The extent of plastic flow in a spherical solid (assumed to be homogeneous and elastically and plastically isotropic), surrounding a concentric rigid sphere was calculated as a function of applied external pressure. The applied pressure necessary to cause plastic deformation throughout the solid was obtained

Nelson, D. A., Jr.

Ruoff, A. L.

English

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NASA(NGR)33-010-189 No. 3.
(NASA-CR-142596) - PLASTIC FLOW AROUND RIGID_ N75-2076-
SPHERICAL .INCLUSIONS (Cornell Univ.), 12 p
HC $ 3.-25 CSCL .20K
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G3/39 18553
PLASTIC FLOW AROUND RIGID SPHERICAL INCLUSIONS
Arthur L. Ruoff and David A. Nelson, Jr.
Department of Materials Science and Engineering,
Materials ,Science Center
Cornell University
Ithaca, New York 14850
Report #2278
Abstract
The extent of plastic flow in a spherical solid (assumed to
be homogeneous and elastically and plastically isotropic) surrounding
a concentric rigid sphere as a function of applied external pressure
is calculated. The applied pressure necessary to cause plastic
deformation throughout the solid is obtained.
https://ntrs.nasa.gov/search.jsp?R=19750012693 2020-03-22T23:22:18+00:00Z
INTRODUICTION
When a solid matrix surrounding a rigid sphere is pressurized,
the rigid particle acts as a stress concentrator. The radial press-
ure at the interface at the rigid sphere exceeds the external
hydrostatic pressure. When the applied pressure is increased
sufficiently, yielding of the matrix begins at the interface. This
is a well known phenomenon and can be observed around inclusions
in solids where dislocations are punched out. 1 The pressure which
causes the onset of yielding is available from earlier calculation
from linear elasticity theory and.Tresca or von Mises yield
criteria. In the present paper we calculate the extent of
plastic flow versus applied pressure and finally obtain the
applied pressure necessary to cause plastic deformation through-
out the solid.. The present analysis assumes that there is no
strain hardening.
In the present paper external pressurization of a medium
surrounding a rigid spherical inclusion is studied. Two methods
are used. The first, developed here, leads more easily to the
solution than the second, which is the technique used by Hill 2
for studying the internal pressurization of a hollow sphere.
THEORETICAL
Consider the matrix of external radius b surrounding a rigid
sphere of radius a. The boundary conditions are that the radial
component of stress is given by
arr= -Pext at r=b (1)
and the radial component of displacement is given b)y
u = 0 at r = a (2)
Elastic Behavior
At sufficiently small values of Pext' the material behaves
everywhere in a linear elastic manner. The stresses then have
the form
a= A + B3 (3)
rr r
a6@= A -2 B (4)
and the radial displacement is given.by
r = r [OO-v(0+ rr) ]  (5)
Here EO is the circumferential strain, E is Young's modulus,
v is Poisson's ratio and we have used the fact that a =a.
From these five equations,the stal:e of stress can be readily
ascertained. The pressure which causes the onset of yielding
can be obtained using Tresca's or von Mises' yield criteria
( which are identical for the sphere):
CT - rr ao (6)
where a is the yield stress in simple tension. This leads to the
result that yielding (designated by an asterisk) begins when
P { (1+v) 2Kj (7)
ext 3 I-2v) 0
where K=b/a.
When yielding begins the contact pressure at r=a for v=1/3
P. = 20 0 (8)Int o
(We note that P. P for large K.)Int 2 ext
Elasto-Plastic
Here the material is elastic for r>c and plastic for r<c. In
the plastic region the stresses are determined by Equation (6)
and the boundary conditions. The elastic strains in the plastic region
are obtained from these stresses using Hooke's laws. It is
assumed that strain hardening is absent, i.e., a0 does not vary
with the extent of plastic deformation. The stresses are continuous
at the elastic-plastic boundary, r=c.
The boundary conditions for Ihe elastic region are
000- rr . 0  at r=c (9)
and
rr = -Pext at r=b. (10)
The solution for the elastic region is
2 c 3 b 30 = -P -3 o 3 (- 3 - 1). (11)
rr ext 3 ob r
and
2 c 3 b 300 =-Pext +0 3 3+1) (12)
The displacement in the elastic region is
u= r[(1-2v)(-P 2 o c3/b 3 )+(1+) a c3/r "3 ] (13)1=  ext+- 0 3 0
The general solution in the plastic region can be obtained
from the equation of radial equilibrium, namely
rdo
drrr + 2( rr-o )=0 (14)
and Equati.on (6). The result is
o = 2o Rnr +D
rr o
where D is an integration constant which can be evaluated by the
requirement of continuity of 0 rr at r=c. Then we have
- 2o nc p 2 c 3
r o r ext 3o 1-3 (i )
The local volume dilation in the plastic region is given by
AV ( (I2v) ( ) +3vr) (16)
V E + arr E rr
The total change in volume of the plastic region (whose inner
radius is constant a and outer radius is c) is therefore
C
(1-2v)(2ao+3orr)4r dr=4 c2 u(c) (17)
a
The value of orr is given by Equation (15) while u(c)
follows from (13) with r set equal to c. It follows that
= (i-v) -2 n /b) (18)ext - ) (c/a) o 3 o
For the fully plastic condition (labelled with double asterisk)
c-b and (for v=1/3)
Pe = 2a oK - 20o nK, (19)
ext o 0
and
Pint - Pext = 2a 0nK. (20)
For the case where K is only slightly greater than one(the thin
£im
shell) we have from (19) Kl Pext = 2o
as required since for a thin shell Orr does not vary across the
shell and the entire shell yields simultaneously as given by (8).
The ratio P /Pext vs K for v=l/3 is shown in Table 1.
ext ext
Table 1. Ratio of pressure needed to cause
plastic flow throughout matrix over pressure
needed for onset of plastic flow versus radius
ratio. Based on Poisson ratio of one-third.
K Pext o Pext/o Pext /Pext
1 1 1 1
2 7.31 0.71 10.3
3 25.90 0.68 38.1
As K becomes modestly large the pressure needed to cause plastic
flow throughout the matrix is enormous relative to the pressure
required for the commencement of plastic flow.
The extent of plastic flow for a given applied pressure and
given K can readily be written in terms of f i, the fraction of
the matrix which has undergone plastic deformation. Here
1-f = (b 3 -c 3 )/(b 3 -a) . (21)
Th.en (for v=1/3) from (18)
next 1 1
2xt (K 3 - l )f m+  -- n[(K 3 -1)fm+1]- (l-fm ) ( 1 - K   (220
Curves of fm vs Pext/20
° for K=1, 2 and 3 are shown in Figure 1.
For the cases considered here, the strains everywhere are five
percent or less (assuming 3(1-2v)o /E< 1/100) so the neglect of
strain hardening is justified unless the matrix has an unusually
rapid rate of strain hardening.
Alternate Method
This problem can also be solved following the technique described
by Hill. 2  The elastic stresses and displacements are identical
as those derived above. In the plastic region, the equation of
radial equilibrium again leads to the radial stress as in
Equation (15). Now following Hill's definition of the
velocity u of a particle,we find
S= /a (23)
in the elastic region; and
3u2u (1-2v) 3+,D( + (24)+r~r - (-- +u)v +2 ) (24)Br r E ac @r rr ee
in the plastic region. These velocities must be continuous at
r=c,the plastic-elastic interface.
Using the equations for the stresses in the plastic region,
(9) and (15),we can solve Equation (24) if we make the same
approximations that Hill makes; i.e. neglecting second and higher
orders of a /E, and neglecting au/ar with respect to 1. Matching
the velocities in the two regions gives the velocity in the plastic
region,
3(1-v) o c 2 2(1-2v)o C 3 C O
= E r (1- 3 +  c r/c (25)E r b 20 ac
when P is here written as a function of c.ext
Since u is defined as dr/dc we can rewrite Equation (25) in
terms of da/dc, the velocity of a particle at the inner boundary.
With an incompressible material in the center da/dc is just zero,
so we are left with an ordinary differential equation.
Solving this we find
et 2 a0 (c/b) 3-2o0 n(c/a)+ (1 v  (c/a) 3 +F (26)ext=3 o o (1-2v) 0o
where F is the constant of integration. We can calculate F
from the boundary condition that Pext=Pext for c=a. This gives
(liv) 2 3 2P _(- (c/a) 20 c1(c/a)+2(c/b) 0 -0 (27)
ext (1-2v) o  o o 3 o
For the fully plastic conditions
P (I-v) a K3 -2 nK (28)
ext TI-2v\) o o
This result agrees with Equation (19) in spite of the approximations
used in obtaining (28).
CONCLUSIONS
The pressure required to cause plastic deformation throughout
a homogeneous elastically and plastically isotropic sphere surrounding
a concentric rigid sphere is shown to be very large relative to
the pressure required for the onset of plas.tic flow. In fact, the
ratio of these pressures approaches 1.5K 3 for no strain hardening,
where K is the radius ratio, for large K. Note that the ratio
already approximates the large K limit for K=2 and K=3. For such
values of K, strain hardening effects usually can be neglected, while
for much larger K strain hardening would have to be taken into
account as would also nonlinear elastic effects.
For values of K of two or larger, the volume fraction of the
matrix,which has undergone plastic deformation, increases nearly
linearly with pressure above the critical pressure for the onset
of yielding.
Acknowledgment
This work was supported by a National Aeronautics and Space
Administration grant. In addition,National Science Foundation
support was received through the use of the facilities of the
Cornell Materials Science Center.
References
1. D. A. Jones, and J. W. Mitchell, Phil. Mag. 3, 1 (1958).
2. R. Hill, The Mathematical Theory of Plasticity, Oxford at
the Clarendon Press, London, (1950) Chapter V, p. 103.
Figure Legends
Figure 1. Volume fraction of matrix which is plastically
deformed versus pressure for vaious K values.
Here v=1/3.
Figure 2. Ratio of internal pressure to external pressure vs
applied pressure for K=2 and 3 for v=1/3. The single
asterisk indicates the point of initiation of plastic
flow. The double asterisk indicates the point at which
.the sphere is fully plastic. The dashed line is the
region in which Equation (20) holds.
K=I
3 K=2
4
K=3
4
I .
0 10 20 30
Pext/2 o
Figure I.
1.5-
1.4
-K.= 3
c 1.2-
K=.2
1.1
I .0
0 IO 20 30
Fext /2o
Figure 2.


Plastic flow around rigid spherical inclusions

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