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Tensile instability in creep damaging solids
~cra metall. Vol. 35, No. 10, pp. 2583-2592, 1987 Printed in Great Britain. All rights reserved
TENSILE
~1-4~~~87 $3.00 + 0.00 Copyright 0 1987 Pergamon Journals Ltd
INSTABILITY IN CREEP DAMAGING SOLIDS A. J. LEVY
Department of Mechanical and Aerospace Engineering, Syracuse University, Syracuse, NY 13244, U.S.A. (Received 8 August 1986) Abstract-Tensile instability in creep damaging bars subject to geometrical and/or material nonuniformities is investigated for the constant load and constant stress tests provided the longwavelength approximation is maintained. The constitutive model employed in the study is valid for the moderate stress-elevated temperature regime which is dominated by completely creep constrained cavity growth. Two definitions of instability are used to study initial rates of imperfection growth and imperfection growth through failure. The first is concerned with the immediate response to a disturbance (stability in the small) while the second is related to a loss of correspondence between increments of perturbed and unpertur~ field variables. A linearized stability analysis based on the first definition reveals that, like the pure viscous creep case, the deformation is unstable however, in contrast to viscous creep, the characteristic time for neck growth is much shorter. A nonlinear analysis based on the second definition associates the onset of instability with fracture occurring at a finite area but infinite local section strain. Histories of strain and area differences reveal the tendency for a highly localized neck to form, however this is countered by stability loss (fracture) at a finite area. R&n&-Nous avons &die l’instabilite en traction dam des barreaux endommages par fluage soumis g des h~t~rog~n~it~s de geombtrie etjou de materiau, dans le cas d’essais a charge constante et a contrainte constante oit l’on maintient ~approximation de la grande longueur d’onde. L’tquation d’etat que nous avons utilisie au tours de cette etude est valable pour le regime des contraintes moyennes et des temperatures &levees qui est control6 par une croissance des cavites sous contrainte de fluage. Nous utilisons deux definitions de l’instabilitt pour etudier les vitesses initiales de la croissance des imperfections, et de la croissance des imperfections au tours de la rupture. La premiere a rapport avec la reponse immediate a une pertubation (stabilite a petite echelle), alors que la seconde est Ii&e a une perte de correspondance entre les accroissements des variables de champ pertube et non pertub& Une analyse de la stabilim linbrisee, basee sur la premiere definition, revPIe que, tout comme dans le cas du fluage visqueux pur, la deformation est instable; cependant, au contraire du fluage visqueux, le temps caractbristique pur la croissance d’une striction est beaucoup plus court. Une analyse non lineaire basic sur la seconde definition associe le debut de l’instabilite a la rupture qui se produit pour une surface finie, mais pour une deformation infinie de la sectin locale. L’evolution des differences de deformation et d’aire revtle la tendance a la formation dune striction trb localiste, malgri: la perte de stabiliti: (rupture) pour une aire finie. Z~~~f~-Die Ins~bili~ten kriechender St&e mit Inhomogenit~ten in der Geometric und/oder dem Material wurden fiir konstante Last und konstante Spannung fur den Fall untersucht, daB die Niiherung grol3er Wellenliingen gilt. Das in dieser Untersuchung verwendete Grundmodell gilt fdr den Bereich der gemalligten Spannungen bei hoheren Temperaturen, bei dem vollstandig durch Kriechen eingeschranktes Porenwachstum vorliegt. Anhand von zwei Definitionen der Instabilitiit k&men die Geschwindigkeiten des Wachstums von Defekten am Anfang und bis zum Bruch untersucht werden werden. Die erste Definition betrifft die unmittelbare Antwort auf eine Stonmg (Stabilitat im kleinen); die zweite Definition hlngt mit dem Verlust der Korrespondenz zwischen den Inkrementen gestiirter und ungestijrter Feldva~ablen zusammen. Aus einer lin~risierten S~bilit~~analyser anhand der ersten Definition folgt, da8 die Verformung wie im Falle des reinen viskosen Kriechens instabil ist, allerdings ist, im Gegensatz zum viskosen Kriechen, die charakteristische &it fur das Wachstum von Einschniirungen vie1 kiirzer. Eine nichtlineare Analyse anhand der zweiten Definition verkniipft den Einsatz der Instabilitat mit einem Bruch, der in einem begrenzten Bereich, aber bei unendlicher Dehnung in einem lokalen Sektor auftritt. Die Geschichte der Dehnung und der Bereichsunterschiede weisen daraufhin, dag eine Tendenz zur Bildung stark lokalisierter Einschniirungen besteht, der allerdings ein Stabilinitsverlust (Bruch) in einem endlichen Bereich entgegensteht.
I. INTRODUCTION
When rate dependent materials are deformed under conditions of uniaxial tension the response is said to ~To characterize the response Hart [IJ uses the difference in Iocal and bulk section areas, Jonas er al. [2] uses the difference in local and bulk section strains and Hutchinson and Neale [3] use the relative difference in local and bulk section areas.
be unstable in the small if the immediate response? to an initial geometrical and/or material inhomogeneity is one of growth [1,4, 51. For materials whose elevated temperature creep behavior is characterized by the power law model (power law creep plus incompressibility condition) the deformation process is one of slow growth leading to the formation of a diffuse neck which terminates when the local area cross section has collapsed to a point [6,7]. This mech-
2583
2584
LEVY:
TENSILE INSTABILITY
anism of failure, known as creep rupture, will occur provided no other failure mechanism intercedes. When polycrystalline metals are deformed at elevated temperatures and within the low to moderate stress regime then, barring an aggressive environment, failure is more often due to the accumulation of damage in the form of grain boundary cavitation. Nucleation and growth of small isolated cavities, on those grain boundaries approximately normal to the tensile axis, occurs throughout the lifetime of the material eventually giving rise to cavity coalescence and microcracking which precedes fracture at a finite area. A previous study [S] on linear tensile instability attempted to account for cavitation in an approximate way by considering it through its effect on gross values of length and area and in conjunction with empirical data. It is the purpose of this paper to study the effect of damage on the stability behavior of bars by employing a physically based constitutive model characterizing completely creep constrained grain boundary cavitation. Although the problem so posed is necessarily a multiaxial and nonuniform one, uniaxial and uniform constitutive models and solutions will be used to study the deformation process by employing the long wavelength approximation [3,7]. In the following section the uniaxial constitutive model will be presented and uniform, perfect bar solutions for the constant load and constant stress tests reviewed. This is followed by a linearized stability analysis which is capable of predicting only the early stages of the response to an initial geometrical and/or material inhomogeneity (stability in the small). In the present work the geometrical inhomogeneity is characterized by an axial variation of area while the material inhomogeneity is characterized by an axial variation of a material parameter (MonkmanGrant constant) which characterizes the way cavities are distributed on the grain boundaries. In order to determine which imperfections (geometrical or material) grow the fastest, and therefore contribute the most to failure, three cases are considered corresponding to the response to geometrical imperfections alone, material imperfections alone and geometrical and material imperfections acting together. For each case characteristic times for neck growth are obtained and compared with the result for viscous power law creep. Section four contains a nonlinear stability analysis capable, within the confines of the long wavelength approximation, of predicting imperfection growth through failure. A form of Hutchinson and Neale’s instability criterion [3], which associates instability with the minimum value of bulk section area attainable, is employed to determine the relationship between the notions of instability, fracture and the variables characterizing them. Critical values of area, associated with the onset of instability, are obtained for different material and geometrical parameter regions. This is followed by a determination of the evolution of an area inhomogeneity through failure.
IN CREEP DAMAGING
SOLIDS
2. THE CONSTITUTIVE RELATION AND PERFECT BAR SOLUTIONS The constitutive relation to be employed in this sequel was developed [9, lo] for polycrystalline metals conditions of elevated temperatures under (0.3 TM - 0.8 TM, TM = melting temperature) and low to moderate stresses i.e. within the regime of completely creep constrained cavity growth. The model, which incorporates such non-classical behavior as dilation in creep and a pressure dependent flow rate, is characterized for uniaxial deformation by the evolution equations P = f?&a/a,~(l - w)-”
(1)
ti = (1 - o)P/nC
(2)
A=WP
(3)
where e is the natural strain, u the true stress, A the volume strain, w the area fraction of cavitated boundaries, C the Monkman-Grant constant, n the power law exponent and uO, t, the reference or initial stress and strain rate, respectively. The Monkman-Grant constant, which is the only material damage parameter in the model, characterizes the way cavities are distributed on the boundaries [lo] [see also equation (1 l)]. By introducing the evolving cross-sectional area A(t) and length L(t) of a bar the definitions of true stress, natural strain and volume strain assume the form P = L/L
(4)
Q = P/A = a,A,/A
(5)
A = (AL)‘/(AL)
(6)
where P is the applied load and A,, the initial area cross section, With the aid of these definitions the constitutive equations (l)-(3) can be written as a set of two coupled differential equations governing the strain and area at any instant in time P = P,(a/ao)n exp(e/C), e(t =O)=O k/A
(7)
= -exp( -e/nC)P,
A(t =O)=A,
(8)
and a/a0 = 1 constant applied stress u,,
(9)
= (A/A,,)-’ constant applied load P = q,Ao. It is evident from (8) that the evolving area cross section is dependent on the load history only through the strain and strain rate histories hence, the first integral of (8) is A/A,
= exp{ -nC[l
- exp(-e/&)1}.
(10)
LEVY:
TENSILE
INSTABILITY
This fact also happens to be true for the evolution of the area fraction of cavitated boundaries o as seen from (2) and so w(e) = 1 - exp(-e/K). An alternative
(11)
form of (10) can be written as A/A, = exp(-nCo).
(12)
where use has been made of (11). Fracture is defined to occur when all the boundaries become cavitated i.e. when w -+ 1 which, by (1 1), is equivalent to an infinite fracture strain. Although this is obviously not correct physically (bars fracture at finite strains and with less than 100% of their boundaries cavitated?) it does lead to reasonably accurate values of the fracture time and fracture area owing to the large strain rates which occur at relatively moderate values of w. The area at fracture can be obtained directly from (12) by letting w -+ 1, then A, = A, exp( -nC).
(13)
It follows that the fracture area is independent of load history and depends only on the initial area cross section, the power law exponent and the Monkman-Grant constant. In fact, for the limiting case of C + co, corresponding to pure power law creep (no damage), we recover the result of zero area at fracture. When the applied load is a constant stress the strain rate equation is independent of the area and the strain can be obtained by simply integrating (7) to get e = ln[l - (P,/C)t]-C
(14)
by employing the above mentioned fracture criterion, the fracture time is easily seen to be given by t/= C/P,
(15)
which is precisely the Monkman-Grant relationship which is known to correlate creep fracture data for many materials within the elevated temperaturemoderate stress regime. The determination of the area as a function of time follows directly by combining (10) and (14). When the applied load is constant, a single equation governing the strain can still be found by combining (10) with (9) and (7). An analytical solution to
iExperimental evidence indicates that anywhere from l/3 to l/2 of the boundaries are cavitated at fracture. $For integral values of power law exponent it can be shown [IO] that
IN CREEP
DAMAGING
this equation can be obtained for integral values of power law exponent [lo] however, for simplicity, the solution for all values of the exponent will be represented symbolically by t = t,F(e,
n, C)
The fracture time in the constant load test is given by t,
=
p(n,
c)/k,$>
§Quantities written
referred to the local section with a circumflex.
will henceforth
pn
<
1
(18)
and is obtained from (16) by letting e + co. Hence all three critical times tf, t,, t, are seen to obey a Monkman-Grant type relationship, i.e. critical times inversely proportional to strain rate. For typical values of parameters it follows that the fracture time at constant load (tF) is less than the fracture time at constant stress (tr) which is less than the rupture time (fR). The evolution of the strain and area with time are obtained by integrating the governing equations (7) (8) numerically. Figures 1 and 2 depict these solutions graphically along with solutions for the constant stress test. The behavior of the pure power law creep model is also shown. 3. THE TENSILE INSTABILITYLINEAR ANALYSIS The analysis outlined in this section is similar in some respects to previous investigations [1, 3,4,7] which were concerned with the response of a general class of rate dependent incompressible material models to geometrical imperfections. Just as in those investigations it will be assumed that the wavelength of the nonuniformity is large in comparison with a characteristic length of the bar cross section thereby eliminating the necessity of a multiaxial analysis. In the present study however, slight variations in material damage are accounted for and moreover, the material model can no longer be regarded as incompressible. Because of this it was shown that fracture of a perfect bar occurs at a nonzero area. We can therefore anticipate that the instability behaviour of the present model will be qualitatively different from those previously considered. Consider a tensile bar of finite length to have a section (local section) with area and material damage property (as measured by C) which differs slightly from the rest of the bar (see Fig. 3). Let II, which is a measure of the initial area nonuniformity, be defined by 2 -A, q=_“-
(194
where &, is the initial local section area and A, is the initial bulk section areas. Let i, which is a measure of the material nonuniformity, be defined by
(n - l)! (n2Cy-
(16)
where t, is Hoff’s critical time for ductile rupture which is defined by t, = l/Q&). (17)
All
-exp(-n*C)(-l)“-I--_
2585
SOLIDS
’ be
/f- c-c C
(19b)
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2586
TENSILE INSTABILITY IN CREEP DAMAGING
SOLIDS
By neglecting all but terms linear in 6AfA and 6e in (22), a system of linearized equations governing strain and area differences can be obtained, sk - (n - 1)A [Ge/nC - &A/A f ne/nC] = 0 (24a) & - n@[Ge/& - sA/A f ne/nC] = 0. (24b) Furthermore, it is not difficult to show that the system (24) has a first integral and that, because of this, the mathematical problem can be reduced to finding the solution to a single, linear first order equation. By combining (24a) and (24b) and noting that Time
ri =&A/A --.&A/A~
Fig. 1. Effect of test and material model on creep response.
(25)
we obtain - ci = [exp( - e /nC)se
where 1 may be positive, negative or zero (no material inhomogeneity). Introduce the area difference and strain difference 6A=A”--A
GOa)
&=2-e
+ n(e + nC) exp(-e/nC)]‘. By taking into account the initial condition (19a) and noting that Se(t = 0) is zero and 1 is independent of time it then follows that
POW
and denote the relative area difference at any time by a = 6AjA.
(3)
Governing equations (7), (8) can be written for both the local and bulk sections provided we require that the tensile load acting on each cross section be the same. By combining (7), (8), (19) and (20) the equations governing the strain and area differences can be written 6k - A(1 -t SA/A)-“+’ exp[(n - 1) (22a)
(se + Ae)jnC(l --A)] = -A 6P - P(1+6A/A)-”
exp(-e/&)&e
+ A[(e + nC)
x exp(-e/nC)
-r&T] = --(a + q).
Unlike the analogous relation for pure creep (obtained by letting C + co) (26) is history dependent and, because of this, the strain difference and relative area diReren= are no longer equivalent measures of imperfection growth. This is true even if material nonuniformities are neglected. By combining (26) with (24a) while taking into account (25) the following equation governing the relative area difference can be obtained, ti + h(t)a = k(6)
exp[(& + Ae)/C( I-
A)] = -P
(26)
(27a)
(22b)
where &j =‘$-‘&j &&-&
3ulk
section
and
(234
6A(t = 0) = -VA,
GW
6e(t = 0) = 0.
&crJl section
6
IO
12
14
16
76
20
22
Time
Fig. 2. Evolution of area with time for different models.
Fig. 3, Imperfect bar with geometric and material nonuniformity.
LEVY:
TENSILE
INSTABILITY
with h(t) =
n-l no
-e
+ n exp(-e/nC)
IN CREEP
k(t) = (n - I)Pfq/nC -I x [I - exp(-e/nC)]j.
(27~)
The equation, although independent of the applied load explicitly, is influenced by it through the dependence on the strain and strain rate of the bulk section of the bar. For the problem as outlined above three cases can be identified depending on the stability test desired, they are, stability with respect to an initial area imperfection, stability with respect to an initial area imperfection and a material imperfection and, stability with respect to a material imperfection only. 3. I. Case A. Geometrical imperfection For this case, 1 = 0 and (27a, b, c) reduce to n-l = nC
(284
Vi
with n-l no
h(t) = -t; A formal solution given by
+ g exp(-e/nC)
to (28) by integrating
1 .
(28b)
factor is
u(t) = -9 exp -H(r) x [l-5
j:e(r)expX(r)dr]
(29)
with ’ h(t)dz s0 and it is immediately seen that, unlike the pure creep H(t) =
case, exponential growth or decay no longer depends solely on the sign of h. From (29) it can be shown that the magnitude of a increases and so the deformation
is always unstable in the small. This is true regardless of whether the inhomogeneity is measured by the relative area difference a or the normalized area difference 6A/A,. For pure power law creep, with values of power law exponent on the order of unity, it has been shown [7J that the characteristic time? associated with ex-
ponential growth or decay is similar to the time required for the development of axial strains of unit magnitude in the perfect bar. Hence, although the deformation is always unstable in the small (provided n 2 1) a visible neck may not appear until the later stages of deformation. This result has been confirmed experimentally for a strain hardening, rate sensitive material []I]. Employing the creep damage model withC-0.1 andn - 1 it follows from (28b) and (29) that the characteristic time for exponential growth is on the order of the time required for development of axial strains of magnitude 0.1. Therefore, on the basis of this linear analysis, we should expect to see neck development substantially sooner than for the pure power law creep case. 3.2. Case B. Geometrical and material imperfection
Here, q # 0 and 1 # 0 and although q is positive h may be either positive or negative depending upon whether the local value of C is larger or smaller than the bulk value. The solution to (27) by integrating factor, is I a = exp[ - H(t)] expW(r ll4~ W - v [S 0 I with H(t) given above. In order to determine the relative contribution to the response of a geometrical and a material nonuniformity consider the expression for k(t) as given by (27~). For 1, on the same order of magnitude as q. k(t) is dominated by the q,/nC term as opposed to the 1 - I - exp( -e/m)] term (recall that C - 0.1). Hence area inhomogeneities will contribute, through k(t) more to the response by almost an order of magnitude. If I is of an order of magnitude greater than Y)then contribution to k(t) will be of similar strength. Negative values of i, corresponding to smaller local damage rates, will act to retard growth while positive values of A will act to enhance growth. For practical values of 1 (A - 0.01) however, this will be a second order effect since the inhomogeneous term(k(t)) will not generally dominate the behaviorf. Hence the instability behavior, for typical values of material parameters, is governed by geometrical and not material inhomogeneities and characteristic time for neck growth is on the order of the time required for development of axial strains of magnitude 0.1. 3.3. Case C. Material imperfection A material inhomogeneity by itself (PJ= 0, il # 0) may still cause unstable behavior although, as indicated above, the behaviour will not be as severe as for the other cases. The solution to (27) is given by
tThe characteristic time for exponential decay is defined to be that time at which the relative area difference has decreased from an initial value of -q to a value of -_9 exp(l). $It
2587
SOLIDS
1 W’b)
and
Li + h(t)a
DAMAGING
is possible to have a reversal of the rate of growth of the area perturbation, however this would occur at unrealistically large negative values of I and always at some time greater then zero.
a = exp( - H(t)]
’ exp[ici(s)]k(r)dr [S 0
1
with k(t) = -(n
- I)&[1 - exp(-e/nC)]
and H(t) as given above. By changing slightly the
2.588
LEVY: TENSILE INSTABILITY IN CREEP DAMAGING SOLIDS
r
aa (o)r.oo 0.6
where use has been made of (17). Although the linear analysis can predict the initial state of imperfection growth it is incapable of describing the developing interaction of dilation due to cavitation with the geometrical constriction of area at increasing longitudinal strain. Therefore, in order to obtain local (finite) and bulk areas at fracture, a fully nonlinear analysis is required.
c
4. THE TENSILE INSTABILITYNONL~~R ANALYSIS
0.6 t
Fig. 4. (a) Response to geometrical nonunifo~ity-linear analysis. (b) Response to material nonunifo~ity-linear analysis. definition of characteristic time to be that time at which a has decreased from a value of zero to a value of -q exp(1) an examination of a and k given above indicates that the characteristic time is just about the same order of time required for development of axial strains of magnitude 0.1. Figure 4 (a, b) depicts the evolution of an area inhomogeneity for various values of parameters and for two of the cases just discussed. Case B curves depicting the effect of material and geometrical imperfections were not drawn since it was determined that, for relative variations in C up to lo%, the effect was negligible. Hence the curves were essentially the same as those given for case A. Only constant load test curves are shown but constant stress test curves exhibit similar behavior except that they are displaced to the right. The curves were obtained by straightforward numerical integration of (7), (S), (9) and (27a, b, c) and are measured in terms of nondimensional time defined by
It has already been noted previously that a response which has been determined to be unstable in the small may not be the kind of catastrophic behavior normally associated with our intuitive notion of instability. The fact that this is true can be seen from a number of simulation studies concerning the rate dependent flow of metal bars [6,10, 121. Because of this an alternative definition of instability has been proposed [3j which associates the onset of instability in both rate dependent and independent materials to the maximum value of bulk section strain attainable i.e. instability is said to occur when the unique correspondence between increments of local and bulk section strain is lost. In mathematical form the instability criterion is de/de” = 0 or, alternatively dA/dA” =0
(30a)
in which case instability is associated with the minimum value of bulk section area attainable. In order to obtain expressions for dA /dA” (or de{dF) governing equations (7), (10) are written for the local and bulk sections of the bar together with the connecting relation P(t) = a( = ci(t)A(t) which simply states that, at any instant, local and bulk bar sections support the same load. Then (:I*‘[1
+ln($r($)
=(I -?r(~)“-‘[l+ln(3’“‘T’
A” *
0
X*
r = t/t, = m&t
(314
A0
which has, at least numerically, a first integral A@).? A straight-forward application of the instability criterion (30a) would then yield the local area cross section at which instability occurs. A simpler approach would be to use (31a) directly to get
TFor integral values of n,
F(~,n,C)=(l--1)“F(%.n,C) W) where (n - I)! (n -r
-
[(i-(1
l)!(n2cy
1. ‘- *
+ln($-J-11
(32)
LEVY:
TENSILE
INSTABILITY
Depending on the values of the parameters q, C, c instability may occur at either a/&
= exp( -nZI) = A,/&
(33a)
when (30a) is satisfied, or when A/A,
= exp(-nC)
= A,/A,.
For the latter case the instability dJ/dA
DAMAGING
SOLIDS
2589
bar solution is approached and recovered when c = C. For this case then, stability loss will occur at the section (local, bulk) in which cavitation is most severe. Figure 5(b) depicts this behavior for the range of values of c and C.
Wb)
criterion becomes
= 0
(30b)
and stability loss is associated with the minimum value of local section area. That instability may be associated with critical values of either local or bulk section area may be seen from the following three cases which describe the behavior for different ranges of parameters. 4.1. Case A. Geometrical
IN CREEP
D
0.9
$ OB (1V_1*‘lncrMrl”O, _
imperfection
For this case the Monkman-Grant constant is uniform (2; = C) while the relative area imperfection q may take on any value. By employing (32) the initial conditions.
“r(b)
and noting that the function f given by
f=z n-‘[l +lnzarm’, B>l,n>l,O 0 instability occurs when A”/& = exp( -nC)
t---
l
E-c
while for q < 0 instability occurs when A/A, = exp( - nC). Hence, when the initial local area cross section (A”,) is less than the initial bulk section area (A,), instability (necking) is associated with critical values of local section area &. Likewise for & > A,, instability (bulging) is associated with A,. For q = 0, instability occurs when the uniform bar area is a minimum. Figure 5(a) illustrates this behavior for different values of the parameter 9. 4.2. Case B. Material
imperfection
Here, the bar is geometrically uniform at the initial instant (q = 0) but the Monkman-Grant constant is nonuniform (c # C). The function f given above decreases monotonically with decreasing values of z and increasing values of p and so (32) together with (34) imply that for c < C instability occurs when the local area is A”/& = exp( - nC) while for c > C instability occurs when the bulk area is A/A,, = exp( -nC).
AS r??approaches C from above or below the uniform
Fig. 5. (a) Stability with respect to geometrical nonuniformity (c = C). (b) Stability with respect to material nonuniformity (q =O). (c) Stability with respect to geometrical and material nonuniformity.
2590
LEVY: TENSILE INSTABILITY IN CREEP DAMAGING SOLIDS
4.3. Case C. Material and geometrical imperfection
This case is more involved since it is not always possible to predict the critical area without explicitly solving (32). This follows from the fact that, for certain values of parameters (q, (?, C), material nonuniformity and initial geometrical nonuniformity may be competing mechanisms for growth of inhomogeneous deformation. If c < C and q > 0 then (32), (33) and the monotoni~ity off are su~cient to determine that stability loss occurs when the normalized local area A”/& assumes the value exp( -nC). For c > C, q < 0 the same argument implies stability loss at a normalized bulk area A/& of exp(-nC). However, when 2; > C and rl > 0 (or (? < C, q < O),floses its monotoni~ity and (32) must be integrated numerically (or (31 b) must be solved). A quantitative picture of the behavior is given in Fig. 5(c) for a range of values of q, c, C. Clearly a slightly smaller initial local section area does not necessarily imply that stability loss will occur at the local section. To get an idea as to the relative strengths of the competing mechanisms (32) was integrated numerically subject to the initial condition (34) using typical values of parameters (q = 0.01, C = 0.05). It was determined that for L = 0.01 corresponding to a value of c of magnitude 0.0505 the dominating influence was the initial geometrical imperfection i.e. instability occurred at the local section. In fact, in order for the material nonuniformity to dominate, 1 would have to be on the order of 0.1 corresponding to a 10% axial variation in Monkman-Grant constant. Hence, for most practical cases the critical area for instability is governed by the initial geometrical imperfection-a result consistent with the linearized analysis. The critical values of area given by (33) are equivalent, by (1 1), (12), to associating instability with infinite strain or a unit value of area fraction. From this result it therefore follows that loss of stability, as defined by (30), and fracture, as defined by a unit value of area fraction, coincide. This situation is similar in behavior to other rate dependent, but incompressible, materials in which loss of stability and rupture (zero local area cross section) coincide [6,7]. For the rate independent, power law hardening material instability occurs at finite values of local section area and local section strain and is not associated with any definition of failure. Hence, for this case, loss of stability represents a true bifurcation and does not signal the end of the deformation process as it does in the time dependent cases just noted. It should also be noted that the conditions (32) depend only on the material parameters and are, in fact, independent of any specific form of load history. Tensile bars composed of identical material but subject to different loads will lose stability at the same value of area however the time at which this occurs will depend on the specific form of the loading. In order to study the time dependent growth of nonuniform deformation prior to stability loss (frac-
ture) (7), (8) and (31a) may be integrated taking into account the initial conditions
A=&,A
=A,,
att =O.
e=c=O
The relative area difference a follows directly by employing the definition (21) a=A/A
-1.
(21)
The solution, in analytical form, can be obtained for integral values of n however, since the result is cumbersome, a straight-forward numerical procedure has been adopted. The results are depicted in Fig. 6 for various values of parameters and, just as in the linear analysis, are given in terms of nondimensional time z defined in Section 3. The bracketed quantities in Fig. 6 are the values of relative area and bulk section strain at the onset of instability or fracture (note that if stability loss occurs at the bulk section area then the local section area is given). For local section stability loss, the quantity a, was obtained from (21), (33a) and can be written as a,=[A,exp(-nQ]A-‘-
1
(35)
with the value of A obtained from the final data point calculated. The value should be an accurate approximation to the true value since, just prior to fracture, the bulk section area is essentially constant. If stability loss occurs at the bulk section area then an equation analogous to (35) has been used. The bulk section strain (or local section strain as the case may as
I-
(o)r-cm
1 D 1 0.4 t
*I
*
lr?I 0.4 t
Fig. 6. (a) Response to geometrical nonuniformity(b) Response to material analysis. nonlinear nonuniformitv-nonlinear analvsis. i
--=T-rr
- -=‘-- - --I --*’
be) is obtained directly from the expression obtained by combining (35) and (10) e = l+n(~)““’
+ il”.
(36)
For geometrical imperfection only (,?.= 0) a curve in Fig. 6(a) with C = 0.05 and its counterpart with C = 0.075 indicates the effect of cavitation on the flow process. As the ability of a material to damage increases (decreasing C), the magnitude of a, decreases such that the limiting behavior is given by u,= -?I
=a(r
=O),C-,O.
At the same time the strain rate difference increases causing a rapid constriction of the local area cross section while the overall ductility decreases i.e. e, decreases. Taken together the above results imply that, as the ability of a material to cavitate increases, there is more of a tendency for a highly localized neck to form prior to failure however this is countered by fracture at a finite area. This kind of behaviour is in agreement with a flow simulation study of necking in creep cavitating copper bars [lo]. In contrast to this type of behavior the opposite limiting case of creep rupture occurs as C becomes infinite, then a,= -l,C+cc where use has been made of (33). Also, the strain rate difference is maximized for an infinite value of C while the overall ductility is a maximum. Hence for materials deforming solely by power law creep the well known result that the instability process is one of slow growth leading to the formation of a diffuse neck is recovered. In the absence of geometrical inhomogeneities (q = 0), for a given value of C, increasing A i.e. decreasing C, causes a larger magnitude of a, but decreased ductility [Fig. 6(b)]. Although this result may appear to be in conflict with the results from Fig. 6(a), i.e. decreasing C (=c) causes a smaller magnitude of u,., it can be resolved simply by noting that in the former case it is e, and not C, which decreased. Although in both cases the critical time to fracture will be less, the value of area in the bulk section at this time will not since C has not been changed. Hence the magnitude of a, is increased. That the criticai time to fracture for a given C decreases with increasing i follows directly from the fact that increasing i; means decreasing c. It has been shown in the previous section that when a bar is geometrically uniform but materially nonuniform (rl = 0) initial rates of growth are not as severe as those associated with a geometrically nonuniform but materially uniform bar. The nonlinear analysis indicates that critical values of relative area inhomogeneity, in the absence of geometrical nonuniformity, will be smaller in magnitude owing to the dependence of that quantity on the initial local section area (35). Also, the time at which instability
IN CREEP DAMAGING
SOLIDS
2591
is reached is delayed for the geometrically uniform but materially nonuniform bar [Fig. 6(b)]. Taken together the above results indicate that the nonlinear response is governed primarily by geometrical nonuniformities and not material ones. For typical values of parameters (A -0.01, P)-0.01) curves obtained when both effects are considered differ only slightly from those shown in Fig. 6(a). 5. CONCLUSIONS In this paper the effect of completely creep constrained grain boundary cavitation on the tensile stability of metal bars has been investigated. Two definitions of instability have been employed in order to study the effect of both material and geometrical nonuniformities. The first is concerned with the immediate time dependent response to a disturbance whiie the second is related to a loss of correspondence between increments of perturbed and unperturbed quantities governing a process. Based on the first definition (stability in the small) a linearized stability analysis predicts that an area inhomogeneity will always, at least initially, grow in time. Although the rate of growth can change sign it has been shown that this occurs at unrealistically large values of local Monkman-Grant constant and so the response is unstable. For typical values of material parameters, it has been shown that the appearance of visible necks will occur, for the damaging case, at strains on order of 0.1 as opposed to strains on order of unity (cavitation suppressed). This is true regardless of whether the initial defect is geometrical, material or a combination of both although the response is more sensitive to geometrical defects than to material ones. As expected initial rates of imperfection growth increase the greater the ability of the material to damage as measured by decreasing values of Monkman-Grant constant. Also, initial rates of growth are larger in the constant load test then in the constant stress test. A nonlinear analysis based on the second definition of instabiljty predicts that stability loss can occur when either the local or bulk area cross section reaches a critical value. The exact location being dependent on relative magnitude of local and bulk section Monkman-Grant constants as well as initial local and bulk area cross sections. The critical value of area, which occurs at infinite local section strain, coincides with our definition of fracture, i.e. unit value of local damage variable (actually, area fraction of cavitated boundaries). So, by this definition, the deformation is stable prior to the point signaling fracture and the end of the process. By contrast the first definition of instability indicates that the process, beginning at the instant the load is applied, is unstable. An examination of the history of the response subject to an initial area inhomogeneity indicates that the material is more sensitive to geometrical as opposed to material inhomogeneities. Also, as the
2592
LEVY:
TENSILE INSTABILITY
ability of a material to damage increases then (i) the area inhomogeneity at fracture decreases, (ii) the rate of change of area inhomogeneity increases and (iii) the overall ductility decreases. Taken together these characteristics imply that the effect of cavitation on flow is to increase the tendency of a highly localized neck to form at smaller overall strains which is countered by fracture, or stability loss, at finite area. ~ck~ow~edge~en~-The author gratefully acknowledges the helpful comments of Professor M. P. Bieniet. REFERENCES 1. E. W. Hart, Acra metall. 15, 351 (1967).
IN CREEP DAMAGING
SOLIDS
2. J. J. Jonas, R. A. Halt and C. E. Coleman, Acta metall. 24, 911 (1976). 3. J. W. Hut~hinson and K. W. Neale, Acra ~taZ~. 25,839 (1977). 4. J. D. Campbell, J. Mech. Phys. Solids 15, 359 (1967). 5. U. F. Kocks, J. J. Jonas and H. Mecking, Acta metall. 27, 419 (1979). 6. N. J. Hoff, J. Appl. Mech. 20, 105 (1953). 7. J. W. Hutchinson and H. Obrecht, Fracture I977 1, ICF4, Waterloo, Canada (1977). 8. J. J. Jonas and B. Baudelet, Acta metalt. 25,43 (1977). 9. A. J. Levy, J. appl Mech. 52, 615 (1985). 10. A. J. Levy, Acta metall. 34, 1991 (1986). 11. S. Sagat and D. M. R. Taplin, Metals Sci. IO,94 (1976). 12. M. A. Burke and W. D. Nix, Acta metall. 23, 793 (1975).
TENSILE
~1-4~~~87 $3.00 + 0.00 Copyright 0 1987 Pergamon Journals Ltd
INSTABILITY IN CREEP DAMAGING SOLIDS A. J. LEVY
Department of Mechanical and Aerospace Engineering, Syracuse University, Syracuse, NY 13244, U.S.A. (Received 8 August 1986) Abstract-Tensile instability in creep damaging bars subject to geometrical and/or material nonuniformities is investigated for the constant load and constant stress tests provided the longwavelength approximation is maintained. The constitutive model employed in the study is valid for the moderate stress-elevated temperature regime which is dominated by completely creep constrained cavity growth. Two definitions of instability are used to study initial rates of imperfection growth and imperfection growth through failure. The first is concerned with the immediate response to a disturbance (stability in the small) while the second is related to a loss of correspondence between increments of perturbed and unpertur~ field variables. A linearized stability analysis based on the first definition reveals that, like the pure viscous creep case, the deformation is unstable however, in contrast to viscous creep, the characteristic time for neck growth is much shorter. A nonlinear analysis based on the second definition associates the onset of instability with fracture occurring at a finite area but infinite local section strain. Histories of strain and area differences reveal the tendency for a highly localized neck to form, however this is countered by stability loss (fracture) at a finite area. R&n&-Nous avons &die l’instabilite en traction dam des barreaux endommages par fluage soumis g des h~t~rog~n~it~s de geombtrie etjou de materiau, dans le cas d’essais a charge constante et a contrainte constante oit l’on maintient ~approximation de la grande longueur d’onde. L’tquation d’etat que nous avons utilisie au tours de cette etude est valable pour le regime des contraintes moyennes et des temperatures &levees qui est control6 par une croissance des cavites sous contrainte de fluage. Nous utilisons deux definitions de l’instabilitt pour etudier les vitesses initiales de la croissance des imperfections, et de la croissance des imperfections au tours de la rupture. La premiere a rapport avec la reponse immediate a une pertubation (stabilite a petite echelle), alors que la seconde est Ii&e a une perte de correspondance entre les accroissements des variables de champ pertube et non pertub& Une analyse de la stabilim linbrisee, basee sur la premiere definition, revPIe que, tout comme dans le cas du fluage visqueux pur, la deformation est instable; cependant, au contraire du fluage visqueux, le temps caractbristique pur la croissance d’une striction est beaucoup plus court. Une analyse non lineaire basic sur la seconde definition associe le debut de l’instabilite a la rupture qui se produit pour une surface finie, mais pour une deformation infinie de la sectin locale. L’evolution des differences de deformation et d’aire revtle la tendance a la formation dune striction trb localiste, malgri: la perte de stabiliti: (rupture) pour une aire finie. Z~~~f~-Die Ins~bili~ten kriechender St&e mit Inhomogenit~ten in der Geometric und/oder dem Material wurden fiir konstante Last und konstante Spannung fur den Fall untersucht, daB die Niiherung grol3er Wellenliingen gilt. Das in dieser Untersuchung verwendete Grundmodell gilt fdr den Bereich der gemalligten Spannungen bei hoheren Temperaturen, bei dem vollstandig durch Kriechen eingeschranktes Porenwachstum vorliegt. Anhand von zwei Definitionen der Instabilitiit k&men die Geschwindigkeiten des Wachstums von Defekten am Anfang und bis zum Bruch untersucht werden werden. Die erste Definition betrifft die unmittelbare Antwort auf eine Stonmg (Stabilitat im kleinen); die zweite Definition hlngt mit dem Verlust der Korrespondenz zwischen den Inkrementen gestiirter und ungestijrter Feldva~ablen zusammen. Aus einer lin~risierten S~bilit~~analyser anhand der ersten Definition folgt, da8 die Verformung wie im Falle des reinen viskosen Kriechens instabil ist, allerdings ist, im Gegensatz zum viskosen Kriechen, die charakteristische &it fur das Wachstum von Einschniirungen vie1 kiirzer. Eine nichtlineare Analyse anhand der zweiten Definition verkniipft den Einsatz der Instabilitat mit einem Bruch, der in einem begrenzten Bereich, aber bei unendlicher Dehnung in einem lokalen Sektor auftritt. Die Geschichte der Dehnung und der Bereichsunterschiede weisen daraufhin, dag eine Tendenz zur Bildung stark lokalisierter Einschniirungen besteht, der allerdings ein Stabilinitsverlust (Bruch) in einem endlichen Bereich entgegensteht.
I. INTRODUCTION
When rate dependent materials are deformed under conditions of uniaxial tension the response is said to ~To characterize the response Hart [IJ uses the difference in Iocal and bulk section areas, Jonas er al. [2] uses the difference in local and bulk section strains and Hutchinson and Neale [3] use the relative difference in local and bulk section areas.
be unstable in the small if the immediate response? to an initial geometrical and/or material inhomogeneity is one of growth [1,4, 51. For materials whose elevated temperature creep behavior is characterized by the power law model (power law creep plus incompressibility condition) the deformation process is one of slow growth leading to the formation of a diffuse neck which terminates when the local area cross section has collapsed to a point [6,7]. This mech-
2583
2584
LEVY:
TENSILE INSTABILITY
anism of failure, known as creep rupture, will occur provided no other failure mechanism intercedes. When polycrystalline metals are deformed at elevated temperatures and within the low to moderate stress regime then, barring an aggressive environment, failure is more often due to the accumulation of damage in the form of grain boundary cavitation. Nucleation and growth of small isolated cavities, on those grain boundaries approximately normal to the tensile axis, occurs throughout the lifetime of the material eventually giving rise to cavity coalescence and microcracking which precedes fracture at a finite area. A previous study [S] on linear tensile instability attempted to account for cavitation in an approximate way by considering it through its effect on gross values of length and area and in conjunction with empirical data. It is the purpose of this paper to study the effect of damage on the stability behavior of bars by employing a physically based constitutive model characterizing completely creep constrained grain boundary cavitation. Although the problem so posed is necessarily a multiaxial and nonuniform one, uniaxial and uniform constitutive models and solutions will be used to study the deformation process by employing the long wavelength approximation [3,7]. In the following section the uniaxial constitutive model will be presented and uniform, perfect bar solutions for the constant load and constant stress tests reviewed. This is followed by a linearized stability analysis which is capable of predicting only the early stages of the response to an initial geometrical and/or material inhomogeneity (stability in the small). In the present work the geometrical inhomogeneity is characterized by an axial variation of area while the material inhomogeneity is characterized by an axial variation of a material parameter (MonkmanGrant constant) which characterizes the way cavities are distributed on the grain boundaries. In order to determine which imperfections (geometrical or material) grow the fastest, and therefore contribute the most to failure, three cases are considered corresponding to the response to geometrical imperfections alone, material imperfections alone and geometrical and material imperfections acting together. For each case characteristic times for neck growth are obtained and compared with the result for viscous power law creep. Section four contains a nonlinear stability analysis capable, within the confines of the long wavelength approximation, of predicting imperfection growth through failure. A form of Hutchinson and Neale’s instability criterion [3], which associates instability with the minimum value of bulk section area attainable, is employed to determine the relationship between the notions of instability, fracture and the variables characterizing them. Critical values of area, associated with the onset of instability, are obtained for different material and geometrical parameter regions. This is followed by a determination of the evolution of an area inhomogeneity through failure.
IN CREEP DAMAGING
SOLIDS
2. THE CONSTITUTIVE RELATION AND PERFECT BAR SOLUTIONS The constitutive relation to be employed in this sequel was developed [9, lo] for polycrystalline metals conditions of elevated temperatures under (0.3 TM - 0.8 TM, TM = melting temperature) and low to moderate stresses i.e. within the regime of completely creep constrained cavity growth. The model, which incorporates such non-classical behavior as dilation in creep and a pressure dependent flow rate, is characterized for uniaxial deformation by the evolution equations P = f?&a/a,~(l - w)-”
(1)
ti = (1 - o)P/nC
(2)
A=WP
(3)
where e is the natural strain, u the true stress, A the volume strain, w the area fraction of cavitated boundaries, C the Monkman-Grant constant, n the power law exponent and uO, t, the reference or initial stress and strain rate, respectively. The Monkman-Grant constant, which is the only material damage parameter in the model, characterizes the way cavities are distributed on the boundaries [lo] [see also equation (1 l)]. By introducing the evolving cross-sectional area A(t) and length L(t) of a bar the definitions of true stress, natural strain and volume strain assume the form P = L/L
(4)
Q = P/A = a,A,/A
(5)
A = (AL)‘/(AL)
(6)
where P is the applied load and A,, the initial area cross section, With the aid of these definitions the constitutive equations (l)-(3) can be written as a set of two coupled differential equations governing the strain and area at any instant in time P = P,(a/ao)n exp(e/C), e(t =O)=O k/A
(7)
= -exp( -e/nC)P,
A(t =O)=A,
(8)
and a/a0 = 1 constant applied stress u,,
(9)
= (A/A,,)-’ constant applied load P = q,Ao. It is evident from (8) that the evolving area cross section is dependent on the load history only through the strain and strain rate histories hence, the first integral of (8) is A/A,
= exp{ -nC[l
- exp(-e/&)1}.
(10)
LEVY:
TENSILE
INSTABILITY
This fact also happens to be true for the evolution of the area fraction of cavitated boundaries o as seen from (2) and so w(e) = 1 - exp(-e/K). An alternative
(11)
form of (10) can be written as A/A, = exp(-nCo).
(12)
where use has been made of (11). Fracture is defined to occur when all the boundaries become cavitated i.e. when w -+ 1 which, by (1 1), is equivalent to an infinite fracture strain. Although this is obviously not correct physically (bars fracture at finite strains and with less than 100% of their boundaries cavitated?) it does lead to reasonably accurate values of the fracture time and fracture area owing to the large strain rates which occur at relatively moderate values of w. The area at fracture can be obtained directly from (12) by letting w -+ 1, then A, = A, exp( -nC).
(13)
It follows that the fracture area is independent of load history and depends only on the initial area cross section, the power law exponent and the Monkman-Grant constant. In fact, for the limiting case of C + co, corresponding to pure power law creep (no damage), we recover the result of zero area at fracture. When the applied load is a constant stress the strain rate equation is independent of the area and the strain can be obtained by simply integrating (7) to get e = ln[l - (P,/C)t]-C
(14)
by employing the above mentioned fracture criterion, the fracture time is easily seen to be given by t/= C/P,
(15)
which is precisely the Monkman-Grant relationship which is known to correlate creep fracture data for many materials within the elevated temperaturemoderate stress regime. The determination of the area as a function of time follows directly by combining (10) and (14). When the applied load is constant, a single equation governing the strain can still be found by combining (10) with (9) and (7). An analytical solution to
iExperimental evidence indicates that anywhere from l/3 to l/2 of the boundaries are cavitated at fracture. $For integral values of power law exponent it can be shown [IO] that
IN CREEP
DAMAGING
this equation can be obtained for integral values of power law exponent [lo] however, for simplicity, the solution for all values of the exponent will be represented symbolically by t = t,F(e,
n, C)
The fracture time in the constant load test is given by t,
=
p(n,
c)/k,$>
§Quantities written
referred to the local section with a circumflex.
will henceforth
pn
<
1
(18)
and is obtained from (16) by letting e + co. Hence all three critical times tf, t,, t, are seen to obey a Monkman-Grant type relationship, i.e. critical times inversely proportional to strain rate. For typical values of parameters it follows that the fracture time at constant load (tF) is less than the fracture time at constant stress (tr) which is less than the rupture time (fR). The evolution of the strain and area with time are obtained by integrating the governing equations (7) (8) numerically. Figures 1 and 2 depict these solutions graphically along with solutions for the constant stress test. The behavior of the pure power law creep model is also shown. 3. THE TENSILE INSTABILITYLINEAR ANALYSIS The analysis outlined in this section is similar in some respects to previous investigations [1, 3,4,7] which were concerned with the response of a general class of rate dependent incompressible material models to geometrical imperfections. Just as in those investigations it will be assumed that the wavelength of the nonuniformity is large in comparison with a characteristic length of the bar cross section thereby eliminating the necessity of a multiaxial analysis. In the present study however, slight variations in material damage are accounted for and moreover, the material model can no longer be regarded as incompressible. Because of this it was shown that fracture of a perfect bar occurs at a nonzero area. We can therefore anticipate that the instability behaviour of the present model will be qualitatively different from those previously considered. Consider a tensile bar of finite length to have a section (local section) with area and material damage property (as measured by C) which differs slightly from the rest of the bar (see Fig. 3). Let II, which is a measure of the initial area nonuniformity, be defined by 2 -A, q=_“-
(194
where &, is the initial local section area and A, is the initial bulk section areas. Let i, which is a measure of the material nonuniformity, be defined by
(n - l)! (n2Cy-
(16)
where t, is Hoff’s critical time for ductile rupture which is defined by t, = l/Q&). (17)
All
-exp(-n*C)(-l)“-I--_
2585
SOLIDS
’ be
/f- c-c C
(19b)
LEVY:
2586
TENSILE INSTABILITY IN CREEP DAMAGING
SOLIDS
By neglecting all but terms linear in 6AfA and 6e in (22), a system of linearized equations governing strain and area differences can be obtained, sk - (n - 1)A [Ge/nC - &A/A f ne/nC] = 0 (24a) & - n@[Ge/& - sA/A f ne/nC] = 0. (24b) Furthermore, it is not difficult to show that the system (24) has a first integral and that, because of this, the mathematical problem can be reduced to finding the solution to a single, linear first order equation. By combining (24a) and (24b) and noting that Time
ri =&A/A --.&A/A~
Fig. 1. Effect of test and material model on creep response.
(25)
we obtain - ci = [exp( - e /nC)se
where 1 may be positive, negative or zero (no material inhomogeneity). Introduce the area difference and strain difference 6A=A”--A
GOa)
&=2-e
+ n(e + nC) exp(-e/nC)]‘. By taking into account the initial condition (19a) and noting that Se(t = 0) is zero and 1 is independent of time it then follows that
POW
and denote the relative area difference at any time by a = 6AjA.
(3)
Governing equations (7), (8) can be written for both the local and bulk sections provided we require that the tensile load acting on each cross section be the same. By combining (7), (8), (19) and (20) the equations governing the strain and area differences can be written 6k - A(1 -t SA/A)-“+’ exp[(n - 1) (22a)
(se + Ae)jnC(l --A)] = -A 6P - P(1+6A/A)-”
exp(-e/&)&e
+ A[(e + nC)
x exp(-e/nC)
-r&T] = --(a + q).
Unlike the analogous relation for pure creep (obtained by letting C + co) (26) is history dependent and, because of this, the strain difference and relative area diReren= are no longer equivalent measures of imperfection growth. This is true even if material nonuniformities are neglected. By combining (26) with (24a) while taking into account (25) the following equation governing the relative area difference can be obtained, ti + h(t)a = k(6)
exp[(& + Ae)/C( I-
A)] = -P
(26)
(27a)
(22b)
where &j =‘$-‘&j &&-&
3ulk
section
and
(234
6A(t = 0) = -VA,
GW
6e(t = 0) = 0.
&crJl section
6
IO
12
14
16
76
20
22
Time
Fig. 2. Evolution of area with time for different models.
Fig. 3, Imperfect bar with geometric and material nonuniformity.
LEVY:
TENSILE
INSTABILITY
with h(t) =
n-l no
-e
+ n exp(-e/nC)
IN CREEP
k(t) = (n - I)Pfq/nC -I x [I - exp(-e/nC)]j.
(27~)
The equation, although independent of the applied load explicitly, is influenced by it through the dependence on the strain and strain rate of the bulk section of the bar. For the problem as outlined above three cases can be identified depending on the stability test desired, they are, stability with respect to an initial area imperfection, stability with respect to an initial area imperfection and a material imperfection and, stability with respect to a material imperfection only. 3. I. Case A. Geometrical imperfection For this case, 1 = 0 and (27a, b, c) reduce to n-l = nC
(284
Vi
with n-l no
h(t) = -t; A formal solution given by
+ g exp(-e/nC)
to (28) by integrating
1 .
(28b)
factor is
u(t) = -9 exp -H(r) x [l-5
j:e(r)expX(r)dr]
(29)
with ’ h(t)dz s0 and it is immediately seen that, unlike the pure creep H(t) =
case, exponential growth or decay no longer depends solely on the sign of h. From (29) it can be shown that the magnitude of a increases and so the deformation
is always unstable in the small. This is true regardless of whether the inhomogeneity is measured by the relative area difference a or the normalized area difference 6A/A,. For pure power law creep, with values of power law exponent on the order of unity, it has been shown [7J that the characteristic time? associated with ex-
ponential growth or decay is similar to the time required for the development of axial strains of unit magnitude in the perfect bar. Hence, although the deformation is always unstable in the small (provided n 2 1) a visible neck may not appear until the later stages of deformation. This result has been confirmed experimentally for a strain hardening, rate sensitive material []I]. Employing the creep damage model withC-0.1 andn - 1 it follows from (28b) and (29) that the characteristic time for exponential growth is on the order of the time required for development of axial strains of magnitude 0.1. Therefore, on the basis of this linear analysis, we should expect to see neck development substantially sooner than for the pure power law creep case. 3.2. Case B. Geometrical and material imperfection
Here, q # 0 and 1 # 0 and although q is positive h may be either positive or negative depending upon whether the local value of C is larger or smaller than the bulk value. The solution to (27) by integrating factor, is I a = exp[ - H(t)] expW(r ll4~ W - v [S 0 I with H(t) given above. In order to determine the relative contribution to the response of a geometrical and a material nonuniformity consider the expression for k(t) as given by (27~). For 1, on the same order of magnitude as q. k(t) is dominated by the q,/nC term as opposed to the 1 - I - exp( -e/m)] term (recall that C - 0.1). Hence area inhomogeneities will contribute, through k(t) more to the response by almost an order of magnitude. If I is of an order of magnitude greater than Y)then contribution to k(t) will be of similar strength. Negative values of i, corresponding to smaller local damage rates, will act to retard growth while positive values of A will act to enhance growth. For practical values of 1 (A - 0.01) however, this will be a second order effect since the inhomogeneous term(k(t)) will not generally dominate the behaviorf. Hence the instability behavior, for typical values of material parameters, is governed by geometrical and not material inhomogeneities and characteristic time for neck growth is on the order of the time required for development of axial strains of magnitude 0.1. 3.3. Case C. Material imperfection A material inhomogeneity by itself (PJ= 0, il # 0) may still cause unstable behavior although, as indicated above, the behaviour will not be as severe as for the other cases. The solution to (27) is given by
tThe characteristic time for exponential decay is defined to be that time at which the relative area difference has decreased from an initial value of -q to a value of -_9 exp(l). $It
2587
SOLIDS
1 W’b)
and
Li + h(t)a
DAMAGING
is possible to have a reversal of the rate of growth of the area perturbation, however this would occur at unrealistically large negative values of I and always at some time greater then zero.
a = exp( - H(t)]
’ exp[ici(s)]k(r)dr [S 0
1
with k(t) = -(n
- I)&[1 - exp(-e/nC)]
and H(t) as given above. By changing slightly the
2.588
LEVY: TENSILE INSTABILITY IN CREEP DAMAGING SOLIDS
r
aa (o)r.oo 0.6
where use has been made of (17). Although the linear analysis can predict the initial state of imperfection growth it is incapable of describing the developing interaction of dilation due to cavitation with the geometrical constriction of area at increasing longitudinal strain. Therefore, in order to obtain local (finite) and bulk areas at fracture, a fully nonlinear analysis is required.
c
4. THE TENSILE INSTABILITYNONL~~R ANALYSIS
0.6 t
Fig. 4. (a) Response to geometrical nonunifo~ity-linear analysis. (b) Response to material nonunifo~ity-linear analysis. definition of characteristic time to be that time at which a has decreased from a value of zero to a value of -q exp(1) an examination of a and k given above indicates that the characteristic time is just about the same order of time required for development of axial strains of magnitude 0.1. Figure 4 (a, b) depicts the evolution of an area inhomogeneity for various values of parameters and for two of the cases just discussed. Case B curves depicting the effect of material and geometrical imperfections were not drawn since it was determined that, for relative variations in C up to lo%, the effect was negligible. Hence the curves were essentially the same as those given for case A. Only constant load test curves are shown but constant stress test curves exhibit similar behavior except that they are displaced to the right. The curves were obtained by straightforward numerical integration of (7), (S), (9) and (27a, b, c) and are measured in terms of nondimensional time defined by
It has already been noted previously that a response which has been determined to be unstable in the small may not be the kind of catastrophic behavior normally associated with our intuitive notion of instability. The fact that this is true can be seen from a number of simulation studies concerning the rate dependent flow of metal bars [6,10, 121. Because of this an alternative definition of instability has been proposed [3j which associates the onset of instability in both rate dependent and independent materials to the maximum value of bulk section strain attainable i.e. instability is said to occur when the unique correspondence between increments of local and bulk section strain is lost. In mathematical form the instability criterion is de/de” = 0 or, alternatively dA/dA” =0
(30a)
in which case instability is associated with the minimum value of bulk section area attainable. In order to obtain expressions for dA /dA” (or de{dF) governing equations (7), (10) are written for the local and bulk sections of the bar together with the connecting relation P(t) = a( = ci(t)A(t) which simply states that, at any instant, local and bulk bar sections support the same load. Then (:I*‘[1
+ln($r($)
=(I -?r(~)“-‘[l+ln(3’“‘T’
A” *
0
X*
r = t/t, = m&t
(314
A0
which has, at least numerically, a first integral A@).? A straight-forward application of the instability criterion (30a) would then yield the local area cross section at which instability occurs. A simpler approach would be to use (31a) directly to get
TFor integral values of n,
F(~,n,C)=(l--1)“F(%.n,C) W) where (n - I)! (n -r
-
[(i-(1
l)!(n2cy
1. ‘- *
+ln($-J-11
(32)
LEVY:
TENSILE
INSTABILITY
Depending on the values of the parameters q, C, c instability may occur at either a/&
= exp( -nZI) = A,/&
(33a)
when (30a) is satisfied, or when A/A,
= exp(-nC)
= A,/A,.
For the latter case the instability dJ/dA
DAMAGING
SOLIDS
2589
bar solution is approached and recovered when c = C. For this case then, stability loss will occur at the section (local, bulk) in which cavitation is most severe. Figure 5(b) depicts this behavior for the range of values of c and C.
Wb)
criterion becomes
= 0
(30b)
and stability loss is associated with the minimum value of local section area. That instability may be associated with critical values of either local or bulk section area may be seen from the following three cases which describe the behavior for different ranges of parameters. 4.1. Case A. Geometrical
IN CREEP
D
0.9
$ OB (1V_1*‘lncrMrl”O, _
imperfection
For this case the Monkman-Grant constant is uniform (2; = C) while the relative area imperfection q may take on any value. By employing (32) the initial conditions.
“r(b)
and noting that the function f given by
f=z n-‘[l +lnzarm’, B>l,n>l,O
t---
l
E-c
while for q < 0 instability occurs when A/A, = exp( - nC). Hence, when the initial local area cross section (A”,) is less than the initial bulk section area (A,), instability (necking) is associated with critical values of local section area &. Likewise for & > A,, instability (bulging) is associated with A,. For q = 0, instability occurs when the uniform bar area is a minimum. Figure 5(a) illustrates this behavior for different values of the parameter 9. 4.2. Case B. Material
imperfection
Here, the bar is geometrically uniform at the initial instant (q = 0) but the Monkman-Grant constant is nonuniform (c # C). The function f given above decreases monotonically with decreasing values of z and increasing values of p and so (32) together with (34) imply that for c < C instability occurs when the local area is A”/& = exp( - nC) while for c > C instability occurs when the bulk area is A/A,, = exp( -nC).
AS r??approaches C from above or below the uniform
Fig. 5. (a) Stability with respect to geometrical nonuniformity (c = C). (b) Stability with respect to material nonuniformity (q =O). (c) Stability with respect to geometrical and material nonuniformity.
2590
LEVY: TENSILE INSTABILITY IN CREEP DAMAGING SOLIDS
4.3. Case C. Material and geometrical imperfection
This case is more involved since it is not always possible to predict the critical area without explicitly solving (32). This follows from the fact that, for certain values of parameters (q, (?, C), material nonuniformity and initial geometrical nonuniformity may be competing mechanisms for growth of inhomogeneous deformation. If c < C and q > 0 then (32), (33) and the monotoni~ity off are su~cient to determine that stability loss occurs when the normalized local area A”/& assumes the value exp( -nC). For c > C, q < 0 the same argument implies stability loss at a normalized bulk area A/& of exp(-nC). However, when 2; > C and rl > 0 (or (? < C, q < O),floses its monotoni~ity and (32) must be integrated numerically (or (31 b) must be solved). A quantitative picture of the behavior is given in Fig. 5(c) for a range of values of q, c, C. Clearly a slightly smaller initial local section area does not necessarily imply that stability loss will occur at the local section. To get an idea as to the relative strengths of the competing mechanisms (32) was integrated numerically subject to the initial condition (34) using typical values of parameters (q = 0.01, C = 0.05). It was determined that for L = 0.01 corresponding to a value of c of magnitude 0.0505 the dominating influence was the initial geometrical imperfection i.e. instability occurred at the local section. In fact, in order for the material nonuniformity to dominate, 1 would have to be on the order of 0.1 corresponding to a 10% axial variation in Monkman-Grant constant. Hence, for most practical cases the critical area for instability is governed by the initial geometrical imperfection-a result consistent with the linearized analysis. The critical values of area given by (33) are equivalent, by (1 1), (12), to associating instability with infinite strain or a unit value of area fraction. From this result it therefore follows that loss of stability, as defined by (30), and fracture, as defined by a unit value of area fraction, coincide. This situation is similar in behavior to other rate dependent, but incompressible, materials in which loss of stability and rupture (zero local area cross section) coincide [6,7]. For the rate independent, power law hardening material instability occurs at finite values of local section area and local section strain and is not associated with any definition of failure. Hence, for this case, loss of stability represents a true bifurcation and does not signal the end of the deformation process as it does in the time dependent cases just noted. It should also be noted that the conditions (32) depend only on the material parameters and are, in fact, independent of any specific form of load history. Tensile bars composed of identical material but subject to different loads will lose stability at the same value of area however the time at which this occurs will depend on the specific form of the loading. In order to study the time dependent growth of nonuniform deformation prior to stability loss (frac-
ture) (7), (8) and (31a) may be integrated taking into account the initial conditions
A=&,A
=A,,
att =O.
e=c=O
The relative area difference a follows directly by employing the definition (21) a=A/A
-1.
(21)
The solution, in analytical form, can be obtained for integral values of n however, since the result is cumbersome, a straight-forward numerical procedure has been adopted. The results are depicted in Fig. 6 for various values of parameters and, just as in the linear analysis, are given in terms of nondimensional time z defined in Section 3. The bracketed quantities in Fig. 6 are the values of relative area and bulk section strain at the onset of instability or fracture (note that if stability loss occurs at the bulk section area then the local section area is given). For local section stability loss, the quantity a, was obtained from (21), (33a) and can be written as a,=[A,exp(-nQ]A-‘-
1
(35)
with the value of A obtained from the final data point calculated. The value should be an accurate approximation to the true value since, just prior to fracture, the bulk section area is essentially constant. If stability loss occurs at the bulk section area then an equation analogous to (35) has been used. The bulk section strain (or local section strain as the case may as
I-
(o)r-cm
1 D 1 0.4 t
*I
*
lr?I 0.4 t
Fig. 6. (a) Response to geometrical nonuniformity(b) Response to material analysis. nonlinear nonuniformitv-nonlinear analvsis. i
--=T-rr
- -=‘-- - --I --*’
be) is obtained directly from the expression obtained by combining (35) and (10) e = l+n(~)““’
+ il”.
(36)
For geometrical imperfection only (,?.= 0) a curve in Fig. 6(a) with C = 0.05 and its counterpart with C = 0.075 indicates the effect of cavitation on the flow process. As the ability of a material to damage increases (decreasing C), the magnitude of a, decreases such that the limiting behavior is given by u,= -?I
=a(r
=O),C-,O.
At the same time the strain rate difference increases causing a rapid constriction of the local area cross section while the overall ductility decreases i.e. e, decreases. Taken together the above results imply that, as the ability of a material to cavitate increases, there is more of a tendency for a highly localized neck to form prior to failure however this is countered by fracture at a finite area. This kind of behaviour is in agreement with a flow simulation study of necking in creep cavitating copper bars [lo]. In contrast to this type of behavior the opposite limiting case of creep rupture occurs as C becomes infinite, then a,= -l,C+cc where use has been made of (33). Also, the strain rate difference is maximized for an infinite value of C while the overall ductility is a maximum. Hence for materials deforming solely by power law creep the well known result that the instability process is one of slow growth leading to the formation of a diffuse neck is recovered. In the absence of geometrical inhomogeneities (q = 0), for a given value of C, increasing A i.e. decreasing C, causes a larger magnitude of a, but decreased ductility [Fig. 6(b)]. Although this result may appear to be in conflict with the results from Fig. 6(a), i.e. decreasing C (=c) causes a smaller magnitude of u,., it can be resolved simply by noting that in the former case it is e, and not C, which decreased. Although in both cases the critical time to fracture will be less, the value of area in the bulk section at this time will not since C has not been changed. Hence the magnitude of a, is increased. That the criticai time to fracture for a given C decreases with increasing i follows directly from the fact that increasing i; means decreasing c. It has been shown in the previous section that when a bar is geometrically uniform but materially nonuniform (rl = 0) initial rates of growth are not as severe as those associated with a geometrically nonuniform but materially uniform bar. The nonlinear analysis indicates that critical values of relative area inhomogeneity, in the absence of geometrical nonuniformity, will be smaller in magnitude owing to the dependence of that quantity on the initial local section area (35). Also, the time at which instability
IN CREEP DAMAGING
SOLIDS
2591
is reached is delayed for the geometrically uniform but materially nonuniform bar [Fig. 6(b)]. Taken together the above results indicate that the nonlinear response is governed primarily by geometrical nonuniformities and not material ones. For typical values of parameters (A -0.01, P)-0.01) curves obtained when both effects are considered differ only slightly from those shown in Fig. 6(a). 5. CONCLUSIONS In this paper the effect of completely creep constrained grain boundary cavitation on the tensile stability of metal bars has been investigated. Two definitions of instability have been employed in order to study the effect of both material and geometrical nonuniformities. The first is concerned with the immediate time dependent response to a disturbance whiie the second is related to a loss of correspondence between increments of perturbed and unperturbed quantities governing a process. Based on the first definition (stability in the small) a linearized stability analysis predicts that an area inhomogeneity will always, at least initially, grow in time. Although the rate of growth can change sign it has been shown that this occurs at unrealistically large values of local Monkman-Grant constant and so the response is unstable. For typical values of material parameters, it has been shown that the appearance of visible necks will occur, for the damaging case, at strains on order of 0.1 as opposed to strains on order of unity (cavitation suppressed). This is true regardless of whether the initial defect is geometrical, material or a combination of both although the response is more sensitive to geometrical defects than to material ones. As expected initial rates of imperfection growth increase the greater the ability of the material to damage as measured by decreasing values of Monkman-Grant constant. Also, initial rates of growth are larger in the constant load test then in the constant stress test. A nonlinear analysis based on the second definition of instabiljty predicts that stability loss can occur when either the local or bulk area cross section reaches a critical value. The exact location being dependent on relative magnitude of local and bulk section Monkman-Grant constants as well as initial local and bulk area cross sections. The critical value of area, which occurs at infinite local section strain, coincides with our definition of fracture, i.e. unit value of local damage variable (actually, area fraction of cavitated boundaries). So, by this definition, the deformation is stable prior to the point signaling fracture and the end of the process. By contrast the first definition of instability indicates that the process, beginning at the instant the load is applied, is unstable. An examination of the history of the response subject to an initial area inhomogeneity indicates that the material is more sensitive to geometrical as opposed to material inhomogeneities. Also, as the
2592
LEVY:
TENSILE INSTABILITY
ability of a material to damage increases then (i) the area inhomogeneity at fracture decreases, (ii) the rate of change of area inhomogeneity increases and (iii) the overall ductility decreases. Taken together these characteristics imply that the effect of cavitation on flow is to increase the tendency of a highly localized neck to form at smaller overall strains which is countered by fracture, or stability loss, at finite area. ~ck~ow~edge~en~-The author gratefully acknowledges the helpful comments of Professor M. P. Bieniet. REFERENCES 1. E. W. Hart, Acra metall. 15, 351 (1967).
IN CREEP DAMAGING
SOLIDS
2. J. J. Jonas, R. A. Halt and C. E. Coleman, Acta metall. 24, 911 (1976). 3. J. W. Hut~hinson and K. W. Neale, Acra ~taZ~. 25,839 (1977). 4. J. D. Campbell, J. Mech. Phys. Solids 15, 359 (1967). 5. U. F. Kocks, J. J. Jonas and H. Mecking, Acta metall. 27, 419 (1979). 6. N. J. Hoff, J. Appl. Mech. 20, 105 (1953). 7. J. W. Hutchinson and H. Obrecht, Fracture I977 1, ICF4, Waterloo, Canada (1977). 8. J. J. Jonas and B. Baudelet, Acta metalt. 25,43 (1977). 9. A. J. Levy, J. appl Mech. 52, 615 (1985). 10. A. J. Levy, Acta metall. 34, 1991 (1986). 11. S. Sagat and D. M. R. Taplin, Metals Sci. IO,94 (1976). 12. M. A. Burke and W. D. Nix, Acta metall. 23, 793 (1975).