained at various strain rates between 10-3 s-1 and 1.5x102 s-1 shown
in Fig 2a, indicate a moderate strain rate sensitivity of the material. Both the initial yield stress
and flow stress in the post – yield region are seen to increase with increasing strain rate. Beyond
a strain rate of 102 s-1, excessive stress fluctuations are found to occur.
The stress vs. strain curves for specimens tested in tension at a constant strain rate of 10-3 s-1 and
at four different temperatures between 200oC and 500oC are shown in Fig 2b. A distinct decrease
in both the initial yield and flow stresses in the post – yield region is observed.
Fig 2. Stress vs. strain curves at (a) varying strain rates, (b) elevated temperatures
The values of true stress at 5% plastic strain for different strain rates (  ) varying from 10-3 s-1 to
1.5x102 s-1 have been plotted against log (  *) in Fig 3a. The slope of the fitted line gives the
strain rate parameter, c of the Johnson Cook material model.
Fig 3. Variation of stress with (a) strain rate and (b) temperature
With all the model constants known except m, the J – C constitutive relation (4) can be written
σeq = K[1-T*m]
where K = [A + B εpn][1 + c ln  *]
The right hand side of equation (14) is calculated with the constants determined previously.
Rearranging the equation (13) and taking logarithm of both sides, we get,
log (K – σeq) = m log (T*) + log K
which represents a linear equation between the variables, log (T*) and log (K – σeq). The data
obtained from the tensile tests conducted at different temperatures between 27oC and 500oC and
for εp = 1.25% and  = 10-3 s-1, have been fitted to equation (15) as shown in Fig 3b. The slope of
the linear curve gives the value of the temperature dependent material model parameter, m.
Thus, all the five model constants of Johnson – Cook constitutive relation were determined.
It may be noted that the strain values in Figs 2a and 2b are longitudinal strain values. At failure,
they have a relatively low magnitude. However, for calculations, the actual failure strain has
been measured from an area reduction in the “neck” region of the specimen whose value is
considerably higher as seen in Figs 4a to 4c.
In order to determine the Johnson – Cook failure model constants, D2 and D3, the failure strain, εf
is calculated as the area strain at fracture i.e., εf = ln (A0/Af), where A0 is the initial area and Af,
the final area at fracture. To evaluate the triaxiality ratio, σ* at failure, an elasto-plastic FE
analysis has been carried out for the tensile test (at quasi – static strain rate and room
temperature) of specimens having different thickness and notch radii. From the FE analysis, the
mean stress and von Mises stress at the notch region during failure were evaluated and hence, the
stress triaxiality ratio was determined. This exercise was done for the different types of notched
specimens tested and thus, a set of failure strain versus triaxiality ratio data have been obtained.
Using the failure model of Johnson – Cook (11) and analyzing the variation of failure strain with
triaxiality, D2 and D3 have been determined. D1 has been assigned a value 0.05, the strain at
initiation of necking (ultimate point) in cylindrical specimens tested in tensile mode at quasi
static strain rate and room temperature.
The strain rate dependent failure model parameter, D4 is obtained by analysing the variation of
the strain at failure with the strain rate. The effect of temperature on failure strain was found
insignificant within the range of temperatures examined and hence D5 was taken as 0. The effects
of triaxiality ratio, strain rate and temperature on the fracture strain are shown in Fig 4.
The Johnson – Cook material and failure model constants determined from the various uniaxial
tensile tests described in this section are presented in Table 2.
Fig 4. Variation of fracture strain with (a) stress triaxiality, (b) strain rate, (c) temperature
Table 2. Experimentally determined Johnson – Cook material and failure model constants
for armour steel.
Yield stress, A (MPa)
Strain hardening parameter, B (MPa)
Strain hardening exponent, n
Strain rate sensitivity parameter, c
Temperature exponent, m
3.4 FE modeling of tensile tests
To verify the constitutive model calibration and its implementation, three different tensile tests
on the armour material specimens were simulated using the commercial FE code ABAQUS and
the simulated force – elongation curves were compared with the experimental data. Considering
the low strain rates involved in the tests, the inertia effects have been neglected. ABAQUS
Standard was used and the J – C constitutive relation and modified J – C failure models were
implemented as user defined material model (UMAT sub – routine). The values of the
parameters of J – C material and failure models as determined through tensile tests described in
Section 3.3 and presented in Table 2 were used. As already explained, a two – stage damage
evolution rule has also been implemented via UMAT subroutine. The tensile tests that have been
simulated are presented in Table 3.
Table 3. Details of specimens and tensile tests
Thickness or diameter Specimen type
of specimen (mm)
Smooth, flat
Notched, 4mm
Ф = 3.52mm
Strain rate during
tensile test (s-1)
Fig 5. (a) FE mesh for a notched tensile specimen, (b) deformed mesh for notched specimen, (c)
deformed mesh for cylindrical specimen tested at 500oC
The representative FE mesh for a notched specimen and the deformed meshes of the notched and
cylindrical specimens are shown in Fig 5. Considering symmetry of the specimens and loads and
to reduce computational time, one – fourth model has been prepared for the flat specimens, S-1
& N-4-1 and a half – model for the cylindrical specimen, C-S-500. The specimens were
modelled using 8-node hexahedral element (C3D8) with one integration point. The element size
at the critical gauge region in the smooth specimen was optimally chosen as 0.25 x 0.25 mm after
mesh convergence study. A total number of 2160 elements were used for modelling the 3mm
thick smooth flat specimens with 6 elements across the specimen thickness. Similarly, 3316
elements were used to model the 4mm thick flat specimens having notch radius 4mm with 8
elements along the thickness direction. Displacement at the free end was given as input and
reactions were measured from the support end to determine the load.
The FE simulated force – elongation curves for the three tensile test cases are compared with the
experimental observations in Fig 6. Reasonably good agreement is observed with respect to the
maximum load as well as the elongation to fracture.
Fig 6. Comparison of experimental and simulated load (N) vs. displacement (mm) curves for
tensile test of (a) flat, smooth specimen (S-1) at 10-1 s-1, (b) notched specimen (N-4-1) at 10-1 s-1
and (c) cylindrical specimen (C-S-500) at 10-3 s-1 and 500oC.
Also, the damage law implemented in the program is shown in Fig 7 as a plot of damage versus
equivalent plastic strain in the specimens tested.
Fig 7. Damage vs. equivalent plastic strain as per damage growth rule implemented in the
UMAT program
Once the experimentally determined Johnson – Cook material and failure model constants were
found to appreciably represent the armour steel material behavior at strain rates upto 10-1 s-1, high
levels of stress triaxiality and elevated temperatures upto 500oC, their capability to model a
moderately high strain rate phenomenon like Charpy impact test was investigated. An
instrumented Charpy test was conducted on specimen made from the same material and the
results were compared with the FE simulated results of the same test.
4.1 Experimental Charpy Impact test
The Charpy impact test was performed in a Zwick – Roell make test set up. The experimental
set-up consists of the anvils where the standard (ASTM designation, E23) notched specimen is
freely supported and a pendulum with a mass, 30 kg attached to a rotating arm pinned at the
machine body. The characteristic specimen length, height and thickness were 55mm, 10mm &
10mm respectively. The depth of the notch was 2mm. The span length between the anvils was
kept 40mm.
The released pendulum follows a circular trajectory and hits the test specimen at the middle span
transferring kinetic energy to it. In this experiment, the pendulum hammer had a speed of 5.23
m/s while striking. Energy losses due to bearing friction and air resistance have been ignored to
calculate the energy absorbed by the specimen.
The output of the test was obtained in the form of load versus time and displacement as well as
the maximum energy absorbed by the specimen. After the impact, the load quickly rose to a peak
value of around 30 kN within a time interval of around 0.355 ms and thereafter died down to
zero value in about 2 ms. The energy absorbed by the specimen rose to a maximum value of
around 95 J at a time of 1.5 ms after the impact and remained constant after that. Thus, the
armour steel material can be said to have a Charpy impact value of 95 J at room temperature.
From a simple energy balance, the post – impact velocity of the striker can be calculated as 4.62
m/s. Therefore, the average strain rate during deformation is observed to be in the order of 500
4.2 Finite Element Modelling of the Charpy Impact test
FE models of the Charpy impact test specimens were developed in the ABAQUS Explicit
software with the damage model specified through a VUMAT.
Fig 8. FE model of charpy impact specimen
A half model of the Charpy test specimen and the impactor (hammer) with the z-plane as the
plane of symmetry and the z – axis defining the thickness direction was prepared as shown in Fig
8 for reducing the computational time. The span length (distance between the anvils) is
maintained 40 mm as per standard of Charpy test. The impactor was modeled as elastic body and
the specimen as elasto-plastic material. Variable meshing was employed with fine mesh quality
around the notch region of the Charpy test specimen. The mesh size was optimally determined
after a number of runs. The final FE model of the specimen consisted of 40,520 nos. 8 – noded
linear brick, reduced integration, hourglass controlled (C3D8R) elements.
The contact between the anvils and the specimen is assumed as line contact. The nodes on these
lines are constrained in vertical (y) direction throughout the thickness. All the nodes in the z
symmetric plane are constrained in z direction. The nodes on the impactor were constrained to
move only in the vertical direction (y) with impactor velocity given as input. The Johnson –
Cook material model was assigned to the Charpy impact test specimen whereas Johnson – Cook
failure model was assigned to a narrow region over the notch along the height of the specimen (y
– direction) throughout the thickness in order to simulate the crack propagation after impact. The
Johnson – Cook failure model was used as the criterion for deletion of the elements when the
damage parameter, D reached the pre – assigned critical value. The model constants of
constitutive relation and failure model as determined from the tensile tests described in Section 3
and presented in Table 2 were used. This simulation utilized explicit time integration and was run
for a total time of 2 ms. The crack initiation and propagation in the specimen are shown in Fig 9.
Fig 9. Crack initiation at notch tip
4.3. Comparison of the experimental and numerical results
The experimental and numerical results of the Charpy impact test are compared in Fig 10 in the
form of variation of the force exerted on the specimen by the striker with respect to (a) time and
(b) displacement of the striker. The variation of internal energy of the specimen with time has
also been compared (Fig 10c).
Fig 10. Comparison of simulated and experimental plots of (a) force vs. time, (b) force vs.
displacement and (c) internal energy of Charpy specimen vs. time
It is evident that the pattern of the experimental and simulated curves is similar and the time to
reach the peak force value also matches. The drop in the force in both simulated and
experimental plots occurs at the same time and displacement. The nature of the drop also has
been accurately modeled. The successful prediction of the maximum load and the nature of its
drop with time validates the damage distribution as well as the damage growth model used in the
FE simulation. The simulated value of the energy absorbed by the specimen is also found to
appreciably match the experimental value. Thus, the J – C material and modified J – C failure
models as well as the model constants determined are found to satisfactorily represent the
behavior of the armour steel material in a moderately high strain rate event like the Charpy
impact test.
The Johnson – Cook material and failure model constants for armour steel material have
been determined through uniaxial tensile tests conducted over a range of strain rates (10-4
to 1.5x102 s-1) and temperatures (room temperature to 500oC).
Johnson – Cook failure model has been implemented with modification in damage
growth law suitable for armour steel material.
FE simulation of tensile tests at different strain rates and temperatures has produced
results closely matching the experimental data, thus validating the model constants and
the damage growth law.
Finally, the model constants and the damage growth law have been used successfully to
simulate Charpy impact test for the armour steel material.
The authors are thankful to Shri R Appavuraj, Director, Proof & Experimental Establishment,
Chandipur, DRDO for support, encouragement and permission for publishing the work.
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