Effect of the Static Strength of a Metal on Long-Rod Penetration at Low Velocities
The effect of the static strength (yield stress) of a material on major penetration parameters (specific work of deformation and penetration force determined by the pressure at the rod-cavity contact surface) was examined. The analytical equations and quantitative estimates of specific work spent fo...
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Інститут проблем міцності ім. Г.С. Писаренко НАН України
2000
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| Цитувати: | Effect of the Static Strength of a Metal on Long-Rod Penetration at Low Velocities / G.V. Stepanov, V.V. Kharchenko // Проблемы прочности. — 2000. — № 1. — С. 93-110. — Бібліогр.: 17 назв. — англ. |
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irk-123456789-461852013-06-28T16:08:13Z Effect of the Static Strength of a Metal on Long-Rod Penetration at Low Velocities Stepanov, G.V. Kharchenko, V.V. Научно-технический раздел The effect of the static strength (yield stress) of a material on major penetration parameters (specific work of deformation and penetration force determined by the pressure at the rod-cavity contact surface) was examined. The analytical equations and quantitative estimates of specific work spent for formation of spherical and cylindrical cavities and of the pressure at the surface of such cavities at their static expansion were obtained for the rigidplastic and elastoplastic models of the material. The estimates of specific work of deformation at the formation of the spherical cavity in the plate and of the penetration of a rigid rod are shown to be adequate. The influence of strain hardening on a maximum radial stress on the cavity surface was evaluated. The resistance to penetration and the expansion of a cavity are shown to be strongly influenced by the elastic compressibility of a material in the inelastic region at low penetration velocities. The kinetics of the stress-strain state of the material at the formation of the cavity was analyzed for the case of considerable deformation of a rod at penetration. Рассмотрено влияние статической прочности (предела текучести) материала на основные параметры процесса проникания - удельную работу образования каверны и давление на поверхности контакта. Получены аналитические выражения и количественные оценки удельной работы, затраченной на образование сферической и цилиндрической каверн, и давления на поверхности таких каверн при их статическом расширении для жесткопластической и упругопластической моделей материала. Показана адекватность оценок удельной работы деформирования пластины при образовании сферической каверны и при образовании каверны в результате проникания жесткого стержня. Оценено влияние деформационного упрочнения материала на максимальное радиальное напряжение на поверхности каверны. Показано, что при низких скоростях проникания существенное влияние на сопротивление прониканию и расширение каверны оказывает упругая сжимаемость материала в области неупругих деформаций. Проведен анализ кинетики напряженно-деформированного состояния материала при проникании деформируемого стержня. 2000 Article Effect of the Static Strength of a Metal on Long-Rod Penetration at Low Velocities / G.V. Stepanov, V.V. Kharchenko // Проблемы прочности. — 2000. — № 1. — С. 93-110. — Бібліогр.: 17 назв. — англ. 0556-171X http://dspace.nbuv.gov.ua/handle/123456789/46185 539.3 en Проблемы прочности Інститут проблем міцності ім. Г.С. Писаренко НАН України |
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Научно-технический раздел Научно-технический раздел Stepanov, G.V. Kharchenko, V.V. Effect of the Static Strength of a Metal on Long-Rod Penetration at Low Velocities Проблемы прочности |
| description |
The effect of the static strength (yield stress) of a material on major penetration parameters (specific work of deformation and penetration force determined by the pressure at the rod-cavity contact surface) was examined. The analytical equations and quantitative estimates of specific work spent for formation of spherical and cylindrical cavities and of the pressure at the surface of such cavities at their static expansion were obtained for the rigidplastic and elastoplastic models of the material. The estimates of specific work of deformation at the formation of the spherical cavity in the plate and of the penetration of a rigid rod are shown to be adequate. The influence of strain hardening on a maximum radial stress on the cavity surface was evaluated. The resistance to penetration and the expansion of a cavity are shown to be strongly influenced by the elastic compressibility of a material in the inelastic region at low penetration velocities. The kinetics of the stress-strain state of the material at the formation of the cavity was analyzed for the case of considerable deformation of a rod at penetration. |
| format |
Article |
| author |
Stepanov, G.V. Kharchenko, V.V. |
| author_facet |
Stepanov, G.V. Kharchenko, V.V. |
| author_sort |
Stepanov, G.V. |
| title |
Effect of the Static Strength of a Metal on Long-Rod Penetration at Low Velocities |
| title_short |
Effect of the Static Strength of a Metal on Long-Rod Penetration at Low Velocities |
| title_full |
Effect of the Static Strength of a Metal on Long-Rod Penetration at Low Velocities |
| title_fullStr |
Effect of the Static Strength of a Metal on Long-Rod Penetration at Low Velocities |
| title_full_unstemmed |
Effect of the Static Strength of a Metal on Long-Rod Penetration at Low Velocities |
| title_sort |
effect of the static strength of a metal on long-rod penetration at low velocities |
| publisher |
Інститут проблем міцності ім. Г.С. Писаренко НАН України |
| publishDate |
2000 |
| topic_facet |
Научно-технический раздел |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/46185 |
| citation_txt |
Effect of the Static Strength of a Metal on Long-Rod Penetration at Low Velocities / G.V. Stepanov, V.V. Kharchenko // Проблемы прочности. — 2000. — № 1. — С. 93-110. — Бібліогр.: 17 назв. — англ. |
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Проблемы прочности |
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AT stepanovgv effectofthestaticstrengthofametalonlongrodpenetrationatlowvelocities AT kharchenkovv effectofthestaticstrengthofametalonlongrodpenetrationatlowvelocities |
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2025-07-04T05:20:39Z |
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2025-07-04T05:20:39Z |
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1836692478467506176 |
| fulltext |
UDC 539.3
Effect of the Static Strength of a Metal on Long-Rod Penetration at
Low Velocities
G . V . S te p a n o v a n d V. V . K h a r c h e n k o
Institute of Problems of Strength, National Academy of Sciences of Ukraine, Kiev,
Ukraine
Влияние статической прочности металла на проникание
длинного стержня с низкой скоростью
Г . В. С т е п а н о в , В. В. Х а р ч е н к о
Институт проблем прочности НАН Украины, Киев, Украина
Рассмотрено влияние статической прочности (предела текучести) материала на основные
параметры процесса проникания - удельную работу образования каверны и давление на
поверхности контакта. Получены аналитические выражения и количественные оценки
удельной работы, затраченной на образование сферической и цилиндрической каверн, и
давления на поверхности таких каверн при их статическом расширении для жестко
пластической и упругопластической моделей материала. Показана адекватность оценок
удельной работы деформирования пластины при образовании сферической каверны и при
образовании каверны в результате проникания жесткого стержня. Оценено влияние дефор
мационного упрочнения материала на максимальное радиальное напряжение на поверхности
каверны. Показано, что при низких скоростях проникания существенное влияние на сопро
тивление прониканию и расширение каверны оказывает упругая сжимаемость материала в
области неупругих деформаций. Проведен анализ кинетики напряженно-деформированного
состояния материала при проникании деформируемого стержня.
Introduction. The longitudinal force, affecting a long rod at its penetration
in an elastoplastic medium, is usually represented as the sum of two components,
one of them is associated with the hydrodynamic pressure at the contact surface of
a rod head and the other with the resistance of a material to plastic deformation.
Though the real process of penetration should account for the nonlinear
interaction of those two components, their independent representation can greatly
simplify the analysis of penetration.
The static resistance of a plastic medium to the penetration of a long rod is a
parameter widely used for evaluating the strength effect on penetration at high
velocities [1-7]. This component of resistance to penetration and its association
with the characteristics of a material, determining its mechanical behavior, is
examined in a number of publications [8-11]. However, various models of
penetration, models of an elastoplastic material used for its simulation, and
methods of solution of mathematical equations resulted in different values of
resistance and dimensions of a plastic region [3, 5, 8 , 11-14]. It will suffice to
mention that the pressure at the interface at penetration is assumed to be equal to
the pressure necessary for the expansion of a cylindrical or spherical cavity in the
unlimited volume of a material, or to some intermediate value [13, 15, 16].
© G. V. STEPANOV, V. V. KHARCHENKO, 2000
ISSN 0556-171X. Проблемы прочности, 2000, N 1 93
Simplified rigid-plastic or elastoplastic models of a material are often used
for the analysis of penetration. The methods of calculation include computer
simulations and various analytical methods, with some simplifications of real
plastic flow [4, 11, 12]. The main calculated parameters are the pressure onto the
contact surface and the sizes of an elastoplastic interface.
Intense deformations in real materials influence their resistance to
deformation, which varies with strain hardening and/or softening (the latter is the
result of structural changes, damage, and heating of a material at plastic flow).
Therefore, the stress-strain kinetics and the distribution of stress, strain, and a
strain rate near the rod head are important for the evaluation of strength at
penetration, but those effects are not investigated thoroughly enough.
The resistance of a plastic medium to the static penetration of a rod is
determined by its resistance to plastic deformation accounting for a real 3D
stress-strain state in the material near the rod head. Modern models of
homogeneous viscoplastic materials allow one to relate the stress intensity o t , to
the plastic strain intensity e t , the plastic strain rate intensity e ' , the pressure p,
and the temperature T, increasing at penetration as the result of thermal effects at
plastic flow:
o i = o (Ei , e'i, p , T ).
The longitudinal force, affecting the rod at penetration in a plastic material,
is the integral characteristic of the pressure distribution near the contact surface of
interacting bodies. Plastic flow criterion-based approaches [8 , 10] are proposed to
evaluate the magnitude of the pressure and its distribution. According to known
findings, the pressure at the contact surface at long-rod penetration is connected
with the strength of a plastic medium, and the pressure onto the surfaces of a
spherical or cylindrical cavity at its static expansion can be taken as a good
approximation. To get the more reliable value of resistance to penetration, one
should account for the changes in the stress-strain state of the material at its flow
along the rod head. From general representations, in the layer of the material
located between the two planes perpendicular to the rod axis, the cylindrical
cavity formed is caused by the development of axisymmetric flow, which includes
the displacements of the material in radial and axial directions. The analysis of the
flow and changes in the resistance to penetration is one of the main goals
introduced below.
This paper is confined only to the analysis of certain penetration effects due
to the static strength inherent in an elastoplastic medium (without strain
hardening). The penetration is an essentially dynamic process, but at near
threshold velocities, at the last stage of high-velocity penetration, and at the
inertial expansion of a cavity, the plastic flow is mainly controlled by the static
strength of the medium.
1. Pressure onto the Surface of a Spherical Cavity a t Its Expansion
under Static Loading.
1.1. Rigid-Plastic M aterial. According to known approaches, the equation
of static equilibrium for the elementary volume of a spherical layer in the ideally
plastic region (r < re, re is the radius of the elastoplastic interface), using the
G. V. Stepanov, V. V. Kharchenko
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Effect of the Static Strength of a Metal...
condition of plasticity (o r — o t ) = 2 t y ( ty is the yield stress in shear for a
material without strain hardening), can be written in the form:
do r / dr = —2(o r — o t ) / r = — 4 t y / r. (1)
Integrating this equation with the account of o r = —( 4 / 3 ) ty at r = re for the
elastoplastic interface, we obtain the distribution of radial stress components in
the plastic region:
o r = —( 4 /3 ) t y [1— ln(r / re)]. (2)
The stress and plastic strain components in spherical coordinates (Fig. 1)
o 1 = o r ; o 2 = o 3 = o t = o r + 2 tY ; t 12 = t 23 = T 31 = 0;
£ 1 = £r; £2 = £3 = £t = —£r /2 ; £12 = £23 = £31 = 0
correspond to the stress and plastic strain intensities in the plastic region:
o 2 = [(o 1 —o 2 )2 + (o 2 — o 3 )2 + (o 3 — o 1)2 + 3( t2 2 + t 23 + T 21) ] / 2 = 4 t 7 ;
(3)
£ 2 = (2 /9 )[( £1 —£2)2 + (£2 —£3)2 + (£ 3 —£1)2 + 3(£12 + £ 23 + £ 31)]= £;2-
If we take into account that the stress intensity in a material at the initiation of
plastic flow in a 1D stress state is equal to the yield strength o y (at the strain £y)
according to the Mises criterion of plastic flow, it follows from Eq. (3):
t y = o y /2 .
Fig. 1. Stress state near a spherical cavity at its expansion.
Neglecting the elastic volume compression in the plastic region, the
expansion of a spherical cavity to a given radius rc is accompanied with the
expansion of an elastoplastic interface to the radius re. From the equality of the
ISSN 0556-171X. npoôëeubi npounocmu, 2000, № 1 95
3
cavity volume (4n / 3)rc in an incompressible (at plastic flow) material and the
change of the volume inside the elastoplastic interface with the radius re under
the pressure at this interface p e = (2 /3 )o y (it causes the initiation of plastic
flow at the tangential elastic strain Ete), we get:
(rc / re )3 = (3 / 2 )ey ■ (4)
The pressure T3 onto the surface of an expanding spherical cavity is obtained
from Eqs. (2), (3), and (4) in the form:
T3 = - o rc = (2 / 3)o y {1 - ln[(3 / 2)e y ]}■ (5)
This value does not depend on the cavity radius owing to the constant ratio of the
interface radius to the cavity radius.
If at a 1D stress state the conventional yield stress corresponds to the
longitudinal plastic strain Ey = 0 .2 % (usually used for the evaluation of yield
stress), from Eq. (5) we get the pressure onto the cavity surface at its expansion:
T3 = (2 /3)ff Y [1 - ln 0.003]= 4.55oY.
The specific work (per unit volume of a cavity) spent for the formation of a
cavity with the volume W is constant during its expansion and coincides in
magnitude with the pressure T3 necessary for the cavity expansion:
av = dA3 / dW = 4 n R 2dRT3 / 4nR 2dR = T3 . (6 )
1.2. Elastoplastic M aterial.
1.2.1. Evaluation o f the Radius o f a Spherical Elastoplastic Interface. The
radial pressure at the elastoplastic interface (r = re ) under static loading of the
surface of an expanding spherical cavity is calculated from the solution of an
elastic problem, taking into account the compressibility of a material in the plastic
region. The elastic compressibility in the plastic region influences the relative
radius of the elastoplastic interface a = re / rc. The simplified evaluation of this
effect is based on the equality of the mass within the spherical elastoplastic
boundary with the radius re at the final stage (after the formation of a cavity with
the radius rc) and the same mass (before the cavity formation) bounded by the
spherical surface of the radius
re0 = re(1 - E tY )
(EtY = u / re is the tangential strain induced by the radial displacement u of the
material near the elastoplastic interface).
G. V. Stepanov, V. V. Kharchenko
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Effect of the Static Strength of a Metal...
The spherical symmetry of flow results in the elastoplastic transition at the
stresses o r = — (2 /3 )o Y; o t = (1 /3 )o Y .T hus,
e tY = [o t - v (o r + o t ) ] / E = (1+ v)eY■
To determine the distribution of the pressure p along the radius in the plastic
region (rc < r < re), we use the distribution of radial stresses derived for an
incompressible plastic medium:
o r = - ( 2 / 3 ) o Y [ 1 - ln( r / re ) 3 ]; o t = o r - o Y ■
The pressure p = - o r — (2 / 3)o Y = —2o Y ln(r / re) determines the distribution
of the elastic strain e of volume compression and the density p of a material:
e = p / K = 2(o Y / K )ln (r / re),
p = p 0 (1+ e) = p 0 [1 — 2(oY / K )ln (r / re)].
The equation of mass
r ' e
(4 / 3)P 0n re3 (1—etY )3 = / 4nP r 2Pdr
after the Taylor expansion of its left part, the substitution of e tY = (1 + v )eY and
of the equation for density, takes on the form:
• e
(4 /3 )p 0 n re3[ 1 - (1+ v )eY ]= 4p 0 n j [ 1 - 2( <7 Y / K )ln( r / re )] r 2 dr =
= (4 /3 )p o n {(re3 - rc3 )[1 + ( 2 / 3)(7 y / K ) ] - 2(7 y / K )rc3 ln(re / rc)}.
From this equation, the relation
(1 / a 3 )[1 + 2(1 - 2 v)eY(1 + ln a 3 )] = 3eY(1 - v )
r.c
r'c
is derived, which allows for the nonlinear dependence of the relative radius of an
elastoplastic interface on the strain Ey at yield stress and for the influence of
compressibility (displayed as the effect of the strain Ey and the Poisson ratio v).
At a real value of a > 5 (valid for major metals), the latter relation with an
error not exceeding 2 % can be approximated by
1/ a 3 = 3ey ( 1 - v); T3 = 7 rc = (2 / 3)7 Y{1- ln[3(1- v)ey ]},
ISSN 0556-171X. npoôneMbi npouuocmu, 2000, № 1 97
and if v = 0.5, it corresponds to the calculations for a rigid plastic medium
without the account of the volume compressibility. This result coincides with the
equation derived for the final stage of the spherical cavity expansion [8 ].
An increase in the strength of a material (increase in the strain ey ) as well as
the account of compressibility (v < 0.5) leads to a smaller calculated elastoplastic
interface (decrease in the relative radius a) at the formation of a spherical cavity
and a decrease in the pressure T3 at an expanding cavity surface. A decrease in a
and T3 in comparison with a ’ and T3 for a rigid-plastic medium is presented
below:
a ’ / a = [2(1- v)]1/3, T3 / T3 = {1- ln [(3 /2 )ey ] } /{ l - ln [3 (l- v)ey ]}.
For a steel with the yield stress o y = 1.2 GPa, Young’s modulus E = 210 GPa
(ey = 0.0057), and the Poisson ratio v = 0.3, these values are
a ’ / a = 1.12; T3 / T3 = 1.06.
1.2.2. Strain Distribution near the Spherical Cavity in an Ideally Plastic
Medium. The radial displacement of a material in the expansion of a spherical
cavity is accompanied with intense plastic flow. The logarithmic plastic strain of
the material in the tangential direction as the result of its radial displacement u
from the initial position with the radius r0 is determined by the summation of the
true strain increments det = du / ( r0 + u ).
The tangential strain, allowing for the condition of incompressibility
3 3 3 3 3(ro = (r0 + u ) — rc = r — rc ), is calculated from the equation:
e t = J du / ( r0 + u ) = ln[(1 + u / r0) = — (1/3) ln (1— rc3 / r 3 ).
The plastic strain intensity
ei = 2et = — (2 /3 ) ln (1 — r 1 / r ̂
increases from a very small plastic strain value at the elastoplastic interface to an
unlimited one near the surface of a cavity. A real metal exhibits strain hardening
and structural changes, and an increase in the expansion rate causes the adiabatic
heating of the material essential at large strains. These effects can considerably
change the strength of the material and its resistance to penetration. However,
their evaluation requires additional studies. Thus, the simplified elastoplastic
model, if applied to large strains near the cavity surface, should be used with
certain caution.
The elastic volume compressibility of a material in the plastic region reduces
the extent of plastic strain corresponding to the volume changes. This increases a
strain gradient in the plastic region.
G. V. Stepanov, V. V. Kharchenko
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Effect of the Static Strength of a Metal...
1.2.3. Surface Pressure fo r an Expanding Cavity in the Plastic Medium with
Strain Hardening. The effect of strain hardening is examined in many papers,
however, analytical solutions are limited in number. The analysis of linear strain
hardening [8 ] is not applicable to the description of flow at large strains. Really,
an increase in strain should induce the asymptotic growth of resistance to the
utmost value o *, which cannot exceed the theoretical strength of a material,
neglecting softening (as the result of structural changes, damage, and heating).
The stress-strain relation (for the logarithmic plastic strain et = 2et = ln (r / r0)
in the form
o j- = o * — (o * — o y )e x p ( -Aet )
corresponds to o i = o y at small plastic strains ei = 0 (at the initial stage of
strain hardening) and to the utmost stress o t ~ o * at large strains. The relation
for the strain hardening module
M = do j- / de j- = M o exp(—Ae t ), M 0 = A(o * — o Y )
permits to represent the stress-strain relation for spherical symmetry in the form:
oj- = o * — ( M 0 / A)exp(—Aet ) = o * — ( M 0 / A )exp[A ln(r0 / r ) 2 ] =
= o * — (M 0 / A)(r0 / r )2A .
For this relation of strain hardening, the differential equation of equilibrium has
the form:
do r / dr = 2o j- / r = 2{o * / r — (M 0 / A)[1 — (rc / r ) 3 ]2A/3 / r}
3 3 3 3 3(if the equations of incompressibility r — rc = r0 or (r0 / r ) = 1 — (rc / r )
are used). The radial stress in a spherical layer is determined by integration of the
above equation.
For A = 3 /2 (example used only for the evaluation of a strengthening
effect), from the simplified differential equation
d o r / dr = 2 [oY + (o * — o Y)(rc / r ) 3 ] / r ,
we get the stress distribution:
o r = 2 [o Y ln r — (o * — o Y)(rc / r ) 3 /3 ]+ C .
At the elastoplastic interface,
o r = —(2 /3 )o Y = 2 [oY ln re + (o * — o Y)(rc / re ) 3 /3 ]+ C ,
ISSN 0556-171X. npodneMbi npouuocmu, 2000, № 1 99
G. V. Stepanov, V. V. Kharchenko
hence
o r = - ( 2 /3 ) { o Y [ l- ln (r / re ) 3 ] + (a * - a y )[(rc / r )3 - (rc / re ) 3 ]}■
The maximum radial stress at the cavity surface, corresponding to (rc / r ) = 1,
0 r max = - ( 2 /3 ) { o Y [1 - ln( rc / re )3 ]+ ( 0 * - 0 Y ) [ 1 - ( rc / re )3]}
is essentially higher than for an ideal plastic medium, and the behavior of a
material with intense strain hardening is mainly dependent on the utmost stress
0 * ■
2. Pressure onto the Surface of an Expanding Cylindrical Cavity.
2.1. Rigid-Plastic M aterial. For the elementary volume of the cylindrical
layer of a material in the inelastic region (at rc < r < re), using the equation of
static equilibrium and the condition of plastic flow for the case of cylindrical
symmetry (o r - o t ) = 2 t y , we get
do r / dr = - ( o r - o t ) / r = - 2 t y / r (7)
similar to Eq. (1). Integrating this equation (with the account that o re = - T y at
the cylindrical elastoplastic interface, r = re) for the case of the infinite external
boundary of the elastic region, we obtain the distribution of radial stresses in the
form
o r = - t y [1 - ln( r / re ) 2 ]. (8)
The stress and plastic strain components in cylindrical coordinates (Fig. 2)
for a material incompressible at plastic flow (v = 0.5)
o 1 = o r ; o 2 = (o r + o 3 ) / 2 ; o 3 = o t = o r 2 ty ; t 12 = t 23 = t 31 = 0 ;
£1 = £ r; £2 = 0; £3 = £t = - £r ; £12 = £23 = £31 = 0
correspond to the stress and plastic strain intensities in the plastic region:
o 2 = [(o 1 - o 2 )2 + ( o 2 - o 3 )2 + ( o 3 - o 1)2 +
+ 3( t2 2 + t 23 + T 31) ] / 2 = 3 t2 ;
£2 = (2 / 9) [ (£1 £2 ) 2 + (£2 - £3 ) 2 + (£3 - £ 1 ) 2 + (9)
+ 3(£12 + £23 + £31 )] = ( 4 / 3)£2 .
Assuming that the stress intensity o ,• at plastic flow in a 1D stress state
(o i = o y ) and at cylindrical symmetry are equal, we obtain from Eq. (9):
T Y = o Y/ V3.
100 ISSN 0556-171X. npodneMbi npouuocmu, 2000, № 1
Effect o f the Static Strength o f a Metal
Fig. 2. Stress state near a cylindrical cavity at its expansion.
The radius of the elastoplastic interface re is calculated from the equality of
the cylindrical cavity volume with the radius rc and the change of the cavity
volume in an elastic material with the radius re under the surface pressure
p e = TY, which initiates plastic flow:
(rc / re )2 = 2£e = ^ 3 £y ■ (10)
The pressure onto the surface of a cylindrical cavity, formed in the plane layer of
a material by its radial displacement, is derived using Eqs. (8), (9), and (10):
T 2 = - 0 rc = ( o y / 2)[1 — ln(V3 £y )]■ (11)
At eY = 0.002, this pressure
T2 = (o Y /(V3)[1 — ln0.0035]= 3.85o Y
does not depend on the cavity radius (due to the constant ratio between the radii
of the elastoplastic interface and the cavity). The specific work spent for an
increase in the cavity volume
a 2 = dA2 / dW = 2nRdRT2 / 2RdR = T2 (12)
is constant for the expansion of a cavity and is 15% less than the value for the
expansion of a spherical cavity (given in the previous section).
The use of the Tresca criterion of plastic flow results in a lower value of
pressure necessary for the expansion of the cavity. For this criterion, the
maximum shear stress t y = o y / 2 corresponds to the maximum strain in shear
Y y = (3 /2 )e Y, and instead of Eq. (11), we should take
T2 = —o rc = ( o Y / 2){1 — ln[(3/2)£Y ]}. (13)
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G. V. Stepanov, V. V. Kharchenko
At £y — 0.002, the calculated pressure for the cavity expansion is approximately
25% less than T3 in Eq. (5) for a spherical cavity:
2.2. Elastoplastic M aterial.
2.2.1. Evaluation o f the Radius o f a Cylindrical Elastoplastic Interface. The
radial pressure at the elastoplastic interface ( r — re ) under the static loading of the
surface of a cylindrical cavity is determined from a known solution of the elastic
problem for an unlimited medium. The elastic volume compressibility of a
material in the plastic region affects the relative radius of the elastoplastic
interface a — re / rc at the formation of a cavity induced by the radial
displacement of the material. The approximate evaluation of this effect proceeds
from the equation of mass equality within the cylindrical elastoplastic interface
with the final radius re and that of the initial radius re0 :
(£ty = u / re is the tangential strain as the result of the radial displacement u of a
material at the elastoplastic interface).
According to the Mises criterion, for the cylindrical symmetry of plastic flow
with the stresses at the elastoplastic interface o r = — t y ; o t = t y ; o z = 0 and
the tangential strain £ty = (o t — vo r ) / E = t y (1 + v ) / E, we get:
The distribution of the pressure p along the radius in the cylindrical plastic
region (accounting for o r = — t y [1 — 2 ln(r / re )]; o t = o r + 2 tY; o z = o r + t Y )
determines the distribution of volume compression strains and the density of a
material in the plastic region:
T2 = (o Y /2)[1 — ln 0.003] = 3 .4 o Y.
re0 = re(1 — £tY ) = re [1 — V(2 /3 ) £Y ]
T y — o y / ^/3 — E&y / a/3; &tY — (£y / ^/3)(1 + v ).
p — —2t y ln( r / re )
£ — P / ^ — — (2t y / ^ ) l n ( r / re),
P —Po (1+ £) — P 0 [1 — 2(T Y / ^ ) ln ( r / re)].
The equation of conservation of mass for the layer of unit thickness
re
rr.c
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Effect o f the Static Strength o f a Metal
after the Taylor expansion of its left part and the substitution of relations for
tangential strain and density, takes on the form:
• e
P оп г 'Є [1 l£ ty ] = 2p 0n f [1 - 2(Ty / K ) ln (r / Ге )]rdr =
P 0п{(ГЄ — rC )[1 + (TY / K)] + 2(T Y / K )rC ln (rc / re )}.
гС
From the above equation, we have
1 / a 2 = (£Y / л/3)(5 — 4v)/{1 + (ey ̂V3)3(1 — 2v)[1+ l n a 2 ]}.
At a > 5 (valid for major metals), the latter equation is equal to
1/ a 2 = (ey / V3)(5 — 4v ),
with sufficient accuracy, and it coincides with Eq. (10) for a rigid-plastic medium,
if v = 0.5. An increase in the strength of a material (increase in the strain Ey) as
well as the account of the volume compressibility leads to a decease in the radius
of an interface and in the pressure onto the cavity surface necessary for its
expansion. A decrease in a and T2 for an elastoplastic medium in comparison
with a ' and T2 for a rigid-plastic one can be calculated from the relations:
a ' / a = [(5— 4 v )/3 ]1/2; T2 / T2 = [1— l n ( V 3 ) ] / { l — ln[ey (5 — 4v)/>/3)]}.
For a steel with о Y = 1.2 GPa, E = 210 GPa (ey = 0.0057), and v = 0.3, these
values are
a ' / a = 1.12; T2 / T2 = 1.04.
Thus, the effect of compressibility on the radius of an elastoplastic interface
is essential, which does not contradict earlier findings [13]. The results of this
analysis are close to the approximate solution for the dynamic expansion of a
cylindrical cavity, based on the similarity solution for plastic flow [11].
2.2.2. Strain Distribution near the Surface o f an Expanding Cylindrical
Cavity. The radial displacement of a material at the formation of a cavity leads to
intense plastic flow. The logarithmic strain in the tangential direction, caused by
the radial displacement u of a material from the initial radius Г0 , is determined
by summation of the true strain increments dE t = du / ( Г0 + u ). The tangential
2 2component of plastic strain (for zero compressibility, i.e., for r0 = (r0 + u ) —
2 2 2— r0 = r — r0 ) can be calculated by the equation:
Et = f du / ( r0 + u ) = ln [(r0 + u ) / r0 ]= — (1 /2 )ln (1 — r02 / r 2 ).
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The strain intensity et = e t V4 / 3 = ln [(r0 + u ) / r0 ] = — (1 / V3)ln(1 — r02 / r 2 )
is low near the elastoplastic interface. The calculated strains increase to an
unlimited value near the cavity surface. Intense strains near the cavity surface
give rise to strain hardening and an increase in temperature with an increase in a
penetration velocity.
The effects of strain hardening and compressibility are similar to such effects
for the expansion of a spherical cavity.
3. Specific W ork Spent for the Cavity Form ation at Rod Penetration.
The static penetration of a rigid rod in a plastic medium results in a cylindrical
cavity with a diameter approximately equal to the diameter of the rod with the
cross section S . An increase in the penetration depth (at steady-state penetration)
in an infinite plastic medium, accompanied with the radial displacement of a
material (neglecting the friction at the contact surface of the bodies), is
proportional to the strength of the material at plastic flow and is independent of
the shape of the rod head.
An increase in the work dA = SPa dL spent for the deformation of the
medium to increase the penetration depth by dL under the action of the mean
pressure Pa onto the contact surface should be equal to the total work spent for
the formation of a residual cylindrical cavity as the result of axisymmetric plastic
flow in the plane layer of the thickness dL (Fig. 3). The total specific work of the
cavity formation in such a plane layer includes the terms T02 and P02 associated
with the radial and axial displacements of a material in the plastic region.
G. V. Stepanov, V. V. Kharchenko
Fig. 3. Scheme of stresses in the material at long-rod penetration.
The specific works T3 spent for the formation of a spherical cavity (caused
by the radial displacement of a material) and Pa spent for the formation of a
cylindrical cavity at the rod penetration (as the result of the axisymmetric radial
and axial displacements of a material, taking Pa = T02 + P02) can be assumed
equal. From Eq. (5), it follows:
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T3 = Pa = T02 + P02 = (2 / 3)o Y {1 — ln [(3 / 2)eY ]}■ (14)
Hence, the specific work of shear deformation at the cavity formation, as it
follows from (13) and (14), is equal to
which is 25% of the total specific work.
4. Stress-Strain Kinetics for the M aterial near the Contact Surface at
Rod Penetration. The change of the stress-strain state in a volume within the
plastic region is caused by the axial and radial displacements of a material as the
result of its flow at the penetration of a rod. Such displacements are associated
with the stress-strain state if the cylindrical coordinates are used. The radial and
axial displacements and the tensor components of a plastic strain rate for the
material in the elementary ring depend on its initial distance from the axis of the
rod.
The pressure onto the rod head is determined by the stress-strain state of the
material along the zero line of flow adjacent to the contact surface. Along the
section of this line (on the axis of symmetry), from the elastoplastic interface to
the point of branching, the plastic deformation of a material is accompanied with
an increase in the axial compression strain e z and the tangential tensile strains
er = e t . The stress-strain state of the material is similar to that of the expanding
spherical cavity.
The stress and plastic strain intensities from Eq. (3)
determine the specific work (per unit volume of a material) o f plastic
deformation:
With further motion along the zero line of flow away from the point of
branching over the contact surface of interacting bodies (or over the contact
surface with a stagnant zone), the axial and radial displacements of a material
result in the development of 3D strains er , et , ert (strain components er and e t
are not equal, the plastic strain in shear ert is nonzero). It should be noted that
the expansion of a cavity in the radial direction and plastic shear in the axial
direction activate the two groups of main shear planes (rt and rz) with the critical
values of shear stresses which are assumed to be equal:
The stress state in this case is characterized by the tensor components:
P2 = (1 / 6)o y {1 — ln [(3 / 2)eY ]} = (1 / 4)Pa , (15)
o i — o y — 2t y ; e i — eY — 2e
(16)
T rt — (o r o t ) / 2 — t Y; T rz —T Y ■
o r ; o t — o r 2t y ; o z — 0; t rz — t y ,
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G. V. Stepanov, V. V. Kharchenko
which correspond to the stress intensity
° 2 = [(° 1 - ° 2 )2 + (° 2 - ° 3 ) 2 + (° 3 - ° l ) 2 +
+ 6( t 2 2 + T 23 + T 21) ] / 2 = °Y = 6xY
2 2and the shear stress r Y = o ,• / 6 .
The plastic strain tensor components in this case
£r , £t £r , £z 0, £rz, £r̂ £ 0
2 2determine the strain intensity (assuming that £rz = £r )
£ 2 = (2 /9 ) [(£i - £ 2 )2 + (£ 2 - £ 3 )2 + (£ 3 - £ 1)2 +
+ 6( £22 + £T 23 + £21)]= ( 8 / 3 ) £ 2
2 2 and the shear strain £rz = (3 / 8)£,-.
The specific work of plastic deformation in axial shear
ash T rz£rz ^ i£i / 4 , (17)
which is 25% of the total specific work of plastic deformation, in accordance with
the estimation by Eq. (16).
The condition of plastic flow (neglecting strain hardening) is valid for all the
points on the the zero line of flow. This line includes the points on the axis, from
the elastoplastic boundary to the point of branching (with the axial compression
of a material), and the points on the contact surface of interacting bodies (with the
radial and axial displacements of a material). Hence, the resistance of a plastic
medium to the penetration of a rod can be evaluated by the specific work spent for
the formation of a spherical cavity or the work of axial force spent for the
formation of a cylindrical cavity. The evaluation of this characteristic by T3 and
Pa (calculated by Eqs. (14) and (15)) is approximate because of the possible
pronounced effect of strain hardening and the friction on the contact surface. The
account for an error caused by these effects and the real distribution of pressure
would increase the reliability of calculations for determining the penetration depth
in the materials with high strain hardening.
Further investigations of effects, associated with the possible formation of a
stagnant zone at the penetration of rods with a blunt head and studies on the
specific features of plastic flow near the point of branching on the contact surface
are of importance for the reliable prediction of a penetration depth. Some of these
effects were analyzed using the theory of kernel formation [17].
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Effect o f the Static Strength o f a M etal.
Fig. 4. Scheme of plastic flow at long-rod penetration.
5. Deform ation of a Cylindrical Rod at Penetration. The plastic flow in a
long rod at high-velocity penetration leads to its transformation into a tubular
element adjacent to the surface of a cylindrical cavity with the radius rc in a
plastic medium (Fig. 4). The expansion of the tubular element approximately
simulates the deformation of an eroded rod induced by its plastic flow near the
contact surface. The radial stress o r in the wall of this element at its expansion
(under pressure onto the surface of the cylindrical cavity with the radius rt )
corresponds to the solution of Eq. (7). Assuming that the pressure at the external
surface of the tubular element with the radius rcp equals zero (o r = 0 at r = rcp),
the distribution of radial stresses is represented by the relation:
o r = 2 ty ln (rcp / r ) . (18)
From the condition of the incompressibility of a rod material at plastic flow,
the outer and inner tubular radii (rcp and ri ) are related to the initial radius rp of
the rod by
2 2 2 2 / 2 i 2 / 2 r — r- = r or r- / r = 1 — r / ri cp > i • p VL I i / I cp ' p 11 cp'
The pressure Tt at the inner surface of the tubular element, necessary for its
radial expansion according to Eq. (18), is calculated as:
Ti = (o Y / 2) ln (rcp / ri )2 = - (o Y / 2) ln [1 - (rp / rcp )2 ].
The work of expansion of the tubular element per its unit length, which is
assumed to be equal to the work of plastic deformation of the rod, is calculated
by the equation:
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ri
Ap = / 2nriTidri = n (o y / 2 > > [( rc / rp )2 — 1]ln[( rc / rp )2 —1].
0
A relative decrease in the kinetic energy of the rod (for the initial velocity of
the rod V and the densities of the materials of the rod and plastic medium p) at
its transformation into the tubular element (assuming (rc / rp ) = 4) is rather
considerable:
Ap / (n rp p V 2 /2 ) = (o y / 2){[( r c / r p ) 2 — 1]ln[( rc / r p ) 2 — 1]} /(pV 2 /2 ) =
= 3.4 o y / ( p V 2 /2) . (19)
This reduction of the kinetic energy of a steel rod (o y = 1.2 GPa, p =
= 7800 kg/m ), caused by its plastic deformation at the penetration into the same
steel with the initial velocity V = 2000 m/s, exceeds 25% as calculated by
Eq. (19). Hence, the work of deformation of the rod entering in the equation of
energy balance should be accounted for at such velocities.
C O N C L U S I O N S
The static resistance of a plastic material to the penetration of a rigid rod can
be evaluated by the pressure required for the expansion of a spherical cavity as
well as by the mean pressure onto the rod head at penetration. This value does
not depend on the geometry of the head.
The specific work spent for the cavity formation (per its unit volume) is the
same for a spherical or cylindrical cavity formed by the penetration of a rigid or
eroded rod; in the latter case, an increment in the cavity depth causes the
axisymmetric radial and axial displacements of a medium and the deformation
(erosion) of the rod.
For the plastic medium with intense strain hardening, the pressure required
for the expansion of the spherical cavity or the resistance of the medium to the rod
penetration is controlled by the ultimate stress at large strains.
At the low-rate expansion of a spherical cavity and the penetration of a rod in
an elastoplastic medium, the elastic volume compression affects the relative
radius of the elastoplastic interface and the surface pressure for the cavity
expansion (or the mean pressure at the contact surface for the rod penetration).
The evaluation of the resistance of a plastic medium to the penetration of a
rod at low velocities, neglecting the effects of strain hardening, friction on the
contact surface, and the specific flow near the point of branching, can be taken
only as the first approximation. Therefore, the detailed studies on these effects
would be of paramount importance for further progress in this field.
Acknowledgments. The authors would like to express their gratitude to
Mr. Konrad Frank and Mr. William A. Gooch of the U. S. Army Research
Laboratory (ARL), Aberdeen Proving Grounds, MD for their support of the work.
G. V. Stepanov, V. V. Kharchenko
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Effect o f the Static Strength o f a Metal
Р е з ю м е
Розглянуто вплив статичної міцності (границі текучості) матеріалу на основ
ні параметри процесу проникання - питома робота утворення каверни й
тиск на поверхню контакту. Отримано аналітичні вирази й кількісні оцінки
питомої роботи, що затрачена на утворення сферичної й циліндричної
каверн, і тиску на поверхні таких каверн при статичному розширенні
останних для жорсткопластичної та пружнопластичної моделей матеріалу.
Показана адекватність оцінок питомої роботи деформування пластини при
утворенні сферичної каверни та при утворенні каверни в результаті прони
кання жорсткого стержня. Оцінено вплив деформаційного зміцнення мате
ріалу на максимальну радіальну напругу на поверхні каверни. Встановлено,
що при низьких швидкостях проникання суттєвий вплив на опір прони
канню й розширення каверни має пружна стисливість матеріалу в області
непружних деформацій. Проаналізовано кінетику напружено-деформова-
ного стану матеріалу при прониканні деформівного стержня.
R E F E R E N C E S
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2. A. Tate, “A theory for the deceleration of long rods after impact,” J. Mech.
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3. A. Tate, “Further results in the theory of long rod penetration,” J. Mech.
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