Plane Strain Expansion of a Cylindrical Cavity in an Infinite Porous
Rigid/Plastic Medium Obeying a General Yield Criterion
MARINA RYNKOVSKAYA, SERGEI ALEXANDROV, TIMUR ELBERDOV
Department of Civil Engineering,
RUDN University,
6 Miklukho-Maklaya Str., Moscow, 117198,
RUSSIAN FEDERATION
Abstract: - A semi-analytical plane-strain solution for an expanding cylindrical cavity surrounded by an infinite
porous rigid/plastic medium is presented. The constitutive equations are a general yield criterion and its
associated flow rule. The yield criterion depends on the relative density and the linear and quadratic stress
invariants. No restriction is imposed on this dependence, except for the standard requirements imposed on the
yield criteria. The boundary value problem reduces to a Cauchy problem for three ordinary differential
equations. This system of equations must be solved numerically. Numerical results are presented for Green’s
yield criterion. This yield criterion involves two functions of the relative density. The influence of the choice of
these functions on the distributions of the relative density, the radial velocity, and the stress components is
revealed.
Key-Words: - cylindrical cavity, rigid/plastic medium, porous material, general yield criterion, plane strain,
Green’s yield criterion.
Received: April 16, 2023. Revised: October 29, 2023. Accepted: November 28, 2023. Published: December 31, 2023.
1 Introduction
Under certain conditions, the behavior of isotropic
porous and powder ductile materials is successfully
described by plasticity theory, assuming that the
yield criterion depends on the relative density and
stress invariants. The associated flow rule is usually
used as the plastic flow rule. A comprehensive
description of this theory is provided in [1]. In many
cases, elastic strains can be neglected, leading to
rigid/plastic models, [2], [3]. The present paper is
restricted to such models.
The linear stress invariant is responsible for the
plastic compressibility of materials. Therefore, this
invariant must be involved in the yield criterion.
The effect of the cubic stress invariant on the plastic
behavior of porous and powder materials is often
ignored. The corresponding yield criteria have been
proposed in [4], [5], [6], among others. The present
paper assumes an arbitrary dependence of the yield
criterion on the linear and quadratic stress invariants
satisfying the general standard requirements
imposed on the yield criteria. The von Mises yield
criterion is a particular case of this general criterion.
Analytical and semi-analytical solutions to non-
stationary problems are rare in plasticity, even in the
case of rigid perfectly plastic incompressible
materials, [7]. For the class of models specified
above, two solutions for instantaneous flow have
been derived in [8], [9]. The evolution of the
relative density has been considered in these
solutions. Meanwhile, analytical and semi-analytical
solutions that account for the relative density
evolution have theoretical interest. Moreover, such
solutions are important for verifying numerical
codes, [10], [11].
Self-similar processes are an important class of
processes for which analytical and semi-analytical
solutions can be found. The most known solution of
this class is for a spherical cavity expanding in an
infinite medium from a zero radius. An elastic
perfectly plastic solution has been provided in [12].
Several papers have been devoted to dynamic
spherical cavity expansion in various elastic plastic
media. A review of these solutions can be found in
[13]. The process of cylindrical cavity expansion
has attracted less attention. An elastic perfectly
plastic solution has been provided in [12]. The
effect of inertia has been taken into account in [14],
assuming that strains are infinitesimal. A solution
for the quasi-static expansion of a cylindrical cavity
in hypoelastic compressible Mises and Tresca solids
at large strains has been derived in [15]. In contrast
to the solutions above, the present paper considers a
rigid plastic model. The material model provided in
[1], is employed. Neglecting elastic strains changes
the boundary value problem significantly. In
WSEAS TRANSACTIONS on APPLIED and THEORETICAL MECHANICS
DOI: 10.37394/232011.2023.18.30
Marina Rynkovskaya,
Sergei Alexandrov, Timur Elberdov
E-ISSN: 2224-3429
318
Volume 18, 2023
particular, some equations contain the expression
0/0, which requires some analytical treatment of
these equations before using a numerical method.
The numerical solution is based on Green’s yield
criterion, [4]. This yield criterion involves two
functions of the relative density. A review of these
functions is provided in [16]. The functions
proposed in [1] and [4], are adopted in the numerical
solution. The effect of the choice of these functions
on the distributions of the relative density, the radial
velocity, and the stress components is revealed.
2 Statement of the Problem
A cylindrical cavity of a zero initial radius expands
in an infinite porous rigid/plastic medium under
plane strain conditions. The relative density is
uniformly distributed at the initial instant and equals
0
. The flow theory of plasticity is used. A
comprehensive description of this theory has been
provided in [4].
The constitutive equations constitute a yield
criterion and its associated flow rule. The present
paper is restricted to the yield criteria independent
of the third invariant of the stress tensor. Therefore,
the yield criterion can be represented as:
, , 0.
(1)
Here
is the relative density,
is the first
stress invariant,
is the second stress invariant, and
is an arbitrary function of its arguments
satisfying the standard requirements imposed on the
yield criteria. The stress invariants are expressed in
terms of the principal stresses
1
,
2
and
3
as:
1 2 3
3
and
2 2 2
1 2 3
1.
2
(2)
The plastic flow rule associated with (1) is:
12
12
3 , 3 ,
and
3
3
3,
(3)
where
1
,
2
and
3
are the principal strain rates
and
0
. The stress and strain rate tensors are
coaxial. This condition is automatically satisfied in
the problem under consideration. Substituting (1)
into (3) and employing (2) yields:
1 1 2 3
2 2 3 1
3 3 1 2
, 2 , ,
, 2 , ,
, 2 , ,
(4)
where
,
and
,
.
Using a cylindrical coordinate system
,,rz
whose z-axis coincides with the symmetry axis is
natural. The normal stresses in this coordinate
system are denoted as
r
,
, and
z
. These
stresses are the principal stresses. Similarly, the
normal strain rates are denoted as
r
,
, and
z
.
These strain rates are the principal strain rates. One
may choose
1r
,
2
, and
3z
.
Consequently,
1r
,
2
, and
3z
. The
radial velocity is denoted as u. The other velocity
components vanish. Plane strain conditions demand:
0.
z
(5)
The other principal strain rates are expressed
through the radial velocity as:
r
u
r
and
.
u
r
(6)
The third equation in (4) and (5) combine to
give:
, 2 , 0.
zr
(7)
Eliminating
between the first and second
equations in (4) gives:
, 2 , .
, 2 ,
rz
r
zr
(8)
Equations (6) and (8) combine to give:
, 2 , .
, 2 ,
rz
zr
uu
rr
(9)
The only equilibrium equation that is not
identically satisfied is:
0.
rr
rr
(10)
In the case under consideration, the equation of
mass conservation is:
0,
uu
u
t r r r
(11)
WSEAS TRANSACTIONS on APPLIED and THEORETICAL MECHANICS
DOI: 10.37394/232011.2023.18.30
Marina Rynkovskaya,
Sergei Alexandrov, Timur Elberdov
E-ISSN: 2224-3429
319
Volume 18, 2023
where t is the time. One can eliminate the derivative
ur
in (11) using (9) to arrive at:
3, 0.
, 2 ,
zr
u
u
tr
r
(12)
Equations (1), (9), (10), and (12) constitute a
system for determining
r
,
, u, and
.
3 General Solution
A plastic region propagates from the cavity. The
current radius of the rigid/plastic boundary is
denoted as a. The material is rigid in the region
ra
. remainder is rigid. In this region, it is only
necessary to show that a stress field satisfying the
equilibrium equations and not violating the yield
criterion exists. The radial stress, the radial velocity,
and the relative density must be continuous across
the rigid/plastic boundary. The current radius of the
rigid/plastic boundary can be regarded as a time-like
parameter. Moreover, since the material model is
rate-independent, one may put:
1da dt
(13)
without loss of generality. Then, u becomes
dimensionless. Like all similar problems (for
example [4]), the solution depends on the ratio
ra
rather than r and a separately. Taking into
account (13), one can express the derivatives with
respect to r and t in terms of the derivative with
respect to
as:
1d
r a d
and
2.
r d d
t a d a d
a
(14)
Then, equations (9), (10), and (12) become:
, 2 , ,
, 2 ,
rz
zr
du u
d
(15)
0,
r
r
d
d
(16)
and
3, 0,
, 2 ,
zr
u
d
ud
(17)
respectively.
Since the radial velocity and relative density
must be continuous across the rigid/plastic
boundary,
0u
and
0
(18)
for
1
. It is then seen from equations (15) and
(17) that physically reasonable solutions are
possible only if:
, 2 , 0
zr
(19)
for
1
. Equations (1), (7) and (19) allows all the
principal stresses to be found at the rigid/plastic
boundary. This boundary condition and (18) lead to
a Cauchy problem for equations (15) to (17). This
problem must be solved numerically.
Let R be the current radius of the cavity. At the
initial instant, the mass of the material contained in
the unit length of the cylinder of radius a is
determined as:
2
0.Ma
(20)
After any amount of deformation, this mass can
be calculated as:
1
2
2 2 .
a
R R a
M rdr a d
(21)
Equations (20) and (21) combine to give:
1
0
2.
Ra
d
(22)
A numerical solution of equations (15) to (17)
provides the integrand as a function of
.
Therefore, equation (22) determines
Ra
.
4 Green’s Yield Criterion
The yield criterion proposed in [4], can be
represented as
2
22
1 0,
ss
p
(23)
where
s
p
and
s
are functions of the relative
density. An ellipse represents this yield criterion in a
two-dimensional space where the linear and
quadratic stress invariants are taken as Cartesian
coordinates. The length of its major and minor axes
depends on the relative density. The length of the
major axis approaches infinity as the relative density
approaches unity. In this case, Green’s yield
criterion approaches Mises’ yield criterion. It
follows from (2) that:
2
2
,
s
p
and
2
1
,.
s
(24)
WSEAS TRANSACTIONS on APPLIED and THEORETICAL MECHANICS
DOI: 10.37394/232011.2023.18.30
Marina Rynkovskaya,
Sergei Alexandrov, Timur Elberdov
E-ISSN: 2224-3429
320
Volume 18, 2023
Substituting (24) into (7), one gets:
,
zr
(25)
where
2
2
2
4
and
2.
3
s
s
p
(26)
Substituting (25) into (2) yields:
1
3
r
and
22
1 1 2 1 1 .
3
rr
(27)
Equations (23) and (27) combine to give:
22
2 1,
rr
AB
(28)
where
2
2
2
4 1 1 1
12 s
A
and
2
2
2
2 2 1 1 1 .
12 s
B
(29)
Equation (28) is satisfied by the following
substitution:
sin
ra
and
sin cos ,bc
(30)
where
2
1,
1
aA
2,
1
bA
1,cA
and
2
2
2.
21
(31)
Substituting (24), (25), and (30) into (19) results in
2
22
1cos 0
4
s
(32)
at
1
. Since
0
r
, it follows from the first
equation in (30) and (32) that:
2
(33)
for
1
.
Employing (24), (26), (27), and (30), one
transforms equations (15) to (17) to:
22
2
2 cos 3 4 sin
0,
2 1 cos
u
du
d
(34)
22
22
3 4 sin 4 cos
sin cos 0,
41
d da d
a
d d d
(35)
and
2
2
3 4 sin 3 cos
0.
2 1 cos
u
d
ud
(36)
It is advantageous to introduce the following
variable:
ln .
(37)
Then, equations (34) and (36) become:
22
2
2 3 4 tan
0
21
u
du
d
(38)
and
2
2
3 4 tan 3
.
21
u
d
deu
(39)
The derivative
dd
in (35) can be eliminated
by employing (36). Then, upon using (37), equation
(35) becomes:
22
22
3 4 tan 3 3 tan 4
tan 0.
21 1
u
d da
add
eu
(40)
The boundary conditions to equations (38) to
(40) follow from (18), (33), and (37) in the form:
0u
,
0,
and
2
(41)
for
0
.
Some terms in equations (38) to (40) reduce to
the expression
00
at
0
. Therefore, the
solution’s behavior near this point must be
investigated before using a numerical method. A
Taylor series is assumed to represent each unknown
function in the neighborhood of
0
. Then, using
(41),
1,u U o
01 ,o
and
1,
2o
(42)
as
0
. Substituting (42) into equations (38) to
(40) leads to:
12
0
3
1
U
da d
,
1 0 1
U
,
and
WSEAS TRANSACTIONS on APPLIED and THEORETICAL MECHANICS
DOI: 10.37394/232011.2023.18.30
Marina Rynkovskaya,
Sergei Alexandrov, Timur Elberdov
E-ISSN: 2224-3429
321
Volume 18, 2023
2
12
34 .
21
(43)
It is understood here that
and
da d
are
calculated at
0
.
5 Numerical Solution
Equations (38) to (40) have been solved numerically
using (42) and (43). The solution has been
calculated for the functions
s
p
and
s
proposed in [1] and [4]. In particular,
2
2
31
s
k
p
and
32
sk
(44)
and
2 3 ln 1
s
pk
,
3
4
3 1 1
,
3 2 1
s
k
(45)
respectively. In these equations, k is the shear yield
stress of the pore-free material. Equations (26), (29),
(31), (44), and (45) allow the coefficients involved
in equations (38) to (40) to be expressed in terms of
the relative density. Similarly, the coefficients
involved in (42) can be expressed in terms of the
initial relative density.
Figure 1, Figure 2, Figure 3, Figure 4 and Figure
5 illustrate the solution for the functions
s
p
and
s
proposed in [1]. The variation of the relative
density with
is depicted in Figure 1. The
rightmost points correspond to the rigid plastic
boundary where
0
. The leftmost points
correspond to the cavity surface. It can be verified
by substituting the solution for
into (22).
In particular, the dependence of
Ra
on
0
found from this equation is presented in Table 1.
Interestingly, the relative density equals one at the
cavity surface. Figure 2 shows the variation of the
dimensionless radial velocity with
. The rightmost
points correspond to the rigid plastic boundary
where
0u
. Figure 3, Figure 4 and Figure 5
illustrate the distributions of the principal stresses in
the plastic region. The radial stress must be
continuous across the rigid plastic boundary.
Therefore, the values of this stress at the rightmost
points in Figure 3 must be used for extending the
stress field into the rigid region.
Figure 6, Figure 7, Figure 8, Figure 9 and Figure
10 illustrate the solution for the functions
s
p
and
s
proposed in [4]. Qualitatively, the solution
behavior is similar to that for the functions
s
p
and
s
proposed in [1]. However, the solutions
significantly differ quantitatively, emphasizing a
need for accurate representations of the functions
s
p
and
s
for practical applications.
In particular, Table 2 presents the dependence of
Ra
on
0
.
Figure 11, Figure 12, Figure 13, Figure 14 and
Figure 15 show the influence of the choice of the
functions of the distributions of the relative density,
the dimensionless radial velocity, and the principal
stresses at
00.6
.
Table 1. Dependence of
Ra
on
0
for the functions
s
p
and
s
proposed in [1]
0
0.4
0.5
0.6
0.7
0.8
Ra
0.559
0.488
0.416
0.34
0.255
Table 2. Dependence of
Ra
on
0
for the functions
s
p
and
s
proposed in [4]
0
0.4
0.5
0.6
0.7
0.8
Ra
0.34
0.291
0.246
0.201
0.159
Fig. 1: Variation of the relative density with m for
the functions
s
p
and
s
proposed in [1]
WSEAS TRANSACTIONS on APPLIED and THEORETICAL MECHANICS
DOI: 10.37394/232011.2023.18.30
Marina Rynkovskaya,
Sergei Alexandrov, Timur Elberdov
E-ISSN: 2224-3429
322
Volume 18, 2023
Fig. 2: Variation of the dimensionless radial
velocity with m for the functions
s
p
and
s
proposed in [1]
Fig. 3: Variation of the radial stress with m for
the functions
s
p
and
s
proposed in [1]
Fig. 4: Variation of the circumferential stress
with m for the functions
s
p
and
s
proposed in [1]
Fig. 5: Variation of the axial stress with m for
the functions
s
p
and
s
proposed in [4]
Fig. 6: Variation of the relative density with m
for the functions
s
p
and
s
proposed in
[4]
Fig. 7: Variation of the dimensionless radial
velocity with m for the functions
s
p
and
s
proposed in [4]
WSEAS TRANSACTIONS on APPLIED and THEORETICAL MECHANICS
DOI: 10.37394/232011.2023.18.30
Marina Rynkovskaya,
Sergei Alexandrov, Timur Elberdov
E-ISSN: 2224-3429
323
Volume 18, 2023
Fig. 8: Variation of the radial stress with m for
the functions
s
p
and
s
proposed in [4]
Fig. 9: Variation of the circumferential stress
with m for the functions
s
p
and
s
proposed in [4]
Fig. 10: Variation of the axial stress with m for
the functions
s
p
and
s
proposed in [4]
Fig. 11: Influence of the functions
s
p
and
s
on the distribution of the relative density
(Curve 1 corresponds to the functions proposed
in [1] and Curve 2 to the functions proposed in
[4])
Fig. 12: Influence of the functions
s
p
and
s
on the distribution of the radial velocity
(Curve 1 corresponds to the functions proposed
in [1] and Curve 2 to the functions proposed in
[4])
Fig. 13: Influence of the functions
s
p
and
s
on the distribution of the radial stress
(Curve 1 corresponds to the functions proposed
WSEAS TRANSACTIONS on APPLIED and THEORETICAL MECHANICS
DOI: 10.37394/232011.2023.18.30
Marina Rynkovskaya,
Sergei Alexandrov, Timur Elberdov
E-ISSN: 2224-3429
324
Volume 18, 2023
in [1] and Curve 2 to the functions proposed in
[4])
Fig. 14: Influence of the functions
s
p
and
s
on the distribution of the circumferential
stress (Curve 1 corresponds to the functions
proposed in [1] and Curve 2 to the functions
proposed in [4])
Fig. 15: Influence of the functions
s
p
and
s
on the distribution of the axial stress
(Curve 1 corresponds to the functions proposed
in [1] and Curve 2 to the functions proposed in
[4])
6 Conclusion
The quasi-static expansion of a cylindrical cavity of
a zero initial radius in an infinite porous rigid/plastic
medium has been investigated under plane strain
assumptions. The problem has been reduced three
ordinary differential equations. Some of these
equations contain expressions 0/0 at the rigid/plastic
boundary. Therefore, an asymptotic analysis of the
equations has been carried out before using a
numerical method. The numerical solution has been
provided for Green’s yield criterion, giving the
distributions of the relative density, the radial
velocity, and the principal stresses in the plastic
region. The relative density approaches unity at the
cavity’s surface. Green’s yield criterion involves
two functions of the relative density. The choice of
these functions significantly affects the solution’s
behavior.
The solution can be adopted to describe the
expansion of a non-zero initial radius cavity using
the approach proposed in [12]. The resulting
solution may be used to verify the accuracy of
numerical solutions, which is a necessary step for
using such solutions for practical applications, [10],
[11].
The subject of a subsequent investigation is to
consider strain hardening and to reveal the effect of
neglecting elasticity on the solution.
Acknowledgement:
This publication has been supported by the RUDN
University Scientific Projects Grant System, project
№ 202247-2-000.
References:
[1] Druyanov B., Technological mechanics of
porous bodies; Clarendon Press, New-York,
USA, 1993.
[2] Tirosh J., Iddan D., Forming analysis of
porous materials, International Journal of
Mechanical Sciences, 31, 11–12, 1989, pp.
949-965.
[3] Mamalis A.G., Petrosyan G.L., Manolakos
D.E., Hambardzumyan A.F., The effect of
strain hardening in the extrusion of
bimetallic tubes of porous internal layer,
Journal of Materials Processing Technology,
181, 1–3, 2007, pp. 241-245.
[4] Green R.J., A plasticity theory for porous
solids, International Journal of Mechanical
Sciences, 14, 4, 1972, pp. 215-224.
[5] Shima S., Oyane M., Plasticity theory for
porous metals, International Journal of
Mechanical Sciences, 18, 6, 1976, pp. 285-
291.
[6] Gurson A.L., Continuum Theory of Ductile
Rupture by Void Nucleation and Growth:
Part I—Yield Criteria and Flow Rules for
Porous Ductile Media, ASME Journal of
Engineering Materials and Technology, 99,
1, 1977, pp. 2–15.
[7] Richmond O., Plane strain necking of V-
notched and un-notched tensile bars, Journal
of the Mechanics and Physics of Solids, 17,
2, 1969, pp. 83-90.
WSEAS TRANSACTIONS on APPLIED and THEORETICAL MECHANICS
DOI: 10.37394/232011.2023.18.30
Marina Rynkovskaya,
Sergei Alexandrov, Timur Elberdov
E-ISSN: 2224-3429
325
Volume 18, 2023
[8] Monchiet V., Kondo D., Exact solution of a
plastic hollow sphere with a Mises–
Schleicher matrix, International Journal of
Engineering Science, 51, 2012, pp. 168-178.
[9] Pindra N., Leblond J.B., Kondo D., Limit-
analysis of a circular cylinder obeying the
Green plasticity criterion and loaded in
combined tension and torsion, Meccanica,
53, 2018, pp. 2437–2446.
[10] Roberts S.M., Hall F., Van Bael A., Hartley
P., Pillinger I., Sturgess E.N., Van Houtte P.,
Aernoudt E., Benchmark tests for 3-D,
elasto-plastic, finite-element codes for the
modelling of metal forming processes,
Journal of Materials Processing Technology,
34, 1992, pp. 61–68.
[11] Abali B.E., Reich F.A., Verification of
deforming polarized structure computation
by using a closed-form solution, Continuum
Mechanics and Thermodynamics, 32, 2020,
pp. 693–708.
[12] Hill R., The mathematical theory of
plasticity; Clarendon Press, Oxford, UK,
1950.
[13] Cohen T., Durban D., Hypervelocity Cavity
Expansion in Porous Elastoplastic Solids,
ASME Journal of Applied Mechanics, 80,
2013, Article 011017.
[14] Druyanov B.A., Sokolova L.E., Problem of
the expansion of a circular aperture in an
infinite plate, Soviet Applied Mechanics, 13,
1977, pp. 487–491.
[15] Masri R., Durban D., Cylindrical cavity
expansion in compressible Mises and Tresca
solids, European Journal of Mechanics -
A/Solids, 26, 4, 2007, pp. 712-727.
[16] Doraivelu S.M., Gegel H.L., Gunasekera
J.S., Malas J.C., Morgan J.T., Thomas J.F., A
new yield function for compressible PM
materials, International Journal of
Mechanical Sciences, 26(9–10), 1984, pp.
527-535.
Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
The authors equally contributed to the present
research, at all stages from the formulation of the
problem to the final findings and solution.
Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
This publication has been supported by the RUDN
University Scientific Projects Grant System, project
№ 202247-2-000.
Conflict of Interest
The authors have no conflicts of interest to declare.
Creative Commons Attribution License 4.0
(Attribution 4.0 International, CC BY 4.0)
This article is published under the terms of the
Creative Commons Attribution License 4.0
https://creativecommons.org/licenses/by/4.0/deed.en
_US
WSEAS TRANSACTIONS on APPLIED and THEORETICAL MECHANICS
DOI: 10.37394/232011.2023.18.30
Marina Rynkovskaya,
Sergei Alexandrov, Timur Elberdov
E-ISSN: 2224-3429
326
Volume 18, 2023
