Nuclear Engineeringand Design 116 (1989) 329-342
North-Holland, Amsterdam
329
C O M P U T A T I O N A L T E C H N I Q U E S F O R INELASTIC S T R E S S AND FRACTURE ANALYSES
S.N. A T L U R I
Center for the Advancement of Computational Mechanics, Georgia Institute of Technology, Atlanta, Georgia 30332-0356, USA
Received 27 October 1988
Details of a unified creep-plasticity theory, based on the concept of an internal time, are presented, and a geometric
interpretation of such a theory is given. A' tangent-stiffness'formulation,for the finite-element/boundary-elementcalculation
of weak solutions of the strain-history in the structure, is presented. An implicit algorithm of generalized mid-point radial
mapping for computing the stress-history at a material point for a given strain-history is given. Appropriate crack-tip
parameters that may be used to correlate the creep-crack-growthdata for single-dominate-flawsin structures operating at
elevated temperatures are discussed.
1. Inlroduction
In several of the general-purpose computer programs commercially available today, the non-elastic
strain in a solid(/structure) operating at elevated temperatures is usually treated as the sum of 'a
time-independent plastic strain' and ' a time dependent creep strain', as long as the strains are infinitesimal. A similar formal summation of the 'rates' is considered in a large-strain problem. However, in as
much as the underlying physical processes of dislocation glide, climb, pileup, etc, are common to both
plasticity and creep phenomena, a 'unified' constitutive relation is more preferable to characterize
plasticity and creep as well as creep-plasticity interaction. Such unified theories are justified on the basis
of experimental evidence available on the behaviour of structural alloys at elevated temperatures.
Several unified creep-plasticity relations have been proposed in the recent literature. Examples of these
are the works of Bodner and Partom [4]; Bodner et al. [5]; Chaboche [9]; Hart et al. [11]; Krieg et al. [12];
Lee and Zaverl [14]; Robinson [21]; Stoufer and Bodner [24]; Walker [25]; and Watanabe and Atluri
[26,27]. In this paper, a further elaboration of the unified creep-plasticity model given in Watanabe and
Atluri [26,27], based on the concepts of an endochronic theory of Valanis [28], is given. Emphasis is placed
on: (i) a geometric interpretation of the unified theory [26,27] and (ii) an implicit, 'forward gradient'
algorithm of generalized mid-point radial mapping for the determination o f stress-history based on this
unified theory, for a given strain history, and (iii) appropriate 'tangent-stiffness' finite-element and
field-boundary-element spatial discretization algorithms for determining the strain-history. These three
algorithmic steps constitute the key ingredients for a successful implementation of the present unified
creep-plasticity theory for inelastic stress-analysis of pressure-vessels and piping operating at elevated
temperatures and moderately high stress levels.
Another important topic in the life prediction of structures operating at high temperatures is that of
creep-growth of a single-dominant-flaw (crack) in such structures. In this paper, a summary of various
crack-tip parameters such as C*, C(t), 7"* that have been proposed as candidates to correlate the
creep-crack-growth rates with [i.e., d a / d t = fl( )~ where ( ) is the candidate crack-tip parameter] is
given. Some of these parameters [C*, C(t)] are linked to the use of a specific material model, to
characterize creep, such as the Norton's power law (i - o") type steady state creep. Others such as 2P* and
~/'c, are independent of the material model employed and thus are suitable when creep and plasticity are
present near the crack-tip. A detailed discussion of the suitability and generality of each parameter in
situations of non-steady creep, as well as methods for their evaluation in structural problems, are given.
0029-5493/89/$03.50 © Elsevier Science Publishers B.V.
S.N. Atluri / Computationaltechniques
330
2. A unified creep-plasticity relation based on the concept of an internal time
We use the notation that boldface characters denote a second-order tensor; for instance, d , / d e n o t e s the
differential of inelastic strain which includes the effects of both creep and plasticity. The notation of
(dT! : dT/) is used to denote the tensor trace product, i.e., d~"2 = dT/ijd,/ij. The differential of internal time
dz is defined through the relation (cf, [28]);
dz 2 = ~'-----~2
d
+ t---~2
d
[ f(~,)]z
g2 '
(2.1)
where d~" is the differential of equivalent inelastic strain to within a scalar constant (the actual equivalent
inelastic strain differential, dT/= i/~/3 d~'), and dt is the differential of the Newtonian time. In the above,
f ( ~ ) is a monotonically increasing function of ~', with f ( 0 ) = 1. Further, g is a 'scaling function', defined
(see [26,27]) as
]
m > 1,
(2.2)
where B is a constant; s is the deviator of the Cauchy stress in the material-particle; r is the deviator of
the back-stress (which is an internal variable); ry° is an initial yield-stress; and m is an exponent, with
m > 1. Further IIs - rH 2 = ( s - r ) : ( s - r ) . In an initially (t = 0) stress-free solid, it is seen that g(0) tends
to ~ . We take the initial conditions at t = 0, to be ~"= 0, z = 0. Following [28], we assume that s has the
integral representation:
s=2OfoP(Z-:')aT,
z
d~ dz,,
(2.3)
where #0 is the initial shear modulus, and p is a kernel of z. We assume that p has the form:
p ( z ) = Po 8 ( z ) +
pl(z),
(2.4)
where ~(z) is a Dirac function centered at z = 0. Use of (2.4) in (2.3) results in:
d*l d z '
+ 2/*0f0zp~(z - z') dz----7
s = 2#o0o-~
- r ° d ~ + r(z),
Y dz
(2.5a)
(2.5b)
wherein the definitions of r ° and r(z) are apparent. It can be seen that the kernel pl(z) characterizes the
back-stress r(z), and thus pa(z) characterizes the kinematic-hardening or the translation of the yield
surface. From (2.5b) we obtain the relation:
dll
dz
(s - r)
r°
(2.6)
From (2.6), one has:
IIs - rll 2
d~ 2
(,o/2
d:2
(2.7)
S.N.Atluri/ Computationaltechniques
331
Use of (2.1) in (2.7) results in the relation:
[Is-- r][2
(
70----2 =
[~y/(~-)]
1 dt2 )
1
(2.8)
.
? -dz'
In a continuing inelastic process, by the definition of internal time as in (2.1), (dz/dt) is always positive.
Thus, the right-hand side of 92.8) is always less than 1 in a continuing inelastic process. From eq. (2.8) it is
also evident that the function f(~) characterizes the expansion of the yield-surface or the isotropic
hardening. From (2.8), one has:
d,
1[
,,s
,,,2
.
(2.9)
(::
From (2.6)and (2.9),one has:
d~_ (s-r)
(s:r)
:
:
111
-- 1 / 2
IIs - rll 2
(¢:)2
(2.10)
Using the definition of g given in (2.2), one has:
-- 1 / 2
d~!
dt
IIs - rll
IIs - rll B
~.yOf
IIs - rll 2 )
1
(2.11)
(,o: 1'
-NG(~, ¢),
(2.12)
where N is a unit director in the stress-space, defined by:
i v - (s - r).
IIs- rll '
N: N= 1
(2.13)
and G is a material-characterization curve such that, for uniaxial tension,
d~/dt = G(ff, ~'),
(2.14)
d~l/dt = ~ - G ( f f , ~),
(2.15)
i.e., G(o, 71) characterizes the strain-hardening behaviour in a uniaxial creep test, and
G(ff, ~) -= B ~.of(~) ]
1
[~.yof(~,)]2
.
(2.16)
In the case of a uniaxial tension creep test;
S 1 1 = "2~ O l l ,.
$22 =
- - -1~ O l l , .
$33=
1o11.
(2.17)
Also, since dTI is purely deviatoric in the present theory, it follows, from the definition of r that is evident
from cqs. (2.5a,b), that:
r22 =
1
-- ~rll
.
,
/'33 = -- ½r11.
(2.18)
S.N. Atluri / Computational techniques
332
From eqs. (2.15), (2.16), (2.17) and (2.18) it follows that in a uniaxial creep-plasticity test, the equivalent
strain-rate is given by:
O_._~_~= ¢~-~B l o l l -- 3rll I
dt
rof(~ ")
lm[
1
I°n
-- 3rll
0
]
12 -1/2
2
(2.19)
Eq. (2.19) can be used to fit material test data, once the growth-law for the back-stress r n is specified.
Recall from eq. (2.5) that:
, ( ~ ) = 2~,0f0zp,(z _ z ,,)h-~Tz,
d,7 dz'.
(2.20)
A convenient choice for p l ( Z ) is an exponential function of z, i.e.,
p,(z)
=
~.Pu e x p ( - a , z ) ,
(2.21)
i
where a t are constants. Use of (2.21) in (2.20) results in the relation:
d r = E {21~0Pli d~l - r(i)oti dz },
(2.22a)
i
where
r(i) = 2 ~ 0 f0zPli e x p [ - - O l i ( 2 -- 2 ,.,
)l ~d~l d g ' .
(2.22b)
Eq. (2.22a) can be written, upon using (2.9), as:
i. = d , ' / d t
= Ei
air ")
2j, op,,~ - - - ~
1
IIs - rll 2
(,ry°f) 2
(2.23)
Thus the present theory involves a nonlinear kinematic hardening.
For isotropic hardening, we use the exponential relation:
f ( ~ ) = {a + ( 1 - a) e x p ( - 7 ~ ' ) )
a and r are constants.
(2.24)
Thus, the material constants of the present theory are summarized as follows: (i) the constants a and 7 in
the isotropic hardening function f(~) (see eq. (2.24)); (ii) The constants Ply, at (i = 1, 2, 3) in the
kinematic hardening kernel p l ( z ) (see eq. (2.21)); and (iii) the constant B and m in the uniaxial strain-rate
relation G(e, 7/) (see eq. (2.16)).
The determination of the above material constants from uniaxial monotonic and cyclic plasticity data;
as well as creep data at high temperatures and at varying stress levels; has been illustrated, in detail, for
the case of 304 stainless steel in [27]. Further, it has been shown in Watanabe and Atluri [27] that the
present theory models the m a x i m creep data at stress levels both below and above the initial yield limit,
rather exceptionally well.
In summary, once the uniaxial-creep-characterization of the material is modeled by the function
G(5, ~) as in eqs. (2.15) and (2.19), one obtains, for multiaxial creep representation in the present theory.
d~l
a7 = N G ( 5 , ~') =
d
N-~,
(2.25)
S.N. Atluri / Computational techniques
333
where TI is the inelastic-strain, incorporating the unified effects of creep and plasticity and creep-plasticity
interaction.
Also, it should be pointed out that in the present theory, there is no purely elastic regime and thus there
is no hypersurface in the stress-space which demarcates between the elastic and inelastic regions. Rather,
inelastic strains occur as all stress-levels, along with elastic strains. Assuming that the material is
elastically-isotropic, the rate of stress in the present theory is given by:
do = 2/~o de e + ho(d,e : I ) I ,
(2.26)
where de e is the elastic-strain differential, such that:
dc ~ = dc - d~/,
(2.27)
where dc is the total-strain differential. If ds is the deviator of do, one also has:
ds = 2/xo [de - dtNG(~, ~)] = 2/~o de -
N-~ a t ,
(2.28)
where de is the deviator of dc¢ or dc (since the inelastic strain is purely deviatoric here); and
(do: I ) = (E/t o + ho)(dc : I ) ,
(2.29)
where/~o and 2,o are the elastic Lam6 constants.
3. Geometric representation of the present unified theory
To facilitate the interpretation of the present theory as a multi-surface theory, we postulate that the
kernel pl(z) is assumed to be a piecewise-continuous, rather than a continuous function as in eq. (2.21).
Thus, let pl(z) be of the form:
Pl(z) = ~1 e-a~z
in segment 1
(3.1a)
= ~2 e-~2z
in segment 2
(3.1b)
= P3
in segment 3.
(3.1c)
Consider the process of creep deformation when the applied stress is within the first yield-surface f o ) ,
represented by:
[,,',
-
,,',1: :,,1,_,,,,]
Since the kernel
pl(z)
=
-- (2/XoPof) .
(3.2)
is initially characterized by (3.1a) the evolution equation for the back-stress r, is
given, in this first segment, by:
d r 0) = 2po~1 d~l - ~lr O) dz
L ~ Poal I
'-1.t]
(3.3)
(3.4)
Poal I
Since, in the first segment, using (2.6), we have:
s -- r 0)
d ~ - - 2P.oPo
Oz.
(3.5)
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S.N. Atluri / Computationaltechniques
Eq. (3.4) may be rewritten as:
d r 0)
dz
dt = ~ ' ~ [ ~ - s ]
(3.6)
where,
s*=
1+
Pl
s - r(1)).
(3.7)
p0al
The geometrical interpretation of the present kinematic-hardening follows from eqs. (3.6) and (3.7). Since
the considered s is inside the first yield surface, locate a stress-point s (1) on f(1) such that:
[ s - r ( ' ) ] = k [ s (1)-r(l>],
O_<k<l.
(3.8)
We now introduce a second yield surface f(2) such that it may expand but not translate
The radius of f(2) is chosen such that:
R '2) =
1+
= 2g o
until a later time.
+ __Z f .
a0 ]
(3.9)
We locate the stress-point s t2) on f a ) , at which the normal to f¢2) is parallel to the normal to ftl) at s °)
(with s °) defined as in (3.8)). That is to say,
$(2)
$(1) -- r(1)
R(2)
~.of
or
$(27 = [ $ ( 1 ) _ ¥(1)] 1 +
01
(3.10)
.
Now we define s * such that
s,=Ks(2)=[s_r(1)](l + PT ).
(3.11)
P0al
From eqs. (3.6), (3.7) and (3.11), the geometric interpretation of the kinematic-hardening theory is evident,
and is seen to be very analogous to the multi-yield-surface theories of Krieg [13] and Mroz [15].
4. An implicit algorithm for determining slress-history at a material point in an inelastic process
Recall that in the present theory,
(4.1)
N =
(s - r)
(4.2)
IIs - rll '
- 1/2
(4.3)
S.N. Atluri / Computationaltechniques
where
pl(z)
335
is an exponential function, and
NG(~, ~) d t ] ,
ds = 2~to[de -
(4.4)
where G(o, 7) is a material property function such that:
~1= ~f~-G(~, ~).
(4.5)
Formally, one may consider the inverse of the relation (4.5) to write:
~ = H ( ~ , ~),
(4.6)
where, ~, the equivalent stress, is defined as:
(s - r ) : (s - r) = ]~2.
(4.7)
Most of the computational algorithms currently utilized today for viscoplasticity may be described as the
'explicit' type. In these algorithms, the weak forms of the momentum balance laws in rate form, viz;
doij,j +
d f/= 0
(4.8)
or alternatively
2/~0 [deij.j ] + [(2/~ + •)/3] ,**.j 8ij --- - dfi + (2/~0N/j,j)G(ff, ~)
(4.9)
are considered. Thus, the effects of the inelastic strain-rates are taken to the right-hand side of the
momentum balance laws as equivalent body-forces. The usual finite-element - or boundary-element-spatial discretization of (4.9) results in a 'linear-elastic-stiffness matrix' from the terms on the left-hand-side
and the inelastic-strains are treated as corrective body-forces. This explicit iterative scheme is often
referred to as 'the initial-strain', elastic-stiffness, iterative approach. The limits on the time-step (At) used
in the incremental analysis (such that Aeij = e i j At, etc), such that numerical stability is retained and
accuracy is maintained, are quite severe.
We now describe an implicit algorithm wherein: (i) a 'tangent-stiffness stress-strain relation may be
used in constituting a tangent-stiffness matrix from finite-element/boundary-element methods, which lead
to approximate solutions for strain-increments, and (ii) a "generalized mid-point radial-mapping" implicit
algorithm is used to determine the stress-increment from the strain-increment'.
Towards this end, we assume that the function G(~, ~), and the attendant material constants, are
determined from unaxial creep tests, as outlined in §2. Let the increment of time be At. Let (~), and
(~l)t+at be the rates of equivalent inelastic strain at times t and (t + At) respectively. Let 0 At[0 _< 0 < 1]
be a 'generalized-mid-point' in the time interval between t and (t + At). We use the linear interpolation:
A~= A/[~'b(1 --O) +
~b+atO].
(4.10)
However, from (4.5) one has:
it+At ~ ~]t "~ " ~ m~ ~- ~
(
0G
0c
(4.11)
Aa ~
-~ Gt -4- - ~ A~ "l- "~o A ~
.
(4.11b)
Using (4.11b) in (4.10), one has:
A~= f~-3 At{ Gt + O(Gn A~ + Go A~) },
(4.12)
336
S.N. Atluri / Computational techniques
where G t is the value of the function G at time t; and
--- 0G/a
;
Go - 0C/0
(4.13)
,
when 0 > 0, we have the so-called forward-gradient method [1,18-20].
From (4.7) we have:
A6= ~ - N : (As-- At).
(4.14)
Upon using (4.3) and (4.4), one may write (4.14) as:
g
(v0j)2
,
(4.15)
wherein, the fact that n : de - N: de; along with eqs. (4.3) and (4.4) has been used.
Using (4.15) in (4.12) one obtains:
A~ =
(4.16)
A~I= ¢r~-NA~.
(4.17)
and
Using (4.16) and (4.17) in the equation:
Ao = 2#[d¢] + ~,(dc : I ) I -
2#o AT!
(4.18)
one obtains the 'tangent' viscoplastic constitutive relation in the form:
Aoij = EiVfkl AC.kl ,
(4.19)
where a%t is the total incremental strain, and Eijk~ is the tangent constitutive matrix, as defined from
(4.16), (4.17) and (4.18). Using (4.19) in the weak-form of the momentum balance law, i.e.,
( E i ~ ' A%,).j + A f / = 0
(4.20)
one may construct a 'tangent-stiffness matrix' from a finite-element or boundary-element type spatial
discretization. From such a F E M / B E M discretization algorithm, one may determine an approximate
solution for A%t.
We now discuss the details of determining (Ao) from the F E M / B E M computed A¢, using an implicit
algorithm of generalized-mid-point-radial mapping. These algorithmic steps are as follows:
(i) during a viscoplastic process, define a parameter 0 </3 < 1 such that A,a =/3 Ac; and define a vector
N~ in the stress-space, such that:
(s + 2g/3 AC) - ( r + / 3 a t )
iI(s +
a-23 -C7
;5,)11 '
(4.21)
S.N. Atluri / Computationaltechniques
337
(ii) determine the inelastic straJ~ increment AT/ from eqs. (4.17) and (4.16), wherein the normal N is
replaced by NB of eq. (4.21).
(iii) The 'correct' stress-increment is computed from (4.18), wherein A~1 is evaluated as per step (ii).
(iv) Compute Ao from (4.15), wherein the tensor N is replaced by Na of (4.21), and AT1 is computed as in
step (ii).
(v) compute Ar from the equation:
{
()_,/2}
Ar = ~'i 2p°Pli A~ - air(i) 1
g
•
IIs- rll
(,i.yOf)2
'
(4.22)
wherein ATI is computed as in step (ii).
The case when/3 = 1 results in a fully-implicit algorithm, while fl = ½ is found, in many experiences, to
result in an optimal algorithm (mid-point radial mapping).
5. Creep-growth of a single dominant crack and life prediction analysis
For structures operating at high temperatures and stress-levels, the linear elastic stress-intensity factor K
is not sufficient to characterize the crack-tip fields when significant inelastic response and stable
crack-growth precedes failure. In addressing situations wherein the structure is mostly undergoing
'steady-state creep' conditions and very slow crack-propagation, and when the creep-damage process zone
is very local to, and therefore controlled by the crack-tip field, Riedel and Rice [20] assume a material
model for creep, of the pure-power law type, ~ - o n, analogous to the power-law hardening elastic-plastic
material: Cp ~ o n. The solutions of Riedel and Rice [20] define a crack-tip quantifying parameter, C(t),
whose limit at infinite time is denoted as C*. There have been several attempts to correlate C* with the
rate of crack-growth in creep, (da/dt), with only mixed results [3].
For pure power-law hardening creeping materials (when the elastic strains in the structure are ignored)
i.e.,
= ,/~n,
_-
(5.1)
(
- is,j,
(5.2)
'
(5.3)
Sij -~ ¢Jij,
t
where oo is the deviator of oo; Riedel and Rice [20] show that, for a stationary crack, the crack-tip stresses
are of the form:
[ C(t)]l/n+l
°iJ = [ B----~,r
50( 0)"
(5.4)
In (5.4), (r and O) are polar coordinates centered at the crack-tip, and ao(O ) are finite functions of 0 and
material properties only i.e., they are independent of the far-field loading conditions. Riedel and Rice [20]
define C(t) to be:
C(t)=~mo{fr,[(n---~)aij,on
,-
Oui] ds},
where, in a two-dimensional problem, F, is a circular path of radius (, encircling the crack-tip and
centered at the crack-tip. In (5.5), n is the exponent in the power law creep relation; x 1 is the Cartesian
338
S.N. Atluri / Computational techniques
coordinate along the crack-axis, and n 1, is the direction cosine along x 1 of a unit outward normal to F,.
Note that (5.5) is defined for the particular material model of pure power-law hardening creep, i - o".
Note also that c(t) is not a path-independent integral i.e., it will not have the same value when integrated
on a far-field path F, while keeping the integrand the same as in (5.5)•
On the other hand, to characterize the crack-tip fields under situations of non-steady creep, arbitrary
material response near the crack-tip as well as in the structure, and arbitrary far-field loading, one may
define a parameter [1], as:
7" * = fr,( lTVn1 - t i -ff-~x
Oiq1 - t.i -~x
~u 1i ~} d F = /
~ h ~i - t i -" ~3xul }i d~F
(14/nl- ti-~x
F+Scr\
[ . alf. ij
• O¢lij ~
+ fVr_v, loiJox, - , i , - ~ x l ] dV.
(5.6a)
(5.6b)
In the above,
(5.7)
= Oij~ij
is the 'stress-power' (elastic plus inelastic), and hence is defined f. or any material response irrespective of the
postulated constitutive relation. Specifically, the rate parameter T* of (5.6) remains valid for crack-growth
in situations wherein elastic, plastic, and creep strains may be simultaneously present, as characterized by
a 'unified' constitutive law of the type described in §2. (In (5.6), F is a far-field path, Scr is that segment
of crack-face between F, and F; and V r and V, are, respectively, the domains enclosed by the curves F and
F,. Thus, /~* as defined in (5.6) has the essential path-independent nature]•
In the limit as 'steady-state creep' conditions attain under constant applied load, i.e., when d i j - 0 and
i i = 0, /~* of (5.6) becomes:
Ts* =
l/Vn 1 - t, d x,
(5.8a)
dr
3fq
r+S~r\
dr- f~_• ~[,ij-~x
[" 0%t
1 ] dF,
(5.8b)
wherein the subscripts (ssc) on /;* indicate 'steady-state-creep'. Once again, (5.8) is independent of the
material-model employed. Note that since W is the stress-power, one has:
•
~g = OijCij ;
~I'~"
3X1
3aiJ "
OiiJ
3X 1 •ij + aiJ OXl '
(5.9)
which has been used, along with the Gauss divergence theorem, to derive (5.8b) form (5.8a).
Goldman and Hutchinson [10] have earlier proposed a parameter C* for the steady-state creep, when
only creep strains are considered, and the creep strain-rate is proportional to the n th power of stress
(Norton's law) which saturates to o at any material particle. This is entirely analogous to the case of
deformation theory of plasticity, wherein % - o". Based on this analogy, Goldman and Hutchinson [10]
define C*, analogous to the deformation theory J, as:
c*= f r,~(W*n, - t'-~x~
oa, }~ dr'
(5.10)
where
~j
W* = f Omn d~mn.
Jo
(5.11)
S.N. Atluri / Computational techniques
339
Note that W* as defined in (5.11) is not the stress-power, it is simply a pseudopotential for ou in terms of
iu, and thus W* has no physical meaning. Note that W* is a single-valued function of iu, just as W in
the deformation theory of plasticity (nonlinear elasticity) is a single-valued function of c u. This, in
steady-state creep:
~W*
Oij = O~ij
~W*
3X 1
~W* ~ij
~ij
3{ij ~X 1 = O;ij 3X 1
(5.11a, b)
Use of the Gauss divergence theorem; eq. (5.11b), and the momentum balance law under steady-state
creep in the absence of body-forces and constant external loads (i.e., gj,j = 0), leads to the 'path-independence' property of C*, i.e.,
C* =
W'n1
dr.
-
(5.12)
It can be seen that, due to (5.8b) and (5.9) when ~," is used instead of W*, one has, even under
steady-state creep,
:
(Wsse-W*)nl=
l(Wssc-W*)nl-
-v, ~u-4--ox, d F ¢ 0.
(5.13)
Recall that while T* may govern crack-tip fields for all inelastic material models, C(t) does not. It can be
shown that while C(t) governs crack-tip fields for the material-model of power-law creep, its definition as
in (5.5) is also such that, under steady-state conditions of constant applied load it reduces at infinite time,
to C *. Towards this end, note that for the material model of (5.2), at infinite time, when the strain-rate is
of pure power-law creep, one has:
Wssc= oij~ij
=
o ~
___n+l
~q~-r%a
Thus, from (5.5),
C ( / ) Issc= t--~aottC ( t ) =
/.,.
=tx/y)
"~l/n[t ( ~. q ) X(n+l)/n .
fF,[(n)(l)l/n[..,(n
~
+ 1)/n
['eq)
(5.14)
~uil
rlI--Oijnj~xlJ ds"
(5a5)
On the other hand, from (5.11), one has:
w*
f iuOmn
= Jo
=
\-n-~1~-7
n ~[ l ll/n( .,,eq),(,+l)/n
I
(5.16)
From (5.10), (5.16), and (5.15) it can be seen that
Lt
C(t)
= C*,
(5.17)
but
t--*O0
vFt• n-~ x !
(5.18)
In as much as the definition of 7~* does not depend on any material model, and hence can accommodate
the unified creep-plasticity behavior at the high-stress levels that are likely to occur near the crack-tip, it
may be a suitable parameter that can be used to correlate creep crack-growth rate, at all times, including
the non-steady-state creep conditions.
340
S.N. Atluri / Computationaltechniques
The computation of 7;* as in eq. (5.6) is often inconvenient, i.e., the computations of line integrals
either near the crack-tip or away from crack-tip in a two-dimensional problem, is often inconvenient. It is
often convenient to evaluate 7;* only through 'domain-integrals' through the 'equivalent-domain-integral'
approach [16,17]. Towards this end, define a function s(xi, X2) in a two-dimensional problem, such that
s(xl, x2) = 1 on ip,,
s(x,, x 2 ) = 0 on F.
(5.19a)
(5.19b)
Apart from the restrictions (5.19), the function s is arbitrary but continuous in the domain Vr -V<
enclosed between F and F~. Several choices for s have been given in Nikishkov and Atluri [16,17]. Thus,
7;* may be redefined as:
0u,
. 0ii~ )s dip,
(5.20a)
r<t
01i~
. Ous ]
= k-c< l'iln 1 - ti-~x 1 - ti-~x 1 )s
dip,
(5.20b)
-
oijS-~xl]ldV.
(5.20c)
Thus, (5.20c) enables the evaluation of 7;* through only domain-integrals. It is in this sense, that the
evaluation of 7;* through (5.20c) is analogous to the 'virtual-crack-extension' method, except now the
actual process of 'virtual extension' is eliminated. Explicit algorithms (i.e., without the necessity for using
numerical quadrature) to evaluate the domain integrals in (5.20c), through the finite-element-method have
been discussed in Nikishkov and Atluri [16,17].
Also, earlier, Stonesifer and Atluri [22,23] considered a crack-tip parameter for non-steady creep, 7;c,
defined as:
7;c =
=
l,i/n1 -
k+ScF
] dr
I~n 1 - t,~xt ) d s -
(5.2la)
-
, ~ i J ~ x'
JVI,- V~k
1]
dv.
(5.21b)
Comparing (5.6a) and (5.21a) it can be seen that for general non-steady conditions,
ti-ff-~xl ) ds,
(5.22)
whereas, under the so-called steady-state creep,
(7;*)ssc- (7;c)s~ = 0 .
(5.23)
From the above discussion, it appears that 7;* as defined in (5.6a), or through its 'equivalent-domainintegral' definition as in (5.20c), has all the desirable attributes as a creep-crack-growth parameter in that it
is independent of the material model employed, it can account for general creep-plasticity-elastic strains
at the crack-tip, and it is valid for non-steady conditions. An appropriate definition of the contour F, over
which 7;* should be evaluated for growing cracks has been discussed by Brust and Atluri [71, Brust et al.
[6,8]. In Brust and Atluri [71, creep-crack-growth test data in a 316 stainless steel notched specimen was
numerically simulated and the variations of various parameters during crack growth were ascertained.
S.N. Atluri / Computational techniques
341
T h e s e results w e r e f o u n d to b e in f a v o u r o f t h e T * p a r a m e t e r in c h a r a c t e r i z i n g c r e e p c r a c k g r o w t h u n d e r
n o n - s t e a d y creep, as w e l l as in s i t u a t i o n s w h e r e i n c r e e p - p l a s t i c i t y i n t e r a c t i o n is s i g n i f i c a n t .
W o r k is u n d e r w a y in t h e U S a n d J a p a n to d e v e l o p e n g i n e e r i n g e s t i m a t i o n p r o c e d u r e s f o r a p p r o x i m a t e
e v a l u a t i o n s o f T * f o r c o m m o n s t r u c t u r a l g e o m e t r i e s , in lieu o f d e t a i l e d f i n i t e - e l e m e n t a n a l y s e s in e a c h
case.
Acknowledgments
T h e f i n a n c i a l s u p p o r t o f t h e U S O f f i c e o f N a v a l R e s e a r c h , a n d t h e e n c o u r a g e m e n t o f D r s . Y. R a j a p a k s e
and A. Kushner are thankfully acknowledge.,
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