1. science
2. mathematics
3. geometry

THE
BEHAVIOR
OF GRAPHITE
BIAXIAL TENSION*
UNDER
W. L. GREENSTREET,
G. T. YAHR and R. S. VALACHOVIC
Oak Ridge National Laboratory,
Oak Ridge, I‘ennessee
37830, I,! .S..I.
(Received I SJune
1972)
AbstractA small-deformation
elastic-plastic
continuum
theory for describing the stressstrain behavior
of graphite
is examined
for specimens
under biaxial loading at room
temperature. The continuum theory takes into account the anisotropy and plastic compressibility of graphite as well as the continuous
nonlinearity
of the stress-strain
response.
The measured
strains for thin-walled
cylindrical
shells loaded by internal pressure
arc
shown to agree \‘ery closely with the predictions
of the theory for both loading and
unloading.
1. INTRODUCTION
This
an
paper
describes
elastic-plastic
was
to
theory
describe
behavior
of
calculated
for combined
artificial
graphite.
cylinders
were
made
from
internally
on loading
pressurized
response,
experimentally
obtained
were compared
with calculated
In the main,
the circular
tested,
Five specimens
while
made
from
used.
In
a single
each
case,
transversely
isotropic
of. isotropic
symmetry
of’the
The
longitudinal
theory
which
material
behavior.
being
results
The
was
G
was also
exhibited
of’
obtained
third
in
to
the
with the axis
in
this
mental
described
specialty
in the next section. The equations
employed
in this stud), were based on approximate,
results
The
batween
for the
t‘rom a
results
and
the specimens
tests
describes
were
com-
the Graphitite-
discussion
theory
f’ourth
of’ the
is also given
section
calculated
gives
and experi-
Graphitite-(;
and the
specimens.
2. MATHEMATICAL
The
use of these
test
stress
continuum
graphite
caused
in the con-
from
Further
section.
were
effects
The
uniaxial
section
comparisons
material,
collection
to be determined
directly
results.
this
parameters
combined
elastic-plastic
in the direction
is briefly
test
data
in properties.
equations
af’ter the
pleted.
were
axis of the specimen.
examined
analogs
of’ \,ariations
with
masking
allowed
combination
speci-
for
overcome
stitutive
Graphitite-(;
graphite,
to
Because
associated
procedures
procedures
curves
cylindrical
simplest,
postulated
material.
by \,ariations
results.
specimen,
the
used
data
of’ this material
a specialty
special
and the
stress-strain
mens were made from extruded
tubing.
Graphitite-G
graphite
to provide
and unloading
the
behavior
in properties
results
Thin-walled
extruded
physical
de\,elopment
of the theory.
The
second
section
describes
the tests
performed
using
specimens
made
from
is a comparison
conditions.
mathematically
the
which
nonlinear
and experimental
stress
of
the
basis of this examination
between
but
examination
continuum
developed
mechanical
The
the
small-deformation
THEORY
elastic-plastic
continuum
theory [ l-31, which was developed
to describe
artificial
graphite
beha\%~r, is
summarized,
and the stress-strain
relations
Wesearch sponsored by the U.S. Atomic Etnergy
Commission
under contract with the Union Carbide Corporation.
43
44
W. L. GREENSTREET,
G. T. YAHR
and R. S. VALACHOVIC
for pressurized
thin-walled
cylinders
are
given in this section. In the theory, as in the
classical mathematical
theory of plasticity,
it is postulated
that the total symmetric
strain tensor, lKL, can be decomposed
into
elastic and plastic components, that is,
Equation
(4) describes
the surface associated with initial or first loading from the
stress-free state of a virgin material. Loading
reversal. occurs when the stress point, which
represents
the stress state in stress space,
moves toward the interior of the current
loading surface.
As a result of loading
reversal, a new surface is generated.
EKL =
E;L -+ c;L,
W,L=
1,%3)
(1)
Consider the case in which the stress point
PKL has moved along an initial loading path in
where l
iL are the elastic components and l
are the plastic components. The elastic strains
stress space until a stress state denoted by
are given by the generalized Hooke’s law
oKL = ugL is reached. A reversal of loading at
will start generating
a loading
WKL = m;L
EiL = SKLMN CMN,
surface given by
(2)
where SKLMN
= SMNKL represents
the elastic
compliances
and oKL = (TLK is the stress
tensor. Equation (2) can be written in incremental form as
dGL
=
SKLMN
(3)
dg,N.
Prior to discussing the stress-plastic strain
relations, loading, or potential, surface behavior must be considered.
In the case of
artificial graphite, a yield surface in the classical sense does not exist; that is, there exists
no surface which encloses a region of purely
elastic response. However, the concept of
a loading surface, distinct from a yield surface, can be introduced. As shown in Refs.
[l] and [2], the loading surface, or plastic
potential surface, for artificial graphite may
be expressed by
f
=
&KLMflKLflMN
=
K,
(4)
where the constants aKLMNsatisfy the relations aKLMN
= aMNKL
and K is a scalar
function
which depends on the history of loading.
Equation (4) represents the simplest mathematical form for describing loading surfaces
appropriate to the behavior of graphite. For
a more general expression for-f, which allows
for translation and growth of the loading
surfaces, the reader is referred to Refs. [l]
and [2].
&
=
hKLMN
(TKL,ll~,\lN,,,
=
K(lh
(5)
where
o
KL,,,
=
(TKL
-
fl$L
(6)
Here ogL represents a pseudovirgin state, and
the stress during loading reversal is measured
from azL in the same way as it is measured
from the zero state during the first loading
of the virgin body.
A new reversal of loading at cr* will again
start generating a new loading surface, which
is given in this case by
where
c
KL,l, =
uKL
-
dL,,,
.
(8)
This process is repeated for every loading
reversal.
Before proceeding with the discussion of
the elastic-plastic
theory, we shall consider
uniaxial loading. In his studies of uniaxial
loading ofgraphite, Jenkins [4] tacitly assumed
the validity of the postulate expressed by
equation (1) and established equations of the
stress vs.
following forms for describing
longitudinal
strain diagrams from uniaxial
tests:
E =
AuS-Bu2;
(9)
THE
BEHAVIOR
OF GRAPHITE
for initial loading,
E-Cm
=
A(rr-a,J
UNDER
BIAXIAL
45
TENSION
d8
-cB(a-urn)2
for unloading from a maximum stress
with the corresponding strain E,, and?
E-~,,=A(tr-u~)-tC~(u-u(,)2
(‘3)
(10)
cm
Similar
expressions
apply for successive
loading reversals. In equation (IS), (’ is the
same as the constant c in equations (10) and
(11). Further,
(11)
cLL,,, = & - lA-l,,
for reloading from (T,,with the coresponding
strain E,, when CTc Urn. In equation
(9),
(lo), and (ll), A (= l/E) is the elastic compliance, H is a material constant which characterizes the plastic deformation,
and c is a
constant. E is the elastic modulus.
-Jenkins’ development [4] resulted in a value
of one-half for c, with the value being independent of orientation with respect to the
grain, that is, independent
of whether the
specimen is oriented in the against-grain or
with-grain direction. In addition, this constant does not depend upon whether the
specimen
is being unloaded or reloaded.
Since his equations were shown[4,6-81
to
give good to exceptional representations
of
graphite
behavior,
the plastic strains are
taken as quadratic functions of the stresses
in the mathematical analog to elastic-plastic
behavior used here. However, the results of
our investigations show that c does not necessarily take the value of one-half as implied
by Jenkins, and it is for this reason that we
have included it as an unknown constant in
equations (10) and (11).
Following the developments
in Refs. [I]
and [2], the relation between plastic strain
increments
and stress increments
on first
loading is given by
(14)
where c;;, is the plastic strain existing at the
instant of reversal at a;,,
The relation between total strain increments and stress increments on first loading
is
(1.5)
For the first loading reversal the relation is
af‘ - a/’ dtr
.
acrnr,,,MI,,,
+ [?f;:,]l~2
aa
(16)
Here,
de KI.,,,= de,, - de;,.,
where de:,. is the total strain at the instant of
loading reversal at r&.
To simplify the ensuing discussion, we use
contracted notation in which the stress and
strain notations have the following equivalences:
and
The relation for the first loading reversal is
61
=
tin his first paper[4], Jenkins took the specific
case (TV= 0. In a later paper [5], illustrations of
stress-strain curves are given for stresses both in
tension and comoression.
El1
E‘j =
2E 23 =
y23
2E3, = y:<1
E3 = %:3 Eg = 2E 12 = YIZ.
Ep =
$2
E:, =
Note that hK,,(K # L) are
shear strain components.
(18)
the engineering
W. L. GREENSTREET,
46
G. T. YAHR
and R. S. VALACHOVIC
As mentioned
in the Introduction,
the
material is assumed to be traversely isotropic,
and, throughout the subsequent discussions,
the 3-axis is the axis of anisotropic symmetry
and the I- and P-axes lie within the plane of
symmetry,f
In contracted notation the constitutive equation,
equation
(15), becomes
1
s33 =-f
E33
where
(K, L = 1,2,3,4,5,
and the plastic potential
(4), becomes
or 6)
equation,
(19)
equation
(20)
As stated above, the compliances
and the
coefficients, or constants, in the plastic potential equation are symmetrical. In contracted
notations the symmetry relations become
SKI,=
E = Young’s modulus,
G = elastic shear modulus,
VMN = Poisson’s
ratio;
the first subscript
indicates the direction of applied stress
and the second indicates the direction
of induced strain.
Note that the constant A in equations (9), (lo),
and (11) corresponds to sll (= sZ2)and s33.
Through the use of Ref. [l], the following
equations can be written for relating the nonzero agL to parameters
that can be determined from uniaxial test results:
alI = az2= (2B11)2’3,
a12 =
azl = --kc2 mw2'3,
a23 =
al3 =
SLK
and
a31 =
a32 =
--pT3
= -&
a,& = aLn_-
(2B,,)2’3
(2B~3)~‘~,
(22)
a33= (~BxJ~'~,
We are now in a position to discuss the
constants and their relationships to quantities
determined from uniaxial tests. The nonzero
compliances
and their relationships
to the
usual elastic constants
for a transversely
isotropic material aref
S12
zzc
s23 =
s21
=
$32 =
--y12 -_ -- VZI
EII
En’
si3 =
-___--V13 _
f3l E
I1
“.“,‘,s c2.f)
F.
tl and 2 are against-grain directions, while 3 is
the with-grain direction.
$See Love [9] or Hearmon [IO].
a44 =
a55 =
(M23Yf3,
Nere, Btl(=B,,)
is the B value [equation (911
for the against-grain direction, BQ3 is Ihe B
value for the with-grain direction, and B23
is determined from a shear stress versus shear
strain diagram. The &, pT3, and p& are the
plastic strain ratios. [Again, the subscripts
on pgL (K rk L) have the same meanings as
those for VKL(K # L).]
The constitutive equations giving the axial
and circumferential
strains are written in
expanded form below. These equations will
be used in a subsequent section for making
between
predictions
of this
comparisons
theory and measured strains from the thin-
THE
BEHAVIOR
OF GRAPHITE
UNDER
BIAXIAL.
TENSION
17
at the instant of load reversal. Note that the
starred quantities
were designated
by E,,
and crmin equation (10).
walled cylinders. For initial loading,
3. TESTS AND TESTING
PROCEDURES,
GRAPHITITE-G
MATERIAL
(23)
and
(24)
where lq is the circumferential
the axial strain, and
R = 2
L
strain,
E:~is
= constant
is the axial-to-circumferential
The corresponding
equations
reversal ‘Jr unloading are
stress ratio.
for first load
(25)
and
t%,, =
s33(R
-
v:,,)n2,,>
quantities Q,,,
c3,>,,
[= Ku,, I )1are given by
The
E 2,,,=
l.
J,,) =
‘J “,,, =
=
(T ‘\,,>
UP,,,,
and
u3,,,
E2- E$,
Es-E;,
<r2- cr$,
(Tg- (T::.
The stresses a$ and a; are those associated
with the load reversal or unloading point;
the strains E._?and l2 are those that existed
In this section we describe the biaxial tests
conducted
on the thin-walled
cylindrical
specimens as well as uniaxial tests performed
to obtain material property data. The material property data were obtained from uniaxial
tests on the cylindrical specimens following
acquisition
of the biaxial data and from
standard tensile specimens. These data were
used to determine the parameters for use in
equations (23) through (26).
Tests were conducted on five cylindrical
specimens of the design shown in Fig. 1. As
may be seen from the figure, the gage sections of the cylindrical specimens were 6-in.
long, with a 0.092-in. wall thickness, and a
mean radius of 0.922 in. The specimens
were fabricated from tubes having a liin.
inside diameter and a 2 in. outside diameter.
Short
cylindrical
sleeves were machined
from 2-in.-inside diameter by 2$in.-outside
diameter tube stock and were glued over
each end of IO-in.-long cylinders, which were
machined to form the gage sections. ~l-he
composite
structures
were then machined
to the final specimen configuration.
Since
the specimens were parallel with the withgrain direction of the graphite; the material
was isotropic in planes normal to the axes
of the cylinders.
Each cylindrical specimen was instrumented
on the outside surface with strain gages
oriented
in the axial and circumferential
directions. Type C6-121-A Budd Metalfilm
gages with a 0.125-in.
gage length were
used, and they were mounted in pairs (one
axial gage and one circumferential
gage)
at selected locations. One specimen (no. 1)
had two sets mounted 180-deg apart at the
center of the gage section, while the remaining four had four pairs mounted at SO-deg
intervals around the circumference
at the
48
W. L. GREENSTREET,
G. T. YAHR
and R. S. VALACHOVIC
DRNL_DWG
69_5046force only. A specimen mounted in a loading
fixture for applying internal pressure plus a
compressive force so that no net axial load
is exerted on the specimen is shown in Fig. 3.
[~ t
Each specimen was subjected to biaxial
‘?
! 2~49 loading;
following
this, uniaxial
loading
1 I
conditions were imposed in various sequences.
A summary of the tests and test conditions
is given in Table 1. The loadings used and
the sequences of applications are given in
Fig. 1. Design of thin-walled cylindrical specimen.
column 2 of the table, where the circumferential to axial stress ratio for each loading is
also listed, A cycle consisted of loading to the
center of the gage section. In addition,
maximum pressure or load and unloading
pairs of gages were mounted in the transition
to zero pressure or load. The specimens were
regions at the ends of each specimen. In
unloaded completely before being subjected
two cases, the specimens
(nos. 2 and 3)
to subsequent loadings. The stresses in the
were extensively strain gaged to obtain strain
axial and circumferential
directions
for
distributions as functions of axial position.
Figure 2 shows a cylindrical
specimen
pressure loading were calculated using thinwalled cylinder
equations.
Note that the
mounted in a loading fixture for applying
sequence of loading is not the same in all
internal pressure only or an axial tensile
Table 1. Outline of tests on Graphitite-G thin-walled cylindrical specimens
Specimen
no.
1
2
3
4
5
Load
type
Maximum
internal
pressure
(psi)
Maximum
axial
load
(lb)
Circumferential
(psi)
213
1603
0
1804
0
2134
2 : 1 biaxial
1 : 0 circumferential
0 : 1 axial
200
2004
190
1904
0
2 : 1 biaxial
0 : 1 axial
1 : 0 circumferential
200
2 : 1 biaxial
1 : 0 circumferential
0 : 1 axial
200
190
2 : 1 biaxial
1 : 0 circumferential
0 : 1 axial
200
190
2 : 1 biaxial
0 : 1 axial
1 : 0 circumferential
0 : 1 axial
2 : 1 biaxial
160
Maximum stress
900
180
1000
Axial
(psi)
Number of
cycles
802
1689
0
1
1
2
1877
2
1067
1002
0
1877
1
2
2
0
1904
1002
1877
0
2
2
2
1000
2004
1904
0
1002
0
1877
2
2
2
1000
2004
1904
0
1002
0
1877
2
2
2
1000
2004
1000
190
THE
BEHAVIOR
OF GRAPHITE
cases. Specimen
no. 1 was subjected to a
maximum
internal
pressure
of 160 psi,
while the maximum internal pressure was
200 psi in a11 other cases. In addition, specimen no. 1 was pressurized to failure under
combined stress conditions as a final step.
The other specimens
were not loaded to
Sailure.
The strains measured at the gage locations
along the lengths and around the circumference of specimen nos. 2 and 3 were carefully
examined for each of the three types of loading, that is, internal pressure, axial load, and
internal pressure with no net axial load. To
facilitate this examination, strain distributions
were plotted for various load levels. These
plots showed that the peak strains which
occurred
in the t,ransition regions at the
ends of the gage section diminished rapidly
with distance toward the center of the specimen. Thus, the central regions were not
affected by the discontinuity stresses. There
were small differences
in the strain values
measured around the circumferences
at the
centers of the specimens. As an example
of the results obtained,
the strains for
specimen no. 3 are shown in Fig. 4 for the 200
psi internal
pressure
loading. The strain
values fi.om gages along a single generator
are shown as circles. The strains for other
gage locations around the circumference
of
the specimen are denoted by squares.
Twenty-four
tensile specimens were used
to obtain stress vs. longitudinal
strain and
stress vs. lateral strain data. Eight with-
Fig. 4. Strain distribution
in Graphitite-G
3 under 200 psi internal
pressure.
CAR-Voi.11No.1-D
tube No.
UNDER
RIAXIAI.
39
TENSION
grain and eight against-grain
specimens
were machined from P-in.-diam rod stock,
while eight additional against-grain
specimens were machined from the tube stock,
which had a 2 in. inside diameter and a 23 in.
outside diameter. The latter stock was also
used to make end sleeves for the thin-walled
cylindrical
specimens.
A modified ASTM
specimen design [l 11 was used in each case.
Figure 5 shows the design of the with-grain
the gage section was 0.310 in.
specimen;
in diameter and $in. in length. ‘I’he design
of the against-grain
specimen is shown in
Fig. 6, where the specimen
is shown superposed on the rod stock cross section. ,411 of
the against-grain
specimens had a 0~125in.diam, &in-long gage section. In the case of
the specimens from tube stock, cylindrical
rods and end caps, as shown in Fig. i’(A),
LENGTH
Fig. 5.
DIMENSIONS
IN
With-grain tensile specimen
INCHES
Fig. 6. Against-grain
tensile
specimen.
50
Fig.
W. L. GREENSTREET,
G. T. YAHR
7. Details of against-grain tensile specimen
from tube stock.
were made and assembled to form a composite
structure. This structure was then machined
to the final configuration
illustrated in Fig.
7(B).
Each against-grain
specimen was instrumented with four micro-measurement
type
~A-13-05OAH-120
strain
gages
with a
0*050-in. gage length. Two of the gages were
oriented in the axial direction and two were
oriented
in the circumferential
direction.
The two axial gages were located 180-deg
apart.
The
circumferential
gages
were
mounted to form T configurations
with the
axial gages. Each with-grain specimen was
instrumented
with
four
type
similarly
CG-111 Budd ~etalfilm
strain gages having
a 0*063-in. gage length.
Since the parameters, or constants, in the
constitutive equations cannot be determined
from monotonic
loading alone, the specimens were generally cycled twice between
zero and a constant maximum stress level
before being loaded to failure. Two of the
with-grain specimens and one against-grain
specimen
were loaded to failure without
cycling, while one with-grain specimen was
cycled between zero and 3600 psi prior to
being loaded to failure. The remaining five
with-g-rain snecimens were cycled between
and R. S. VALACHOVIC
zero and 1800 psi as were all of the againstgrain specimens, except the one just mentioned.
A typical stress vs. longitudinal
strain
diagram is shown in Fig. 8. This diagram is
for an against-grain specimen that was made
from rod stock and cycled twice before
loading to failure. The envelope for all stressstrain curves (ignoring the unloading and
reloading, or cyclic, portions), the mean of
these curves, and the fracture points for the
against-grain specimens are shown in Fig. 9.
The circled fracture point is that for the
specimen which was failed without cycling.
The
corresponding
stress-strain
curve
envelope and mean curve as well as the
fracture
points for the eight with-grain
specimens are shown in Fig. 10. The circled
fracture points are those for the two specimens which were loaded, to failure without
cycling, and the square designates the fracture point for the specimen
which was
cycled between zero and 3600 psi. These
figures show that the scatter in data is not
large so far as graphite is concerned and that
cyciing had no apparent influence on failure.
500
q, STRAIN
(%)
Fig. 8. Stress vs. longitudinal strain diagram for
against-grain specimen.
THE
BEHAVIOR
OF GRAPHITE
UNDER
BIAXIAL
equation
4000
TENSION
51
[12]:
(27)
3000
where
“,
2000
&
t?
/Z= total, or apparent,
strain ratio,
E = longitudinal
(= E’ + cl’),
v = Poisson’s
to00
strain
ratio,
strain
ratio in the limit
of zero stress, or strain,
p” = plastic strain ratio.
0
01
0
02
0.3
cL, STRAIN
Fig. 9. Against-grain
04
The
0.5
(-7~)
elastic
tensile data.
the initial
that
were
used
for
of
ing,
c
t
I
0.4
0.2
C1.3
EL ,STRAIN
Fig. 10. With-grain
(%)
The
grain
and
were
used
equation
the
elastic
with-grain
uniaxial
to determine
(9),
(lo),
moduli,
from
and
the
the
(11).
material
were
second
unloading
portions
(see Fig. 8). The
deter-
first reloadof the
value
while
and
reloading
the
two
curves
average
The
from
0.4
the
by Jenkins
The
grain
from
the unloading
is 1.01 X 10” psi. Thus,
values
average
value
are
in good
agree-
value for B is 272 X 10-r”
of one-half
corresponding
specimens
given
results
that
in Table
which
were
in
include
constants
which characterize
the plastic deformations,
and the constant
c. Lateral-to-longitudinal
strain ratio, or total strain ratio, data were
also obtained.
The total strain ratios were
then
decomposed
into elastic
and plastic
components
through the use of the following
was given
for the six withcyclically
loaded
3. The
results
from
were
loaded
to failure
the
without
being cycled are not included.
In
the case of these data, the average
value
for E (= Etltl), as determined
specimens
significantly
which
[4].
two specimens
parameters
which
a value for c of 0.25 which differs
the against-
These
cB values
psi-“, and that for cB is 67 X lO_‘” psi-“, giving
tensile data.
curves
is
[8]. Also given are averages
of E is 1.08 X 106 psi for initial loading,
ment.
RESULTS,
GRAPHITITE-G
MATERIAL
stress-strain
method
determinations
the first unloading,
curves
specimens
The
average
are
4. TEST
loaded.
this value as determined
II
I
and
are listed in Table
these
in Ref.
from
cyclic
and
from
curves,
cyclically
E and
the
mined
=E.J,
as determined
of the 15 against-grain
making
described
E(=E,,
B (=B,,),
loading
2 for each
I
modulus,
plastic constant,
loading
from
curves,
the
is
unloading
from
1.86 X 10” psi,
and
reloading
the initial
and
that
curves
is l-96 X 106 psi. The latter is somewhat higher
than the former,
while in the case of the
against-grain
specimens
the reverse is true.
However,
the agreement
between
the two
values
value
is again good. The average H(=N:I:i)
is 73 X 10~” psi?,
and the average
value for cB is 26 X lo-‘” psiCz. Thus,
the value
W. L. GREENSTREET,
G. T. YAHR
Table 2. Graphitite-G
Initial loading
Specimen
no.
E
( lo6 psi)
B
(10-i’ psiP)
and R. S. VALACHOVIC
against-grain
tensile data
Unloading and reloading
( lo6 psi)
(cBLwe
( 1O-‘2 psi-*)
Stock
material
71
78
76
64
85
80
75
60
Rod stock:
2-in-diam
E ave
RI
R2
R3
R4”
R5
R6
R7
R8
1.13
1.02
1.07
277
288
334
0.98
0.99
1.04
1.14
364
298
284
268
1.01
0.96
0.93
0.92
0.93
0.95
0.95
1.02
Average:
1.05
302
0.96
74
Tl
T2
T3
T4
T5
T6
T7
Average:
1.05
1.11
1.10
1.14
1.15
1.10
1.16
1.11
290
281
268
225
230
220
175
241
0.97
1.02
1.04
1.09
1.10
1.08
1.13
1.06
54
63
72
61
55
63
51
60
Average for combination
1.08
Tube stock:
2-in.-ID X
2Qin.-OD
of rod and tube specimens:
272
1.01
67
*The initial loading portion of the curve obtained was not suitable for
evaluating the constants. The results from this specimen are not included
in Fig. 9.
Table 3. Graphitite-G
Initial
Specimen
no.
E
(IO” psi)
Rl
R2
R3
R4
R5
R6
1.93
1.87
1.78
1.88
1.90
1.80
1.86
Average:
with-grain tensile data
Unloading and reloading
W&e
Stock
material
17
31
17
39
25
28
Rod stock:
2-in.-diam
76
73
87
72
1.84
2.04
1.88
2.09
1.97
1.91
73
1.96
26
E ave
( lo6 psi)
;;
( lo-i2 psim2)
THE
BEHAVIOR
OF GRAPHITE
for c is 0.36, which again is significantly
different from one-half. Because c is assumed
to be a constant that is independent
of the
orientation of the specimen, the average of
the two values, or 0.30, will be used in the
analysis described in this paper.
The average total strain-ratio curve, which
is designated by p,, in Fig. 11, was obtained
from seven with-grain specimens. Since the
elastic strain ratio is equal to the total strain
ratio in the limit of zero longitudinal strain,
vQ1is represented by the straight line parallel
to the l~~l~gitudinal strain axis in this figure
(y:t1= 0.126). Equation (27) was used to calculate the curve for the plastic strain ratio,
&if. In this case, @, is essentially independent
of the longitudinal strain and it is negative
in the range shown. However, the apparent
strain-ratio data lack precision in the small
strain range shown, and additional inaccuracies are introduced
in the decomposition
process. Hence, the value of @, can be assumed constant without introducing
significant error. By giving greater weight to the
values at larger longitudinal strains, a value
of -0.08 was selected for the purposes here.
The total strain ratios, pip, were determined from three against-grain
specimens,
and values for v12 and pf;, were obtained in
the same manner as the vS1and &I values discussed above. These are listed in Table 4
where it can be seen that there are significant
differences
between the numbers for the
three specimens. However, the plastic strain
ratio is negative in each case. The average
O.zr--
.--“- ._~.__--
7
I
-0.1
-0.2
/
0
/____.-_.--..-_
/
/
0.05
ei.
Fig.
0.10
0.15
0.20
LONGITUDINAL STRAIN f%l
~~
0.25
0.30
Il. Average strain-ratio data for seven
Graphitit.e-G with-grain tensile specimens.
UNDER
BIAXIAL.
TENSION
Table 4. Graphit.ite-G strainratio data from against-grain
tensile specimens
Specimen
no.
R7
O-20
0.26
0.14
R8
TS
-0. 13
-0.28
-Q*O5
value for v12 from the three specimens is
0.20, while that for prz is -0~1.5.
Since the strain ratios needed in equations
(23) through
(26) are v13 and P;~, these
quantities were calculated from the above
values for v:$, and @&. Equations (21) give the
following relationship
between the elastic
strain ratios:
E
-__._u
1, 13 -
&
%l>
(28)
and equation (22) give the following relationship between the plastic strain ratios:
(29)
The values thus determined for v~:~and &\
are WO7 and -@03, respectively.
Although sKL and CL%:.
values, as required
in equations (23)-(26), were determined from
the tensile specimens, variations in material
properties
preclude
their use in making
definitive
comparisons
between calculated
and experimental
results for the cylindrical
specimens under internal pressure loading.
The strain-ratio values and the value for c
will be used, however, The uniaxial tests
conducted
on each cylindrical
specimen
provided means whereby the masking effects
due to material variability could be overcome.
But it was necessary to determine the needed
data in such a way that the effects of prior
loading histories did not inHuence the values
obtained.
54
W. L. GREENSTREET,
G. T. YAHR
After loading a specimen in a given direction
in stress
space,?
the stress-strain
response obtained upon loading in a different
direction is not the same as loading a virgin
specimen in the second direction. However,
the model of graphite behavior given in
Refs. [l] and [2] predicts that, once the
specimen has been loaded to a given stress
level in the new direction, unloading and
reloading curves subsequently
obtained by
cycling between
where u& is the stress at the unloading point,
are the same as the unloading and reloading
curves that would have been obtained using
a virgin specimen.
Therefore,
A and CB
values for each of the cylindrical specimens
were obtained from cyclic unloading and
reloading curves produced by the application and release of an axial force and by
cyclic pressurization
with no net end load.
The B values were then calculated using the
value for c from the tensile tests. The average
A and B values obtained in this manner are
tin this case we are concerned
with a twodimensional space in which the axes are labeled
by crz and v~, that is, the axial and circumferential
stress components, respectively, and we further
restrict ourselves to loadings for which the ratio
of a2 to a3 remains constant during a given loading.
Table
Specimen
no.
and R. S. VALACHOVIC
listed
for
each
of the
mens
in Table
5. The
the
cylindrical
and
five cylindrical
overall
tensile
of Tables
for
are
in Table
2 and 3 shows
that,
5
in
the A and B values are larger for the
Also, Table 2 indicates
tensile specimens.
that the B values for specimens made from
rod stock are higher than those from the
tube stock, but the value for c is essentially
the same for the two cases. Finally, it may be
seen that the variations in B values for the
cylindrical specimens are less than those for
the tensile specimens.
Since strain ratios for metals and other
materials are positive, a question naturally
arises regarding
the validity of the stressplastic strain relations for the case where
exist.
The
negative
plastic-strain
ratios
validity in the case at hand is determined
through an examination of the plastic potential function,
which in the mathematical
theory of plasticity must represent a convex
surface in stress space[l3,14].
Therefore,
a
statement
of the requirements
for convexity of surfaces given by equation (20) is
necessary.
Convexity is assured by requiring that the
matrix of coefficients,
general,
bKL/k
be positive
definite.
The
nonzero
5. Data from cylindrical specimens
An
A
(10e6 psi-‘)
(10m6 Tsi-I)
2
3
4
5
0.73
0.78
0.80
0.76
0.77
0.56
0.47
0.58
0.46
0.55
123
133
148
136
135
42
39
50
40
51
Averages:
Cylinder data
Tensile data
0.78
0.92
0.52
0.54
138
272
45
73
1
specimens
given at the bottom of the table.
A comparison
of the results
with those
speci-
averages
B11
(lo-‘*
psiP)
B3
(lo-l2
psi+)
coefficients
THE
BEHAVIOR
OF GRAPHITE
UNDER
BIAXIAL
55
TENSIOK
in this matrix and the equivalences in terms
of measured quantities are given by equation
(22). By imposing
the positive definite
requirement,
the following inequalities are
obtained [15, 161:
B,, > 0.
& > 0,
H14 > 0,
-1 < /..LLz<1.
( 1 - /qz. 1 ’
(30)
%$/4~.
These inequalities show that the strain ratios
can be positive or negative so long as they
remain within the bounds given. The B
values are greater than zero in all cases.
Further, it may be seen that the fourth and
fifth inequalities
are satisfied. Thus,
the
surfaces described by equation (20) are convex.
0
0.10
0.05
o.r5
STRAIN
Fig. 12. Comparisons of measured
curves with
calculated results for Graphitite-G specimen no. I.
2000
5. COMPARISONS
OF CALCULATED
EXPERIMENTAL
RESULTS
0.20
vo)
-CIRCUMFERENTlA
AND
The axial and circumferential
strains for
cylindrical specimens nos. 1 and 2 under the
initial internal pressure loading cycles were
calculated using equations (23)-(26), the constants in Table 5, and the strain ratios and c
value from the tests on tensile specimens.
The experimentally
determined
curves for
these specimens are shown in Figs. 12 and
13 where the points calculated from the
equations are indicated by the open circles.
The experimental
and theoretical
results
shown are in excellent agreement.
Similar
agreement
was found for the remaining
specimens.
Figure 14 shows the comparisons between
theoretical
and experimental
results representing
the averages
from specimens
nos. 2 through 5 when the A and B values
obtained from these cylindrical specimens
are used. Again, the figure shows that the
results are in excellent agreement.
The results of the studies given in this
report demonstrate the need for determining
uniaxial data directly from the specimens
500
-
CALCULATED
0
0
0.05
0.10
STRAIN
0.!5
0 20
Phi
Fig. 13. Comparisons
of measured curves with
calculated results for Graphitite-G specimen no. 2.
tested under
combined
states of stress.
However, investigations using other graphites
have shown that very good agreement
between calculated and experimental results
for graphites exhibiting small variations in
mechanical
properties
are obtained when
data from tensile specimens alone are used
in the analyses. Thus, for some graphites,
56
W. L. GREENSTREET,
G. Y. YAHR
and R. S. VALACHOVIC
were:
E,, =
Ea3 =
V13=
c=
2000
1500
^
::
b
6. CONCLUSIONS
500
t
k
0
1
0.05
0.10
STRAIN
0.t5
0.20
(%)
Fig. 14. Comparisons
of averaged
measured
curves
and averaged
calculated
results
for
Graphitite-G specimen nos. 2-5.
taken
here
in obtaining
the precautions
materials properties data are not necessary,
and the equations may be used with confidence for practical applications when stressstrain data from the uniaxial specimens
alone are obtained.
To illustrate
this point, a comparison
between
calculated
and experimental
results for a specimen made from a specialty
graphite are shown in Fig. 15. The parameters used in the constitutive
equations
1000
2
a
9
ks
‘,
600
400
200
-MEASURED
STRAIN
Fig.
B,, = 99 X IO-12 psi+
Bz3 = 36 X 10-12 psi-”
/_l$= -0.10
where all of the values were determined
directly from tests on uniaxial specimens.
Again, the agreement
between theory and
experiment is very good.
t-z
g 8000
0
127 X IO6 psi
2.37 X 106 psi
046
0.25
i%)
15. Comparison
of measured curves with
calculated results for a specialty graphite.
It is shown that nonlinear
stress-strain
behavior of graphite under combined stress
conditions
at room temperature
can be
accurately described by a small-deformation
elastic-plastic
continuum
theory.
Results
calculated through the use of specia1 constitutive equations appropriate
to the desscription of graphite behavior were shown to
be in excellent agreement with experimental
results corresponding
to initial Ioading and
to unloading under combined states of stress
produced
by internal
pressure
in thinwalled cylindrical
specimens.
Further,
the
calculated residual strains upon unloading
were very close to those determined experimentally.
In the examination
of a theory, such as
that described,
it is essential that experiments be designed to overcome problems
associated with the accurate determinat.ion
of material
constants
and to minimize
masking effects due to material variability.
Because
of the variability
in mechanical
behavior of the primary material used, parameters for use in the constitutive equations
in this investigation were determined using
results from tensile specimens along with
uniaxial data obtained from the thin-walled
cylindrical specimens.
The results derived
establish, in part, the validity of the theory.
Additional
investigations
to examine
this
theory should be designed to study the basic
postulates since the validity has been established, as shown here, in what may be
considered a gross sense.
THE
BEHAVIOR
OF GRAPHITE
Acknowledgements-The
authors
gratefully
acknowledge the assistance of H. A. MacColl in the
data reduction, and J. M. Chapman for his help in
instrumenting
and testing the specimens. Theyalso thank C. E. Pugh for his careful review and
helpful criticism of this work.
UNDER
10.
11
REFERENCES
1. Greenstreet
IV. L.. Ph.D. dissertation, Yale
University (1968).
W. L. and Phillips A., Acta
2. Greenstreet
Mechanica, to be published.
3. Greenstreet,
W. L. and Phillips, A., Proc.
First Int. Co?/. on Structural Mechanics in Reactor
Technology, to be published.
4. Jenkins, G. M., Br.J. Appl. Phys. 13,30 (19fi2).
5. .Jenkins, G. M. and Williamson, G. K.,J. Appl.
Phys. 14,2837 (1963).
6. Rappeneau, J.. Jouquet,
C., and Guimard,
M., Carbon 3,407 (1966).
7. Seldin, E. J., Carbon 4, 177 (1966).
8. Greenstreet,
I&‘. L., Smith, J. E., and Yahr,
G. T., Carbon 7, I5 (1969).
9. Love, A. E. H., A Treatise on the Mathematical
12
13.
14.
15.
16.
BI.4XIAL
TENSION
.55
Theory of Elasticity, 4th Ed., Dover, New York
(1944).
Hearmon, R. F. S., An Introduction to Applied
Anisotropic Elasticity, Oxford University Press,
London (1961).
‘Standard Methods of Tension ‘I‘estinq of
Carbon-Graphite
Mechanical
Mate*‘ials,’
ASTM Designation:
C565-71, pp. 4%-499
in 1971 dnnual
Book of ASTiZl Standards,
Part 13, American Society for Testing and
Materials, Philadelphia (1971).
Shanley.
F. R., Weight-Strength
.4ualysi.\ o/
Aircraft Structures, 2nd Ed.. Dovrl-, New York
(1960).
Prager, W.,j. A#/. Phys. 20,235 (1949).
Drucker,
D. C., Proc. First I’S.
National
C;ongre.\.\ of Applied ;Mwhrcnic.c, pp. 387-491,
New York (1952).
Stein,
F. M., Introduction
to Matrzcas mind
Wadsworth.
Determinants,
100-101,
PP.
Belmont, California (1967).
Born, M. and Huang, K., Dynamical Th,eor)j of
Crystal Lattices, p. 141, Clarendon
Press,
Oxford (1954).