Application of the Martel dynamic hardness to the penetration problem
Penetration of the undeformable kinetic energy projectile (KEP) into the target is considered as the “deep” indentation. It was shown by the example of aluminum alloys that the Martel dynamic hardness HMRd can be used for description of this process. Проникнення недеформівним кінетичним індентором (...
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| Zitieren: | Application of the Martel dynamic hardness to the penetration problem / Yu.V. Milman, V.A. Goncharuk, L.V. Mordel // Электронная микроскопия и прочность материалов: Сб. научн . тр. — К.: ІПМ НАН України, 2012. — Вип. 18. — С. 85-91. — Бібліогр.: 15 назв. — англ. |
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| author | Milman, Yu.V. Goncharuk, V.A. Mordel L.V. |
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| citation_txt | Application of the Martel dynamic hardness to the penetration problem / Yu.V. Milman, V.A. Goncharuk, L.V. Mordel // Электронная микроскопия и прочность материалов: Сб. научн . тр. — К.: ІПМ НАН України, 2012. — Вип. 18. — С. 85-91. — Бібліогр.: 15 назв. — англ. |
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| description | Penetration of the undeformable kinetic energy projectile (KEP) into the target is considered as the “deep” indentation. It was shown by the example of aluminum alloys that the Martel dynamic hardness HMRd can be used for description of this process.
Проникнення недеформівним кінетичним індентором (KEP) в цілому можна розглядати як “глибоке” індентування. На прикладі алюмінієвих сплавів було показано, що динамічна твердість за Мартелем HMRd може бути використана для опису цього процесу.
Проникновение недеформируемым кинетическим индентором (KEP) в целом можно рассматривть как "глубокое" индентирование. На примере алюминиевых сплавов показано, что динамическая твердость по Мартелю HMRd может быть использована для описания этого процесса.
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UDC 620.015
Application of the Martel dynamic hardness
to the penetration problem
Yu. V. Milman, V. A. Goncharuk, L. V. Mordel
Penetration of the undeformable kinetic energy projectile (KEP) into the target is
considered as the “deep” indentation. It was shown by the example of aluminum alloys
that the Martel dynamic hardness HMRd can be used for description of this process.
HMRd for the target can be calculated from the relation
V
mvHMRd 2
2
= , where m is
the mass of the KEP, υ is its rate before impact collision with the target, and V is the
volume of the penetration channel. The ballistic limit vc of the target with a given
thickness l can be calculated by equation
m
ldHMRd
c 2
πυ
2
= for the KEP with a
given mass m and diameter d.
Keywords: Penetration, Martel hardness, hardness from penetration depth, dynamic
hardness.
Introduction
Dynamic hardness was introduced by Martel in 1895 [1]. In experiments,
Martel used a steel ball that dropped from a height h1 onto a smooth metallic
surface and made a spherical indentation on the surface of the sample. It was
shown that A/V = const, where А is the kinetic energy of the ball and V is the
volume of the indentation. Since this relation has the same dimensionality as
pressure [Pa], it can be considered as the dynamic hardness of metals [2, 3].
Thus, the Martel hardness HMR is determined from the relation
V
A
HMR = , (1)
where А is the work to create the indentation and V is the volume of this
indentation.
Since HMR has the same dimensionality as the Meyer hardness (the mean
contact pressure during indentation) and characterizes the same process, it can
be thought that
KHMHMR = , (2)
where K is a dimensionless parameter.
Martel [1] had calculated hardness from equation:
V
mgh
HMR 1= . (3)
But in experiments performed by Martel, after impact of a ball, the elastic
recovery of an indentation occurred, and the ball rebounded to a height h2
(h2 < h1). Since, in these experiments, it was impossible to measure the volume
of indentation under load (to calculate the unrecovered hardness), the recovered
hardness must be calculated. The energy that caused plastic (residual) strain can
85
© Yu. V. Milman, V. A. Goncharuk, L. V. Mordel, 2012
be calculated by subtracting the energy of elastic recovery from the kinetic
energy of the ball.
The mathematical description of the process of dynamic indentation was
given by Tabor [2].
Tabor had obtained equation
( )
V
hhmgHMR 21 8/3-
= , (4)
that take into account the energy of elastic recovery. Equation (3) may be used
if the rebound is not very large, so that h2 is small.
At present, the Martel hardness can also be calculated in static indentation
by a pyramidal indenter by using the instrumental hardness with recording the
“load on the indenter P–displacement of the indenter h” curve. In this case, the work
expended on the formation of the hardness indentation is equal to the area under
the Р—h curve, and the volume of the indentation can be determined from the
contact depth of penetration of the indenter hс; the technique of determination of
hс was developed in [4]. The value of hс can also be determined by standard
microscopy methods via determination the size of the indentation diagonal (in
the assumption that the size of diagonal is not changed during recovery) and
calculation of the height of the indentation pyramid, using the value of the
center line to the face angle of pyramidal indenter.
In the present work, we consider the possibility of applying the Martel
hardness to the problem of penetration a target by a kinetic energy projectile
(KEP) if the KEP is undeformable. It is assumed that, in this case, the process
of penetration of the KEP can be considered as a “deep” indentation. The ratio of
the kinetic energy expended by the KEP on the formation of the penetration
channel in the target to the volume of the penetration channel must correspond
to the Martel hardness according to eq. (1).
The check of this proposition has been performed for several aluminum
alloys. In static indentation, the Martel hardness was determined by the
instrumental indentation method in the microhardness region. The possibility to
use Martel hardness to calculate the ballistic limit for targets from aluminum
alloys is shown.
Experimental Results
The penetration of three aluminum alloys by the KEP was investigated in
the present work, to check the possibility of applying the Martel hardness to the
problem of penetration. The chemical compositions of the alloys, Martel’s
hardness HMR, and Meyer’s hardness HM are presented in table.
Targets of aluminum alloys in the form of sheets 25 mm in thickness were
used. The mass of the KEP was m = 9,6 g, and the diameter of the penetration
channel practically coincided with the diameter of the KEP d = 7,62 mm.
The volume of the penetration channel was calculated as follows:
ldV
4
2π
= , (5)
where l is the depth of penetration of the KEP.
Typical instrumental indentation curves obtained in a Micron Gamma unit
[5] in the Р—h coordinates for three aluminum alloys are shown in fig. 1.
86
Chemical compositions of aluminum alloys investigated in the work, Meyer
hardness HM and Martel hardness HMR; the subscript s corresponds to
static indentation; the subscript d corresponds to the mean hardness
determined in penetration of the target by the KEP, and calculated by (1).
P is the maximum load on indenter
Number
compo-
sition
Chemical composition
НМ, GPa
Р = 150 g
HMRs, GPa
Р = 150 g
HMRs, GPa
Р = 10 kg
HMRd,
GPa
1 Al—4,45Mg—
0,7Mn—0,13Cr
0,99 1,13 1,02 1,18
2 Al—4,45Mg—
0,4Mn—0,3Sc—0,1Zr
1,48 1,68 1,37 1,42
3 Al—6Zn—2,3Mg—
1,5Cr—0,3Sc—0.1Zr
2,07 2,3 2,16 2,0
These curves were used to determine the work of indentation. The volume
of a hardness indentation was calculated by the formula for calculation of a
trihedral pyramid volume.
cShV
3
1
= , where S is the projection area of indentation and hс is the
contact depth of penetration of the indenter.
The kinetic energy of the KEP was calculated as
2
υ2mE = , where υ is the
velocity of the KEP before the impact collision with the target. The energy
expended on the formation of a unit volume of the penetration channel was
considered as the dynamic hardness HMRd and calculated by the formula
V
mHMRd 2
2υ
= , (6)
The values of HMRd are shown in fig. 2 as a function of the velocity of the
KEP v for 3 aluminum alloys.
Fig. 1. Instrumental indentation curves of
aluminum alloys (the numbers of the
alloys correspond to those in table).
Fig. 2. Dependence of the dynamic
Martel hardness HMRd , determined in
penetration of KEP into the targets
according to (6), on the velocity υ.
P, g
H
M
R d
, G
Pa
h, µm , m/s
87
88
The instrumental Martel hardness HMRs and Meyer hardness HM were
determined at a maximum load on the indenter equal to Р = 150 g. The obtained
results (see Table 1) showed that for the investigated aluminum alloys, relation
(2) is satisfied for K ≈ 1,13. The Meyer hardness НМ was also determined with
a Vickers hardness tester under a load of Р = 10 kg, and the Martel hardness
HMRs was calculated by formula (2) under the assumption that K = 1,13 as in
the case of a smaller load on the indenter.
Discussion of Results
The Meyer hardness НМ is the force approach to the indentation problem
when one determines a maximum value of the mean contact pressure at which
the penetration of the indenter terminates. The Martel hardness can be
considered as the energy approach to the indentation problem, for which the
work expended for the formation of the unit of a hardness indentation volume is
determined.
In physics of strength, both the force and the energy approaches are often
applied to the same problem and enable one to reveal more completely the
essence of the process. For instance, in the problem of development of a crack
and the fracture toughness, the energy approach was developed by
Griffith [6] and Orovan [7], and the force approach was developed by Irvin [8].
The force approach is based on the laws of mechanics, and the energy
approach uses the notions of the energy balance.
The use of both the force and the energy approaches extends the possibility
of using the notion of hardness in different physical processes.
For instance, it is practically impossible to compute the Martel hardness by
using the standard methods of measuring the static hardness by a rigid indenter
(the Vickers method, Brinnel method, etc.) because these methods do not enable
one to determine the energy A expended on the formation of an indentation.
An estimate of the quantity A in static indentation (under the assumption that
P ~ dn, where d is the diameter of the indentation of a spherical indenter or a
diagonal of the indentation of a pyramidal indenter) given in [9], does not take
into account that, in the indicated methods, the loading of the indenter by a
maximum load Р is followed by a hold, during which the hardness indentation
increases at Р = const, and some work is expended on this process as well. This
is why the conclusion that HMR = HM (i. e., the conclusion that the constant K
is equal to 1 in formula (2)) made in [9] does not agree with results presented in
table. In other words, K obtained in instrumental indentation is somewhat larger
than 1 precisely due to holding indenter under Р = const. As is seen from table,
for aluminum alloys, we have К ≈ 1,13. It is clear, that in penetration of the
KEP into a target, the notion of Meyer hardness loses its meaning because this
hardness is determined by the value of residual (plastic) strain at the moment
when the indenter under a load Р stops, but in the penetration process the KEP
does not stop in the surface layer.
However, the efficiency of using the fairly simple technique of measuring
the Meyer hardness by different rigid indenters (spheres, trihedral or tetrahedral
regular pyramids, cones, etc.) for the characterization of properties of the
materials is beyond any doubt.
Static indentation by a rigid indenter with the determination of the Meyer
hardness makes it possible to determine the average contact stress and calculate
not only the flow stress from it, but also determine a number of other mechanical
characteristics of materials, e. g., to construct stress-strain curves, determine
the strain hardening and plasticity characteristic, estimate the fracture
toughness, etc. [10, 11]. In [12], it was shown that, in aluminum alloys, around
the channel of penetration, a disperse granular nonequiaxial structure and a
dislocation cellular substructure, which are typical for metals deformed in
compression by 70%, are formed. However, this does not enable us to
determine the Meyer hardness НМ in the case of penetration a target by a KEP.
At the same time, experimental results obtained in the present work show
that the Martel hardness can be used to describe the process of penetration of
the KEP.
It is seen from fig. 2 that the Martel hardness HMRd, that is determined in
the process of penetration the target by the KEP , is practically independent on
the velocity υ of the KEP in the investigated range of used values of υ. This
enables us to determine the critical velocity of penetration (ballistic limit) from
the relation
m
ldHMR
c 2
πυ
2
d= . (7)
The critical thickness of the target l, which will be penetrated at a velocity υ
of the KEP, can be determined from the same relation at a known velocity υ . At
the same time, it follows from fig. 2 that there exists some tendency to decrease
of HMRd as the velocity υ of the KEP diminishes. A comparison of the value of
HMRs (obtained under static loading of the indenter) and HMRd shows that for
alloys 1 and 2, HMRd is somewhat higher than HMRs (table). Under static
loading of the indenter, its velocity can be estimated from the relation
t
h
=sυ
(where h is the maximum displacement of the indenter and t is the loading
time). The estimate shows that υs = 5×10-7 m/s, i. e., it is smaller by 9 orders
than that in the case of penetration of the target by the KEP.
As is seen from (2), HMR ~ HM. However, it is known that НМ is
proportional to the flow stress σs and that σs increases with the strain rate [10,
13]. For this reason, the fact that HMRd is somewhat larger than HMRs seems to
be natural. The indicated relationship between HMRd and HMRs is observed for
alloys 1 and 2, whereas for the hardest alloy 3, HMRd is even slightly lower than
HMRs. It can be assumed that, for this alloy, the condition of determination of
the Martel hardness cannot be satisfied, i. e., the KEP cannot be absolutely
undeformed.
At the same time, there exists one more factor that can lead to a difference
between the values of HMRd and HMRs, namely, the scale dependence of the
hardness (indentation size effect), which manifests itself to the highest degree
for nanohardness [14], but some decrease in НМ with increasing load Р is
observed in micro- and macro hardness regions as well. For the hardest alloy 3,
the scale dependence of the hardness must be stronger than that for the softer
alloys 1 and 2 because the size of indents for harder alloys is smaller than that
for softer alloys.
It should also be noted the important results of work [15], in which ballistic
limit for some aluminum alloys was determined for two different KEPs.
89
The values of HMRd calculated by eq. (6) for the data of work [15] were found
to be practically equal for KEPs of different diameter and mass. For instance,
for 5083 aluminum alloy, at υс = 722 m/s, HMRd appears to be equal for the
KEP with m = 44,9 g, d = 12,9 mm and l = 59,7 and the KEP with m = 10,4 g,
d = 7,84 мм, and l = 35. For both KEPs, HMRd ≈ 1,43 GPa. Note also that alloy
2, used in the present work, has a chemical composition close to that of 5083
alloy and its value of HMRd (1,42 GPa) is practically equal to that of 5083 alloy.
Conclusion
It has been proposed to consider the penetration of the target by the
undeformable kinetic energy projectile as a process of "deep" indentation and
determine the Martel dynamic hardness HMRd for this target from the relation
V
mvHMR
2
2
d = , where m is the mass of the KEP, υ is its rate before impact
collision with the target, and V is the volume of the penetration channel. By the
example of aluminum alloys, it has been shown that HMRd depends slightly on
the velocity of the KEP υ. The static Martel hardness HMRs is related to the
Meyer hardness HM by the simple relation HMRs = КHM, where K is somewhat
larger than 1.
If HMRd of the target has been determined , the ballistic limit for this target
with a given thickness l can be calculated by equation (7) for the KEP with a
given mass and diameter.
1. Martell R. Commision de Methodes d’Essai des Materiaux de Construction. —
Paris. — 1895. — 3. — 261 p.
2. Tabor D. The hardness of metals. — Oxford: Clarendon Press, 1951.
3. Kohlhöfer W., Penny R. K. Dynamic hardness testing of metals // Ynt. Y. Pres. Ves.
and Piping 61. — 1995. — P. 65—75.
4. Oliver W. C., Pharr G. M. A new improved technique for determining hardness and
elastic modulus using load and sensing indentation experiments // J. of Mater.
Res. — 1992. — 7, No. 6. — P. 1564—1582.
5. Gnatowicz S. R., Shmarov V. N., Zakiev I. M., Maistrenko Y. N. Control of physical
and mechanical properties of solid surfaces by methods of scanning local and micro
/ nano indentation // Nanotechnology: Sat. Reports. Kharkov Nanotechnology
Assembly 2008. — Kharkov: KIPT, 2008. — 1. — P. 243—246.
6. Griffith A. A. The phenomena of rupture and flows in solids // Phil. Trans., Roy.
Soc. — 1920. — A 221. — P. 163.
7. Orovan E. Fracture and strength of solids rep // Progr. Phys. — 1949. — 12. — P. 185.
8. Irvin G. R. Fracturing of Metals. — Cleveland. Ohio: ASM, 1948. — 147 p.
9. Grigorovich V. K. Hardness and Microhardness of Metals [in Russian]. —
Moscow: Nauka, 1976.
10. Milman Yu. V. Physics of the hardness of materials and new possibilities of the
indentation technique // Sintering. — 2000. — P. 133—146.
11. Milman Yu. V. New methods of micromechanical testing of materials by local
loading with a rigid indenter // Advanced Mater. Science: 21st century. —
Cambridge international science publishing, 1998. — P. 638—660.
12. Milman Yu. V., Chugunova S. I., Goncharova I. V. et al. Physics of deformation
and fracture at impact loading and penetration // Int. J. Impact Engineering. —
2006. — 33. — P. 452—462.
13. Trefilov V. I., Milman Yu. V. The influence of temperature and structure factors on
mechanical properties of refractory materials // Proc. of the 10th Plansee Seminar,
1981. — 2. — P. 15—26.
90
14. Milman Yu. V., Golubenko A. A., Dub S. N. Indentation size effect in nanohardnes //
Acta Mater. — 2011. — 59. — P. 7480—7487.
15. Tyrone L. Jones and Richard D. DeLorme. Development of a ballistic specification
for magnesium alloy AZ31B // Army Res. Laboratory USA—ARL—TR—4664. —
December 2008.
Застосування динамічної твердості за Мартелем
до проблеми проникнення
Ю. В. Мільман, В. А. Гончарук, Л. В. Мордель
Проникнення недеформівним кінетичним індентором (KEP) в цілому можна
розглядати як “глибоке” індентування. На прикладі алюмінієвих сплавів було
показано, що динамічна твердість за Мартелем HMRd може бути використана
для опису цього процесу. HMRd може бути розрахована із співвідношення
V
mv
dHMR
2
2
= , де т — маса KEP, υ — швидкість перед зіткненням з мішенню
і V — об’єм каналу проникнення. Балістичну межу VC мішені із заданою
товщиною l можна розрахувати по рівнянню
m
lddHMR
c 2
2π
υ = для KEP з даної
масою т і діаметром d.
Ключові слова: проникнення, твердість по Мартелю, твердість від глибини
проникнення, динамічна твердість.
Применение динамической твердости по Мартелю
к проблеме проникновения
Ю. В. Мильман, В. А Гончарук, Л. В. Мордель
Проникновение недеформируемым кинетическим индентором (KEP) в целом
можно рассматривть как "глубокое" индентирование. На примере алюминиевых
сплавов показано, что динамическая твердость по Мартелю HMRd может
быть использована для описания этого процесса. HMRd может быть
рассчитана из соотношения
V
mvHMRd 2
2
= где т — масса KEP, υ — скорость
перед столкновением с мишенью и V — объем канала проникновения.
Балистический предел VC мишени с заданной толщиной l можно рассчитать по
уравнению
m
ldHMRd
c 2
πυ
2
= для KEP с данной массой т и диаметром d.
Ключевые слова: проникновение, твердость по Мартелю, твердость от глубины
проникновения, динамическая твердость.
91
|
| id | nasplib_isofts_kiev_ua-123456789-63536 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | XXXX-0048 |
| language | English |
| last_indexed | 2025-12-07T18:21:19Z |
| publishDate | 2012 |
| publisher | Інститут проблем матеріалознавства ім. І.М. Францевича НАН України |
| record_format | dspace |
| spelling | Milman, Yu.V. Goncharuk, V.A. Mordel L.V. 2014-06-03T13:35:51Z 2014-06-03T13:35:51Z 2012 Application of the Martel dynamic hardness to the penetration problem / Yu.V. Milman, V.A. Goncharuk, L.V. Mordel // Электронная микроскопия и прочность материалов: Сб. научн . тр. — К.: ІПМ НАН України, 2012. — Вип. 18. — С. 85-91. — Бібліогр.: 15 назв. — англ. XXXX-0048 https://nasplib.isofts.kiev.ua/handle/123456789/63536 620.015 Penetration of the undeformable kinetic energy projectile (KEP) into the target is considered as the “deep” indentation. It was shown by the example of aluminum alloys that the Martel dynamic hardness HMRd can be used for description of this process. Проникнення недеформівним кінетичним індентором (KEP) в цілому можна розглядати як “глибоке” індентування. На прикладі алюмінієвих сплавів було показано, що динамічна твердість за Мартелем HMRd може бути використана для опису цього процесу. Проникновение недеформируемым кинетическим индентором (KEP) в целом можно рассматривть как "глубокое" индентирование. На примере алюминиевых сплавов показано, что динамическая твердость по Мартелю HMRd может быть использована для описания этого процесса. en Інститут проблем матеріалознавства ім. І.М. Францевича НАН України Электронная микроскопия и прочность материалов Application of the Martel dynamic hardness to the penetration problem Застосування динамічної твердості за Мартелем до проблеми проникнення Применение динамической твердости по Мартелю к проблеме проникновения Article published earlier |
| spellingShingle | Application of the Martel dynamic hardness to the penetration problem Milman, Yu.V. Goncharuk, V.A. Mordel L.V. |
| title | Application of the Martel dynamic hardness to the penetration problem |
| title_alt | Застосування динамічної твердості за Мартелем до проблеми проникнення Применение динамической твердости по Мартелю к проблеме проникновения |
| title_full | Application of the Martel dynamic hardness to the penetration problem |
| title_fullStr | Application of the Martel dynamic hardness to the penetration problem |
| title_full_unstemmed | Application of the Martel dynamic hardness to the penetration problem |
| title_short | Application of the Martel dynamic hardness to the penetration problem |
| title_sort | application of the martel dynamic hardness to the penetration problem |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/63536 |
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