- Email: [email protected]
Non-linear one-dimensional continuum damage theory
Pergamon
1m J. Mech,S(i. Vol. 38. No. 10, pp. 1139 1150,1996
Copyright I~ 1996ElsevierScienceLtd Printed in Great Britain.All rights reserved 0020-7403/96 $15.00+ 0.00
0020-7403(95) 00106-9
NON-LINEAR
ONE-DIMENSIONAL
CONTINUUM
DAMAGE
THEORY
VEADYSLAV P. GOLUB Department of Creep Mechanics, Institute of Mechanics, Ukrainian Academy of Sciences 3, Nesterov str., Kyiv 252057 Ukraine (Received 20 June 1994; and in revised form 2 October 1995)
Abstract--The background of Non-Linear Continuum Damage Mechanics is presented for the case of one-dimensional tensile stresses. A brief critical analysis of the existing methods of non-linear damage models construction is given. A new approach to the damage theory development based on the "separability" principle is suggested. The condition of non-linearity and the criterion of long-term failure are formulated. The non-linear damage constitutive equations for static and cyclic loading, allowing one to take into account the loading history, are proposed on this basis. Within the framework of the approach suggested the problems of total and residual lifetime calculation under additional loads and partial unloads are solved. The predictions are compared with experimental data obtained using some particular heat-resistant materials. Copyright ;(~ 1996 Elsevier Science Ltd. Keywords: damage kinetics, non-linearity conditions, separability principle, momentary damage, temporal damage, non-linear damage summation law.
NOTATION
o) g~ f~, WR zu tR nR C, D, /~, k, m, n
applied stress scalar damage parameter integral momentary damage parameter integral temporal damage parameter specific fracture energy ultimate strain time to rupture number of cycles to rupture material coefficients 1. I N T R O D U C T I O N
Structural components subjected to long-term loading, especially at high temperature, suffer non-elastic deformation and damage as a result of micro-cracks and cavities nucleation and their growth and coalesence into macroscopic cracks in the materials. Modern methods for the analysis of the progressive material strength deterioration are based on the concepts of Continuum Damage Mechanics [1 3]. This approach has been developed further for the many phenomena involved in the problems to be solved, including coupling between damage and creep [2, 4, 5], low and high cyclic fatigue [6, 7], creel>fatigue interaction [6-8], crack growth [6, 9-13] and components lifetime prediction [4, 12, 14]. Two main problems arise in the description of the structural reaction to the applied loads in terms of Continuum Damage Mechanics. The first one is what kind of coupling between classical governing equations of Solid Deformable Mechanics and damage evolution equation should be used to take into account the effect of damage kinetics on the stress-strain behaviour. This problem is usually solved using the "effective stress concept", proposed by Kachanov [-1] and Rabotnov [2]. The generalization of this hypothesis for three-dimension (3D) conditions has been made by Hayhurst [14] and Murakami and Ohno [15]. The second problem has been associated with the identification of a damage parameter and the construction of adequate damage constitutive equations. This problem is usually solved from the phenomenological point of view and the damage parameter is considered as an internal variable of the damage process. The corresponding evolution equations are formulated in a differential form so that their structure enables the boundary conditions to be readily satisfied. The damage kinetics between limiting values remain unknown and are not taken into account in the equations. Most 1139
1140
V.P. Golub
existing damage evolution equations therefore fit the linear damage summation law which has not been confirmed experimentally [16]. The objectives of this paper are to introduce briefly the non-linear uniaxial damage theory, to review its specific advantages campared to the existing models and to illustrate it by results obtained with some structural materials under long-term static and cyclic loading. 2. N O N - L 1 N E A R I T Y C O N D I T I O N S
Before any non-linear damage cumulative theory is introduced it is necessary to define what is meant by "linearity" and "non-linearity" in Continuum Damage Mechanics and what kind of criteria may be used for their identification.
2.1. Damage linearity notion Under the classical Miner Robinson definition the linearity of the damage process is usually associated with the linear law of the damage accumulation over time (Figs l(a) and l(b), straight lines). The physical meaning of this definition is that equal time intervals At lead to equal damage increments Ao) and the total lifetime under stepped loading will be always equal to 1. Analytically this condition is given by the relation -
~
-1
tR(a)
~
-1,
(1)
tR(0")
i: 1 tR(ai)
which is well known as the linear damage summation law and the graphical interpretation is shown by the straight line in Fig. l(c) which is independent of the loading history. Here a is the applied stress and tR(') is the time to rupture under constant load. According to Kachanov's approach [-1] the damage kinetics are described by the evolution equation d(,) _ C
(2)
dt
'
from which the following expression for the damage parameter ~o is obtained ~,)= 1 -
1-
t
T+,,
(3)
tR(O-
where C and m are material coefficients to be determined. In this case, the damage kinetics is assigned in a non-linear form (Figs l(a) and (b), concave) and equal time intervals At lead to different damage increments Aco. However, differentiating Eqn (3) with respect to time and summing up over all values of damage increments, the damage cummulative law is written as follows d~,)
(1 + m) 1
tk(
tR(~)~
tR(~)
1,
(4)
which can be seen to coincide with the linear damage summation law (1). The characteristic feature is that, Kachanov's equation, Eqn (2) describes damage kinetics in a non-linear form over time and leads to the same results under stepped loading as a linear equation. All other known damage evolution equations, in particular Rabotnov's equation [2] d°) dt-C
(or) ~
m
1 "~o ~
(5)
and the Lemaitre Plamtree equation [6, 8] d(o _ C dt
"
(6) '
gives estimations similar to those mentioned above. The corresponding damage kinetic diagrams will differ from those assigned by Eqn (1) only by their curvature and these equations will fit the linear damage summation law adequately, too [17].
Non-linear one-dimensional c o n t i n u u m damage theory
1141
(a) 1.0
0.3
i
~o 0.5
0.0
(b)
(c) 1.fl
1.0
~]~
0.5
0.0
ti(*l)
tr(al) t
tR(O'l)
o.~
0.5
0.0
0.5 tit R
1.0
ti/tRi
Fig. 1. Schematic illustration of damage kinetics in (a) physical, (b) normalized time-scale, and (c) of damage s u m m a t i o n law.
2.2. Definition of damage non-linearit,y As can be shown in the previous section, non-linear damage kinetics-time relation is a necessary, but not a sufficient condition of damage non-linearity on the whole. So the damage non-linearity is defined by the conditions in which damage kinetics are given by a non-linear function over time, and the normalized total lifetime under non-stationary loading will be different from I (Fig. l(c), dashed lines). In analytical representation the damage non-linearity condition is written as follows d~o =f(o-, t) dt (7)
ti (O-i, t)d, ~] ~'of i= 1
tK
#1 " ,Ll ti = ~"
where i indicates the number of the different stress levels. 2.3. Principle of separability The analysis of the reasons for the absence of the difference between both approaches mentioned in Section 2.1 in the problem of lifetime calculation under stepped loading shows that, they both give the unified damage kinetics curve in the normalized coordinates system which does not depend on stresses (see Fig. l(b)). Hence, to satisfy the damage non-linearity condition (7), it is necessary to assign the damage kinetics in the normalized time scale as that depending on stresses. This condition, which may be interpreted as a sufficient condition of damage non-linearity, will be termed the "principle of separability". In analytical representation the separability principle may be given by relation d~.9 = fT [0"{ T), m, C t, C> ..., Ck } d T or in form
d
--
do" T = const
= var,
(8)
(9)
1142
V.P. Golub
where f x ( ' ) is the damage function, T is the normalized time, and Ci(i = 1, 2 , . . . , k) are material coefficients to be determined. Hence, all damage evolution equations, which do not satisfy condition (8) cannot give the damage as the non-linear process. In particular, replacing the independent variable t by T for initial linear damage equation from (1), for Kachanov's Eqn (2), for Rabotnov's Eqn (5), and for the Lemaitre-Plumtree Eqn (6) one obtains the following expressions correspondingly dco - 1 dT dco dT
(10)
1 (1 + m )
1 (1-co)"
(11)
dco_ _J ['(I--CO)m - dco
(12)
dco 1 dT=(1 +m+q)
(13)
dT
(1 - CO)mJO
COp
1 (1--co)re+q"
AS can be seen, all these equations do not contain stresses in their right sides and therefore, as it was shown above, fit the linear damage summation law adequately. 3. CONSTITUTIVE EQUATIONS OF THE THEORY There are many possible ways to construct the damage evolution equations which satisfy the non-linearity condition, Eqn (7). Consider long-term static and cyclic loading regimes using the separability principle. All results are presented and discussed in the normalized time scale. 3.1. Assumed relations First of all it is necessary to select and to justify the structure of the initial damage evolution equation. The detailed analysis of existing evolution equations and the non-linear forms of the damage models is presented in Ref. [17]. Here it is emphasised that the initial equation must contain two terms with damage variable of different power index (one of them must depend on stresses). In the general case, the initial damage differential equation is written in the form dco =ft' [a(t), co(t), C~, C2)f," [co(t), C3(a)] dt,
(14)
where ft'(.) and ft"(') are damage functions and C3(') is the stress dependent power index• The structure of evolution Eqns (5) and (6) satisfy initial Eqn (14). In order to specify Eqn (14) let us assume that damage accumulates not only over time but also under each moment of change in stresses including initial loading. In such a case, let us introduce two damage parameters and assign them by their integral measures. The first one reflects the value of the cumulative damage under each moment of change in stresses and has been defined as the "momentary" (time-independent) damage f2,. The second or traditional parameter reflects the value of the cumulative damage over time under constant stress and has been defined as the "temporal" (time-dependent) damage ~t. Each of these components will be identified with the corresponding specific fracture energy. We will also assume that energy spent on fracture due to the total damage accumulation is a constant value for the given material and temperature and is equal to the fracture energy of the single momentary deformation. In this case, the condition of balance between integral measures of momentary and temporal damage at the moment of failure is written as follows
n.+n,=
w.
f2~
w-S+W
f~t
:I,
(153
where W~ is the specific energy of single momentary fracture. The value of integral momentary damage f ~ is determined by the stress-strain diagram and corresponds to the work done by the applied stress a acting through the ultimate strain eu [Fig. 2(a), shaded area]. The integral temporal damage Ot may be associated with the energy dissipated in the process of the new internal surfaces formation. Its value may be determined by the normalized
Non-linear one-dimensional continuum damage theory
(a)
1143
(b) 1.0
0.5
tO m
0
r~
0.0
EB
0.5
1.0
tit R Fig. 2. Integral m o m e n t a r y (a) and temporal (b) damage measures.
damage kinetic diagram [Fig. 2(b)] and directly proportional to the area (shaded plot) bounded by the damage curve, the rule change of which is unknown. The value of total fracture energy wR is determined by the stress strain diagram too, and corresponds to the work done by stress limit OrB acting through the ultimate strain ~,o, so that WR =
~Oo(~:1d~,
(16)
where q~o(e) is an equation of the stress-strain diagram. Analytical expressions for momentary fl~ and temporal f~ damage will be dependent on loading conditions. 3.2. Damage kinetics under static loadin9 In the case of static loading following the previous definitions for the value of integral momentary damage f ~ [see Fig. 2(a)] the following relation is obtained fQ =
{Po(O&:+ (~:~- c,,~){ri.
(17)
)
Here a~, is the strain on the stress strain diagram corresponding to the applied stress ai {i = 1,2 . . . . .
n).
To specify the second summand in Eqn (15) it will be assumed that, the dimensionless function ~,/WR is some invariant relative to the time to rupture tR quantity depending on the damage parameter e). The average value 0,,, of the function (o(t) on the interval 0 ~< t ~< tR may be chosen so that
R-- Om
--
(18)
--O'"CVRd
where COc(') is the value of the damage parameter under static loading. The balance condition (15) between the momentary and temporal damage components at the moment of failure, taking relations (16)-(18) into account, may be written in the form J'o ~o(r,)d~ - J','i][{po(r,) - {hi d~:
+
(oc
d
= 1,
(19)
S~° ~0o(~)d~:
or, after simple transformations, in the form ~o¢
d
"
qOo(Odc -
]-(po(C)- csi] d~: = O.
which may be considered as a long-term fracture criterion under static loading.
(20)
1144
V . P . Golub
Then, using the differential equation
d(~)c
[
o_i ] m r ~ l ~ , ~ ,
(=,,
=CL _I Ll- o/
as the initial damage evolution equation and replacing t by T = t/tR, on integration of Eqn (21) for the initial conditions coc = 0 at T = 0, one obtains the following expression g
c9c
=1-
1-
t
1 + m + qW)
,
(22)
1 C , a~'"
(23)
where tR is the time to rupture given by tR
1 (1 + m + q(cr)) Ca'/'
=
On substitution of Eqn (22) into criterion (20) and variation of an expression for the q(a) index is obtained that together with the evolution Eqn (21) gives the constitutive equations of the nonlinear damage theory. In the normalized time scale this set of equations is written as follows
1
.[
1 ]
d-T = [1 + m + q(a) q(a)
(24)
J'~ (Po(~:)dc
- (2 + m),
= -
The graphical interpretation of the damage kinetics calculated from Eqn (24) is shown in Fig. 3(a). The fundamental difference between the results obtained and those previously developed [see Fig. l(b)] is that the damage kinetics in the normalized time-scale depends on stress. The corresponding (a)
(b)
2
1.(l
0.5
0.0
0.5
1.0
0.0
Tt
0.5
T2 1.0 t/t R
t/tR
(c)
(d)
I.O
2
I'0 /
o. 1 ~
" '
| ~" 0.5
0.0
0.5
t/t R
1.0
0.0
TI
0.5
T 2 1.0
t/tR
Fig. 3. Damage kinetic diagrams under static [(a) and (b)] and cyclic [(c) and (d)] loading calculated from non-linear damage theory.
Non-linear one-dimensional continuum damage theory
1145
damage diagrams displace down in the integral measure of the temporal damage with the stress rising, and up in the integral measure of the momentary damage. The effect of separation of normalized damage kinetic diagrams in the stress parameter allows one to take into account the loading history under non-stationary conditions. As an example, Fig. 3(b) shows the graphical solution of the classical problem of residual lifetime calculation under additional loading (a2 < as) and partioned unloading (as > ai ). As can be seen, the total lifetime will be less than I (curve I) and greater than ! (curve 2), respectively. Qualitatively these results are in good agreement with the majority of experimental data. 3.3. Damage kinetics under cyclic loading Under cyclic loading the discrete time assigned with the number of cycles of change in stresses n is used as the argument for the damage function. In order to pass to the natural time t it is sufficient to know the loading frequency v = dn/dt since n = yr. Under sufficiently large values of n it can be assumed that the smoothed processes of discrete loading satisfy differential Eqns (2), (5) and (7). In the case of cyclic loading, the expression for the integral momentary damage f2~ [see Fig. 2(a)] may be written in the form fl~ = ae,~ -
fo (a) da.
(25)
0
or, taking into account the loading rate, in the form df~
f~a -
-
e.u
da
-)Co(a,),
(26)
where fo(a) is an equation of the stress-strain diagram [fo(a) = q0o 1(~)]. The average value con, of the function ~o(n) on the interval 0 ~< n ~< nR by analogy with Eqn (18) may be written as follows
f~' = cgm~]-
WR
~°'o)f(n)dn -- f] (of ( ~ ) d (-~R) , nR
(27)
where o)f(. ) is the value of the damage parameter under cyclic loading and nR is the number of cycles to failure. The balance condition (15) between the momentary and temporal damage components at the moment of failure, taking Eqns (16), (26) and (27) into account, may be written in the form ~:.
- Yo(a~)
£u
+ f l o ) f \n~/ ( n ~ d ( n )~
= 1,
(28)
= O,
(29)
or, after simple transformations in the form (,..If
d
"
0
,%-- f(ai)
which may be considered as a long-term fracture criterion under cyclic loading. Here it is assumed that, WR = dWR/da = r,u. Then, using the equation do)f dn = D
V
ai
I k '
Ll_--2~)fj
V1 l/~'',
(30)
ko)~J
as the initial damage evolution equation and replacing n by N = n/nR, on integration of Eqn (30) for the initial conditions o)f = 0 at N = 0, one obtains the following expression n --
nR
(o)f)
=
F(2 + k - fi(a)) fi"' ([ 2 o)f_)k do)f, Y(1 + k)F(1 - fi(a)) (e)f)/~/~)
(31)
1146
V.P. Golub
where nt is the number of cycles to rupture and is given by
nR =
F(1 + k)F(1 - fi(a)) 1 1 F(2 + k - fi(cr)) Da~' - D,a~ "
(32)
Here F(-) is a gamma-function. On substituting Eqn (31) into criterion (29) and varying a an expression is obtained for the fi(a) index that together with evolution Eqn (30) gives the constitutive equations of nonlinear damage theory. In the normalized time scale this set of equations is written as follows d(~__j f =-((of)I~(~l . ~.l (1 - (of)k dr:of dN (1 - (of)k )o ((of)/~l~' (33) fl(a) = ~:u - f 0 ( a l ) ( 2
+ k)
~:u - f o ( " , ) The graphical interpretation of the damage kinetics calculated from Eqn (33) is shown in Fig. 3(c). As can be seen, the fatigue damage kinetics depends on stresses in the normalized time scale, too. However, unlike the static loading, the cyclic damage diagrams displace up in the integral measure of the temporal damage with the stress rising, and down in the integral measure of the momentary damage. In such a case, the total lifetime will be greater than I [Fig. 3(d), curve I] under additional loading (~rl < ors) and less than I [Fig. 3(d), curve 2] under partional unloading (~4 > al ). Qualitatively these results also coinside with experimental data. 4. T O T A L A N D R E S I D U A L
LIFETIME
PREDICTION
Now consider the capabilities of the damage theory suggested as an example of residual and total lifetime calculation of prismatic rods subjected to non-stationary static and cyclic loading at high temperature. In this case, creep and fatigue processes occur. The rods are made of refractory and heat-resistant alloys. 4.1. Determination of the theory coefficients In order to solve the problems on the basis of the damage theory developed it is necessary to know two groups of coefficients. The first one includes the coefficients m, C and k, D which specifies a material resistance to the damage accumulation over time. The second group includes the function q)o(~), the a u - ~-u characteristics which specifies a material short-term strength. So the base experiment involves one long-term strength curve or one fatigue curve with symmetrical cycle and stress-strain diagram. The value of stress independent m and k exponents is first determined using Eqns (23) and (32), respectively. These equations are fitted to the uniaxial long-term strength and fatigue data using a least-squares technique, and errors are minimized with respect to the corresponding coefficients. The function ~o0(e) and all corresponding characteristics (~, cru, ~:u) are determined from the tension tests data, presented in the form of the stress strain diagrams. Then, the stresses dependent q(a) and fl(a) exponents are calculated from the second relations of Eqns (24) and (33) using previous determined m and k exponents and q~o(e,) function. Finally, the C and D coefficients are determined from Eqns (23) and (32) using the values of m, k, q(a) and [3(~) exponents. As an example, Table 1 shows values of the coefficients m, C,, k, D, and values of the characteristics au, ~, and Fig. 4 shows the relationship of q and fi exponents with the stresses for some materials calculated from the above presented procedure. 4.2. Lifetime under stepped loading The residual and total lifetime calculation under stepwise loading is carried out using a step-bystep integration of Eqns (24) and (33) with varying limits of integration at each stage. The sequence of integration is given by the familiar loading programme. The value of load within the limits of each
Non-linear one-dimensional continuum damage theory
1147
Table 1. Values of calculated coefficients and parameters Alloy
EI437B 0 (°C)
C,(MPa-~-hr m
-1)
D,(MPa-k'(hr'f)
750
800
1 . 8 6 × 1 0 ~l* 4.6
3 . 5 5 x 1 0 -12 4.3
-1 )
--
k a, (MPa) ~
5.9 680 0.07
5.4 520 0.08
E 1698VD 750
900
950
VJL12U 1000
1 . 5 5 x 1 0 -z5 8.2 2.51 × 10 T M 9.6 775 0.10
1.38x10-: 6.3 4.57 × 10 - . 1 13.6 580 0.06
2.00xlO -t3 5.2 3.02 × 10 -3'~ 11.1 370 0.06
5.37x10 -17 6.5 2.69 × 10 -28 9.6 475 0.04
(a)
(b)
4
!
I
2
I
(3
0.2
I
,
0.4
0.6
'~0.2
0.4
N
/
N
/
I
\
e
\
\
"\
\
_
j" f
/
0.8
x
\
f
I
0.6 or ,,, orB
X
/
/
4
"~,
/
°B
2
I
I 110'.8!
"
6
E 1867
\
9
,f 12-
8
IO
t
15
i
Fig. 4. Relationship of power indexes q (a) and fl (b) on stresses for heat-resistant EI437B (dashed lines) and EI698VD (dot dash lines) alloys at 750°C. stage is assumed to be constant. The fracture condition is given by the relation
•
~oi(ai, t) ,= ~, = 1,
(34)
i=a
where i is the n u m b e r of loading stages. So in the case of static loading integrating Eqn (24) with the initial condition ~oc = 0 at t = 0, the value of d a m a g e coi f o r / - s t a g e of loading is determined from the expression o9i=1-I(1-o9i
a) a + ,, +q(,) _
t
] l + m 1+ q(~,t
(35)
/R(O'i)J
and the residual lifetime t* for this stage from the expression t* = tR(~ri) [1 -- COi_ t] 1 + " + qI~j~,
(36)
where it is assumed o3c = ca and ~o(crj_ 1) = ~o(ai) at t = ti. The sequential solution of Eqns (35) and (36) with allowance for E q n (34) determines the j-stage of loading on which fracture takes place, so that the total lifetime tRZ under an arbitrary n u m b e r of loading stages is determined by the relation tax=t1
Here ta,
t2, ...,
+ t2 +
... + t R ( 6 j ) [ l
are loading durations on each stage.
-- ~ O ) - l ] l+m+q(~j)
(37)
1148
V.P. Golub
In the specific case of two stepwise loading (i = 2), for the total time to rupture t~z we will obtain the following from Eqn (37) tRX = t I + t 2
l
tR(0"1)J
(38)
or after some simple transformations for the creep residual lifetime t* one obtains
V t* l :'(~)
tt
t.(0-,~+ L ~ J
=
i,
(39)
where 7(0") is a loading history parameter. For additional loading (0"t < 0"2) and partial unloading (61 > 0-2) one has, respectively,
1 + m + q(0-1)
7(0-) =
1 + m + q(0"2)
< 1 and 7(0")=
1 + m + q((rl) 1 + m + q(0"2)
(40)
> 1,
because q(O'l) < q(O'2) at at < 0-2 and q(0-1) > q(a2) at 0-1 > 0"2" As an example, Fig. 5(a) compares experimental data (points) with calculated lifetime curves (broken lines) for E1437B alloy under additional (0-t = 300 MPa, 0"z = 400 MPa) loading (o) and partial (ai = 400 MPa, 0"2 = 300 MPa) unloading (©) at 75OC. In the case of cyclic loading, integrating Eqn (33) with the initial condition coj = 0 at n = 0 and using the procedure mentioned above for the total number of cycles to failure nRz, under two stepwise
(b)
(a) 1"° ~..x~ ~ .
o"\
[ '
0.5 -
I.O ,,.,..
\.
\\
"'.
\
• 0.0
\
\,,o
~ \\ 0.5
•
~.
\
\\N
0.5
N\
.\
\
~
*\
\\
°\\
\~ 1.0
0.0
/
I
"-..o~ _
0.5
1.0
n/np.(eq)
t/tR(~O
(c)
(d)
1.0[~- -~..~
1.0
~
I o.,_
0.5
[A,A ~ / V M''-"
~''~.
\
°), I
0.0
0.5 t/tR(tr)
1.0
0.0
0.5
! .0
n/nR(o')
Fig. 5. Comparison of experimental data (points) with calculated life-time curves (broken lines) under stepped static (a) and cyclic (b) loading and under creel~fatigue (c) and fatigue~reep (d) alternation.
Non-linear one-dimensionalcontinuum damage theory
1149
loadings we will obtain the following nRZ = nt + n2
1
nR(at}
(41)
or after some simple transformations for the residual fatigue lifetime n~ one obtains nR(a 1) + [ _ ~ j
= 1,
(42)
where r/(a) is a loading history parameter, too. For additional loading and partial unloading we have, respectively, tl(a)-
1 +k+fl(a,) 1 +k+fi(ax) > 1 and ~/(a)< 1, 1 + k + fl(a2) 1 + k + fl(O'2)
(43)
because fl(al) > fi(a2) and ol < o2 and fl(a0 < fi(a2) at al > a2. The resulting comparison between experimental data (points) and predicted lifetime curves (broken lines) for E1437B alloy under additional (a~ = 200 MPa, a2 = 300 MPa) loading (e) and partial (at = 300 MPa, a2 = 200 MPa) unloading ((2)) at 8 0 0 C is shown in Fig. 5(b). 4.3. Lifetime under creep-fatigue alternation Now consider the conditions when static and cyclic loads act consistently. In this case, the damage kinetics has been determined by the simultaneous development of creep and fatigue processes. Using the superposition principle, the initial differential damage equation may be written as follows dcoz d l d d~- - d t (am, O)E) -1- --V ~ (O'a' ")~)'
(44)
where cox = cot + o)f is the value of the total cumulative damage. In Eqn (44) it is also assumed that, the creep damage kinetics has been given by the static component am of the load and the fatigue damage kinetics by its cyclic component era. We will consider two types of creep and fatigue processes alternation. In the first case, a portion of life tc is spent in creep and the remaining life t* being calculated under fatigue. Following integration of Eqn (44) with the initial condition ~o~ = 0 at t = 0 one obtains the following expression for the total damage co* cumulated to the moment of failure co~ = 1 -
1
6
| i = ~
t~'
tR~m)~
;l+,n+~l,°.,
/R(Ga))
,
(45)
from which assuming that, the fracture occurs at the damage value o)~ = I for the total lifetime tRXwe have the relation
7 1-k+fiqao) tRX = tc + t* = tc + /Rf
1
to [ c ; ~
(46)
tRcJ
where tRC = tR(O'm) and tRf = tR(G) are the time to rupture under pure creep and pure fatigue, respectively. In the second case, a portion of life tf is spent in fatigue and the remaining life t* being calculated under creep. So by analogy with the previous case one obtains the following for the total damage co~' cumulated to the moment of failure co~ = 1 --
1
tf
1+ .... ql~) _ _ _
tR~a)
t + k-/~/~,
(47)
tR(O'm)
and for the total lifetime tRX one obtains the relation
I tRX =
~f -]- t~ = tf -v tRC
where tRC = tR(am) and tRf = fR(O'a) as above. MS 38-10-H
~1 .... + ql~mi 1 -- tt- ~-; ~ , tRfJ
(48)
1150
V.P. Golub
The resulting c o m p a r i s o n between experimental data (points) a n d predicted lifetime curves (broken lines) for E1867 alloy u n d e r creel~fatigue a n d fatigue-creep a l t e r n a t i o n at 900°C a n d am = 300 M P a , aa = 200 M P a is s h o w n in Figs. 5(c) a n d 5(d), respectively. 5. CONCLUSIONS The n o n - l i n e a r d a m a g e c u m u l a t i v e theory has been developed for the uniaxial static a n d cyclic loading. The d a m a g e n o n - l i n e a r i t y c o n d i t i o n s are e n u n c i a t e d o n the basis of the "separability principle" giving the d a m a g e kinetics d e p e n d i n g on the stresses in the n o r m a l i z e d time scale. The constitutive e q u a t i o n s of the theory have been c o n s t r u c t e d a n d a technique is p r o p o s e d for the d e t e r m i n a t i o n of its coefficients for two structures of initial d a m a g e e v o l u t i o n a r y equations. W i t h i n the framework of the theory suggested, the p r o b l e m s of the total a n d residual lifetime calculation u n d e r stepped l o a d i n g a n d u n d e r creep fatigue a l t e r n a t i o n c o n d i t i o n s are solved. T h e o r y predictions are in good agreement with experimental results o b t a i n e d for some heat-resistant alloys. It must be emphasized that, all predictions have been m a d e using material coefficients which have been d e t e r m i n e d only from pure creep or (and) pure fatigue tests a n d stress-strain diagrams. REFERENCES 1. 2. 3. 4. 5. 6. 7.
L. M. Kachanov, On time to rupture under creep conditions (in Russian), lzv. AN SSSR, OTH, 8, 26 (1958). Yu. N. Rabotnov, Creep problems in structural members (Russian translation). North-Holland, Amsterdam (1969). J. Janson and J. Hult, Fracture mechanics and damage mechanics--a combined approach. J. Mech. Appl. 1, 69 (1977). F. A. Leckie and D. R. Hayhurst, Creep rupture of structures, Proc. R. Soc., Lond. A240, 323 (1974). J. Hult, Creep in Continua and Structures, Topics in Applied Continuum Mechanics. Springer, Vienna (1974). J. Lemaitre, How to use damage mechanics, Nucl. Engng Design 80, 233 (1984). J.-L. Chaboche, Continuous damage mechanics- a tool to describe phenomena before crack initiation. Nucl. Engng Design 64, 233 (1981). 8. J. Lemaitre and J.-k Chaboche, A non-linear model of creep-fatigue damage cumulation and interaction. In Mechanics of Visco Plastic Media and Bodies, p. 291. Springer, Berlin (1975). 9. R. Billardon and J. Lemaitre, Numerical prediction of crack growth under mixed-mode loading by continuum damage theory. Proc. of AUM Congress, Lyon (1983). 10. K. C. To, A phenomenological theory of subcritical creep crack growth under constant loading in an inert enviroment. Int. J. Fract. 11(4), (19751. 11. D. R. Hayhurst, P. R. Brown and C. J. Marrison, The role of continuum damage in creep crack growth. Phil. Trans. R. Soc. Lond. A311, 131 (1984). 12. D. R. Hayhurst, P. R. Dimmer and C. J. Morrison, Development of continuum damage in the creep rupture of notched bars. Phil. Trans. R. Soc. Lond. A311, 103 (1984). 13. S. Murakami, M. Kawai and H. Rong, Finite element analysis of creep crack growth by a local approach. Int. J. Mech. Sci. 30(7), 491 (1988). 14. D. R. Hayhurst, Creep rupture under multiaxial states of stress. J. Mech. Phys. Solids 20, 381 (1972). 15. S. Murakami and N. Ohno, A continuum theory of creep and creep damage. Proc. of3rd I U T A M Syrup. Creep in Struetures, p. 422. Springer, Berlin (1981). 16. V. P. Golub, A. D. Pogrebnjak and A. V. Romanov, The applicability of the linear damage summation hypothesis in creep and fatigue problems (in Russian). Prohl. Proch. 10, 20 (1993). 17. V. P. Golub and A. V. Romanov, Kinetics of damage in isotropic materials under creep conditions. Soy. Appl. Mech. 25(12), 1264 (1989).
1m J. Mech,S(i. Vol. 38. No. 10, pp. 1139 1150,1996
Copyright I~ 1996ElsevierScienceLtd Printed in Great Britain.All rights reserved 0020-7403/96 $15.00+ 0.00
0020-7403(95) 00106-9
NON-LINEAR
ONE-DIMENSIONAL
CONTINUUM
DAMAGE
THEORY
VEADYSLAV P. GOLUB Department of Creep Mechanics, Institute of Mechanics, Ukrainian Academy of Sciences 3, Nesterov str., Kyiv 252057 Ukraine (Received 20 June 1994; and in revised form 2 October 1995)
Abstract--The background of Non-Linear Continuum Damage Mechanics is presented for the case of one-dimensional tensile stresses. A brief critical analysis of the existing methods of non-linear damage models construction is given. A new approach to the damage theory development based on the "separability" principle is suggested. The condition of non-linearity and the criterion of long-term failure are formulated. The non-linear damage constitutive equations for static and cyclic loading, allowing one to take into account the loading history, are proposed on this basis. Within the framework of the approach suggested the problems of total and residual lifetime calculation under additional loads and partial unloads are solved. The predictions are compared with experimental data obtained using some particular heat-resistant materials. Copyright ;(~ 1996 Elsevier Science Ltd. Keywords: damage kinetics, non-linearity conditions, separability principle, momentary damage, temporal damage, non-linear damage summation law.
NOTATION
o) g~ f~, WR zu tR nR C, D, /~, k, m, n
applied stress scalar damage parameter integral momentary damage parameter integral temporal damage parameter specific fracture energy ultimate strain time to rupture number of cycles to rupture material coefficients 1. I N T R O D U C T I O N
Structural components subjected to long-term loading, especially at high temperature, suffer non-elastic deformation and damage as a result of micro-cracks and cavities nucleation and their growth and coalesence into macroscopic cracks in the materials. Modern methods for the analysis of the progressive material strength deterioration are based on the concepts of Continuum Damage Mechanics [1 3]. This approach has been developed further for the many phenomena involved in the problems to be solved, including coupling between damage and creep [2, 4, 5], low and high cyclic fatigue [6, 7], creel>fatigue interaction [6-8], crack growth [6, 9-13] and components lifetime prediction [4, 12, 14]. Two main problems arise in the description of the structural reaction to the applied loads in terms of Continuum Damage Mechanics. The first one is what kind of coupling between classical governing equations of Solid Deformable Mechanics and damage evolution equation should be used to take into account the effect of damage kinetics on the stress-strain behaviour. This problem is usually solved using the "effective stress concept", proposed by Kachanov [-1] and Rabotnov [2]. The generalization of this hypothesis for three-dimension (3D) conditions has been made by Hayhurst [14] and Murakami and Ohno [15]. The second problem has been associated with the identification of a damage parameter and the construction of adequate damage constitutive equations. This problem is usually solved from the phenomenological point of view and the damage parameter is considered as an internal variable of the damage process. The corresponding evolution equations are formulated in a differential form so that their structure enables the boundary conditions to be readily satisfied. The damage kinetics between limiting values remain unknown and are not taken into account in the equations. Most 1139
1140
V.P. Golub
existing damage evolution equations therefore fit the linear damage summation law which has not been confirmed experimentally [16]. The objectives of this paper are to introduce briefly the non-linear uniaxial damage theory, to review its specific advantages campared to the existing models and to illustrate it by results obtained with some structural materials under long-term static and cyclic loading. 2. N O N - L 1 N E A R I T Y C O N D I T I O N S
Before any non-linear damage cumulative theory is introduced it is necessary to define what is meant by "linearity" and "non-linearity" in Continuum Damage Mechanics and what kind of criteria may be used for their identification.
2.1. Damage linearity notion Under the classical Miner Robinson definition the linearity of the damage process is usually associated with the linear law of the damage accumulation over time (Figs l(a) and l(b), straight lines). The physical meaning of this definition is that equal time intervals At lead to equal damage increments Ao) and the total lifetime under stepped loading will be always equal to 1. Analytically this condition is given by the relation -
~
-1
tR(a)
~
-1,
(1)
tR(0")
i: 1 tR(ai)
which is well known as the linear damage summation law and the graphical interpretation is shown by the straight line in Fig. l(c) which is independent of the loading history. Here a is the applied stress and tR(') is the time to rupture under constant load. According to Kachanov's approach [-1] the damage kinetics are described by the evolution equation d(,) _ C
(2)
dt
'
from which the following expression for the damage parameter ~o is obtained ~,)= 1 -
1-
t
T+,,
(3)
tR(O-
where C and m are material coefficients to be determined. In this case, the damage kinetics is assigned in a non-linear form (Figs l(a) and (b), concave) and equal time intervals At lead to different damage increments Aco. However, differentiating Eqn (3) with respect to time and summing up over all values of damage increments, the damage cummulative law is written as follows d~,)
(1 + m) 1
tk(
tR(~)~
tR(~)
1,
(4)
which can be seen to coincide with the linear damage summation law (1). The characteristic feature is that, Kachanov's equation, Eqn (2) describes damage kinetics in a non-linear form over time and leads to the same results under stepped loading as a linear equation. All other known damage evolution equations, in particular Rabotnov's equation [2] d°) dt-C
(or) ~
m
1 "~o ~
(5)
and the Lemaitre Plamtree equation [6, 8] d(o _ C dt
"
(6) '
gives estimations similar to those mentioned above. The corresponding damage kinetic diagrams will differ from those assigned by Eqn (1) only by their curvature and these equations will fit the linear damage summation law adequately, too [17].
Non-linear one-dimensional c o n t i n u u m damage theory
1141
(a) 1.0
0.3
i
~o 0.5
0.0
(b)
(c) 1.fl
1.0
~]~
0.5
0.0
ti(*l)
tr(al) t
tR(O'l)
o.~
0.5
0.0
0.5 tit R
1.0
ti/tRi
Fig. 1. Schematic illustration of damage kinetics in (a) physical, (b) normalized time-scale, and (c) of damage s u m m a t i o n law.
2.2. Definition of damage non-linearit,y As can be shown in the previous section, non-linear damage kinetics-time relation is a necessary, but not a sufficient condition of damage non-linearity on the whole. So the damage non-linearity is defined by the conditions in which damage kinetics are given by a non-linear function over time, and the normalized total lifetime under non-stationary loading will be different from I (Fig. l(c), dashed lines). In analytical representation the damage non-linearity condition is written as follows d~o =f(o-, t) dt (7)
ti (O-i, t)d, ~] ~'of i= 1
tK
#1 " ,Ll ti = ~"
where i indicates the number of the different stress levels. 2.3. Principle of separability The analysis of the reasons for the absence of the difference between both approaches mentioned in Section 2.1 in the problem of lifetime calculation under stepped loading shows that, they both give the unified damage kinetics curve in the normalized coordinates system which does not depend on stresses (see Fig. l(b)). Hence, to satisfy the damage non-linearity condition (7), it is necessary to assign the damage kinetics in the normalized time scale as that depending on stresses. This condition, which may be interpreted as a sufficient condition of damage non-linearity, will be termed the "principle of separability". In analytical representation the separability principle may be given by relation d~.9 = fT [0"{ T), m, C t, C> ..., Ck } d T or in form
d
--
do" T = const
= var,
(8)
(9)
1142
V.P. Golub
where f x ( ' ) is the damage function, T is the normalized time, and Ci(i = 1, 2 , . . . , k) are material coefficients to be determined. Hence, all damage evolution equations, which do not satisfy condition (8) cannot give the damage as the non-linear process. In particular, replacing the independent variable t by T for initial linear damage equation from (1), for Kachanov's Eqn (2), for Rabotnov's Eqn (5), and for the Lemaitre-Plumtree Eqn (6) one obtains the following expressions correspondingly dco - 1 dT dco dT
(10)
1 (1 + m )
1 (1-co)"
(11)
dco_ _J ['(I--CO)m - dco
(12)
dco 1 dT=(1 +m+q)
(13)
dT
(1 - CO)mJO
COp
1 (1--co)re+q"
AS can be seen, all these equations do not contain stresses in their right sides and therefore, as it was shown above, fit the linear damage summation law adequately. 3. CONSTITUTIVE EQUATIONS OF THE THEORY There are many possible ways to construct the damage evolution equations which satisfy the non-linearity condition, Eqn (7). Consider long-term static and cyclic loading regimes using the separability principle. All results are presented and discussed in the normalized time scale. 3.1. Assumed relations First of all it is necessary to select and to justify the structure of the initial damage evolution equation. The detailed analysis of existing evolution equations and the non-linear forms of the damage models is presented in Ref. [17]. Here it is emphasised that the initial equation must contain two terms with damage variable of different power index (one of them must depend on stresses). In the general case, the initial damage differential equation is written in the form dco =ft' [a(t), co(t), C~, C2)f," [co(t), C3(a)] dt,
(14)
where ft'(.) and ft"(') are damage functions and C3(') is the stress dependent power index• The structure of evolution Eqns (5) and (6) satisfy initial Eqn (14). In order to specify Eqn (14) let us assume that damage accumulates not only over time but also under each moment of change in stresses including initial loading. In such a case, let us introduce two damage parameters and assign them by their integral measures. The first one reflects the value of the cumulative damage under each moment of change in stresses and has been defined as the "momentary" (time-independent) damage f2,. The second or traditional parameter reflects the value of the cumulative damage over time under constant stress and has been defined as the "temporal" (time-dependent) damage ~t. Each of these components will be identified with the corresponding specific fracture energy. We will also assume that energy spent on fracture due to the total damage accumulation is a constant value for the given material and temperature and is equal to the fracture energy of the single momentary deformation. In this case, the condition of balance between integral measures of momentary and temporal damage at the moment of failure is written as follows
n.+n,=
w.
f2~
w-S+W
f~t
:I,
(153
where W~ is the specific energy of single momentary fracture. The value of integral momentary damage f ~ is determined by the stress-strain diagram and corresponds to the work done by the applied stress a acting through the ultimate strain eu [Fig. 2(a), shaded area]. The integral temporal damage Ot may be associated with the energy dissipated in the process of the new internal surfaces formation. Its value may be determined by the normalized
Non-linear one-dimensional continuum damage theory
(a)
1143
(b) 1.0
0.5
tO m
0
r~
0.0
EB
0.5
1.0
tit R Fig. 2. Integral m o m e n t a r y (a) and temporal (b) damage measures.
damage kinetic diagram [Fig. 2(b)] and directly proportional to the area (shaded plot) bounded by the damage curve, the rule change of which is unknown. The value of total fracture energy wR is determined by the stress strain diagram too, and corresponds to the work done by stress limit OrB acting through the ultimate strain ~,o, so that WR =
~Oo(~:1d~,
(16)
where q~o(e) is an equation of the stress-strain diagram. Analytical expressions for momentary fl~ and temporal f~ damage will be dependent on loading conditions. 3.2. Damage kinetics under static loadin9 In the case of static loading following the previous definitions for the value of integral momentary damage f ~ [see Fig. 2(a)] the following relation is obtained fQ =
{Po(O&:+ (~:~- c,,~){ri.
(17)
)
Here a~, is the strain on the stress strain diagram corresponding to the applied stress ai {i = 1,2 . . . . .
n).
To specify the second summand in Eqn (15) it will be assumed that, the dimensionless function ~,/WR is some invariant relative to the time to rupture tR quantity depending on the damage parameter e). The average value 0,,, of the function (o(t) on the interval 0 ~< t ~< tR may be chosen so that
R-- Om
--
(18)
--O'"CVRd
where COc(') is the value of the damage parameter under static loading. The balance condition (15) between the momentary and temporal damage components at the moment of failure, taking relations (16)-(18) into account, may be written in the form J'o ~o(r,)d~ - J','i][{po(r,) - {hi d~:
+
(oc
d
= 1,
(19)
S~° ~0o(~)d~:
or, after simple transformations, in the form ~o¢
d
"
qOo(Odc -
]-(po(C)- csi] d~: = O.
which may be considered as a long-term fracture criterion under static loading.
(20)
1144
V . P . Golub
Then, using the differential equation
d(~)c
[
o_i ] m r ~ l ~ , ~ ,
(=,,
=CL _I Ll- o/
as the initial damage evolution equation and replacing t by T = t/tR, on integration of Eqn (21) for the initial conditions coc = 0 at T = 0, one obtains the following expression g
c9c
=1-
1-
t
1 + m + qW)
,
(22)
1 C , a~'"
(23)
where tR is the time to rupture given by tR
1 (1 + m + q(cr)) Ca'/'
=
On substitution of Eqn (22) into criterion (20) and variation of an expression for the q(a) index is obtained that together with the evolution Eqn (21) gives the constitutive equations of the nonlinear damage theory. In the normalized time scale this set of equations is written as follows
1
.[
1 ]
d-T = [1 + m + q(a) q(a)
(24)
J'~ (Po(~:)dc
- (2 + m),
= -
The graphical interpretation of the damage kinetics calculated from Eqn (24) is shown in Fig. 3(a). The fundamental difference between the results obtained and those previously developed [see Fig. l(b)] is that the damage kinetics in the normalized time-scale depends on stress. The corresponding (a)
(b)
2
1.(l
0.5
0.0
0.5
1.0
0.0
Tt
0.5
T2 1.0 t/t R
t/tR
(c)
(d)
I.O
2
I'0 /
o. 1 ~
" '
| ~" 0.5
0.0
0.5
t/t R
1.0
0.0
TI
0.5
T 2 1.0
t/tR
Fig. 3. Damage kinetic diagrams under static [(a) and (b)] and cyclic [(c) and (d)] loading calculated from non-linear damage theory.
Non-linear one-dimensional continuum damage theory
1145
damage diagrams displace down in the integral measure of the temporal damage with the stress rising, and up in the integral measure of the momentary damage. The effect of separation of normalized damage kinetic diagrams in the stress parameter allows one to take into account the loading history under non-stationary conditions. As an example, Fig. 3(b) shows the graphical solution of the classical problem of residual lifetime calculation under additional loading (a2 < as) and partioned unloading (as > ai ). As can be seen, the total lifetime will be less than I (curve I) and greater than ! (curve 2), respectively. Qualitatively these results are in good agreement with the majority of experimental data. 3.3. Damage kinetics under cyclic loading Under cyclic loading the discrete time assigned with the number of cycles of change in stresses n is used as the argument for the damage function. In order to pass to the natural time t it is sufficient to know the loading frequency v = dn/dt since n = yr. Under sufficiently large values of n it can be assumed that the smoothed processes of discrete loading satisfy differential Eqns (2), (5) and (7). In the case of cyclic loading, the expression for the integral momentary damage f2~ [see Fig. 2(a)] may be written in the form fl~ = ae,~ -
fo (a) da.
(25)
0
or, taking into account the loading rate, in the form df~
f~a -
-
e.u
da
-)Co(a,),
(26)
where fo(a) is an equation of the stress-strain diagram [fo(a) = q0o 1(~)]. The average value con, of the function ~o(n) on the interval 0 ~< n ~< nR by analogy with Eqn (18) may be written as follows
f~' = cgm~]-
WR
~°'o)f(n)dn -- f] (of ( ~ ) d (-~R) , nR
(27)
where o)f(. ) is the value of the damage parameter under cyclic loading and nR is the number of cycles to failure. The balance condition (15) between the momentary and temporal damage components at the moment of failure, taking Eqns (16), (26) and (27) into account, may be written in the form ~:.
- Yo(a~)
£u
+ f l o ) f \n~/ ( n ~ d ( n )~
= 1,
(28)
= O,
(29)
or, after simple transformations in the form (,..If
d
"
0
,%-- f(ai)
which may be considered as a long-term fracture criterion under cyclic loading. Here it is assumed that, WR = dWR/da = r,u. Then, using the equation do)f dn = D
V
ai
I k '
Ll_--2~)fj
V1 l/~'',
(30)
ko)~J
as the initial damage evolution equation and replacing n by N = n/nR, on integration of Eqn (30) for the initial conditions o)f = 0 at N = 0, one obtains the following expression n --
nR
(o)f)
=
F(2 + k - fi(a)) fi"' ([ 2 o)f_)k do)f, Y(1 + k)F(1 - fi(a)) (e)f)/~/~)
(31)
1146
V.P. Golub
where nt is the number of cycles to rupture and is given by
nR =
F(1 + k)F(1 - fi(a)) 1 1 F(2 + k - fi(cr)) Da~' - D,a~ "
(32)
Here F(-) is a gamma-function. On substituting Eqn (31) into criterion (29) and varying a an expression is obtained for the fi(a) index that together with evolution Eqn (30) gives the constitutive equations of nonlinear damage theory. In the normalized time scale this set of equations is written as follows d(~__j f =-((of)I~(~l . ~.l (1 - (of)k dr:of dN (1 - (of)k )o ((of)/~l~' (33) fl(a) = ~:u - f 0 ( a l ) ( 2
+ k)
~:u - f o ( " , ) The graphical interpretation of the damage kinetics calculated from Eqn (33) is shown in Fig. 3(c). As can be seen, the fatigue damage kinetics depends on stresses in the normalized time scale, too. However, unlike the static loading, the cyclic damage diagrams displace up in the integral measure of the temporal damage with the stress rising, and down in the integral measure of the momentary damage. In such a case, the total lifetime will be greater than I [Fig. 3(d), curve I] under additional loading (~rl < ors) and less than I [Fig. 3(d), curve 2] under partional unloading (~4 > al ). Qualitatively these results also coinside with experimental data. 4. T O T A L A N D R E S I D U A L
LIFETIME
PREDICTION
Now consider the capabilities of the damage theory suggested as an example of residual and total lifetime calculation of prismatic rods subjected to non-stationary static and cyclic loading at high temperature. In this case, creep and fatigue processes occur. The rods are made of refractory and heat-resistant alloys. 4.1. Determination of the theory coefficients In order to solve the problems on the basis of the damage theory developed it is necessary to know two groups of coefficients. The first one includes the coefficients m, C and k, D which specifies a material resistance to the damage accumulation over time. The second group includes the function q)o(~), the a u - ~-u characteristics which specifies a material short-term strength. So the base experiment involves one long-term strength curve or one fatigue curve with symmetrical cycle and stress-strain diagram. The value of stress independent m and k exponents is first determined using Eqns (23) and (32), respectively. These equations are fitted to the uniaxial long-term strength and fatigue data using a least-squares technique, and errors are minimized with respect to the corresponding coefficients. The function ~o0(e) and all corresponding characteristics (~, cru, ~:u) are determined from the tension tests data, presented in the form of the stress strain diagrams. Then, the stresses dependent q(a) and fl(a) exponents are calculated from the second relations of Eqns (24) and (33) using previous determined m and k exponents and q~o(e,) function. Finally, the C and D coefficients are determined from Eqns (23) and (32) using the values of m, k, q(a) and [3(~) exponents. As an example, Table 1 shows values of the coefficients m, C,, k, D, and values of the characteristics au, ~, and Fig. 4 shows the relationship of q and fi exponents with the stresses for some materials calculated from the above presented procedure. 4.2. Lifetime under stepped loading The residual and total lifetime calculation under stepwise loading is carried out using a step-bystep integration of Eqns (24) and (33) with varying limits of integration at each stage. The sequence of integration is given by the familiar loading programme. The value of load within the limits of each
Non-linear one-dimensional continuum damage theory
1147
Table 1. Values of calculated coefficients and parameters Alloy
EI437B 0 (°C)
C,(MPa-~-hr m
-1)
D,(MPa-k'(hr'f)
750
800
1 . 8 6 × 1 0 ~l* 4.6
3 . 5 5 x 1 0 -12 4.3
-1 )
--
k a, (MPa) ~
5.9 680 0.07
5.4 520 0.08
E 1698VD 750
900
950
VJL12U 1000
1 . 5 5 x 1 0 -z5 8.2 2.51 × 10 T M 9.6 775 0.10
1.38x10-: 6.3 4.57 × 10 - . 1 13.6 580 0.06
2.00xlO -t3 5.2 3.02 × 10 -3'~ 11.1 370 0.06
5.37x10 -17 6.5 2.69 × 10 -28 9.6 475 0.04
(a)
(b)
4
!
I
2
I
(3
0.2
I
,
0.4
0.6
'~0.2
0.4
N
/
N
/
I
\
e
\
\
"\
\
_
j" f
/
0.8
x
\
f
I
0.6 or ,,, orB
X
/
/
4
"~,
/
°B
2
I
I 110'.8!
"
6
E 1867
\
9
,f 12-
8
IO
t
15
i
Fig. 4. Relationship of power indexes q (a) and fl (b) on stresses for heat-resistant EI437B (dashed lines) and EI698VD (dot dash lines) alloys at 750°C. stage is assumed to be constant. The fracture condition is given by the relation
•
~oi(ai, t) ,= ~, = 1,
(34)
i=a
where i is the n u m b e r of loading stages. So in the case of static loading integrating Eqn (24) with the initial condition ~oc = 0 at t = 0, the value of d a m a g e coi f o r / - s t a g e of loading is determined from the expression o9i=1-I(1-o9i
a) a + ,, +q(,) _
t
] l + m 1+ q(~,t
(35)
/R(O'i)J
and the residual lifetime t* for this stage from the expression t* = tR(~ri) [1 -- COi_ t] 1 + " + qI~j~,
(36)
where it is assumed o3c = ca and ~o(crj_ 1) = ~o(ai) at t = ti. The sequential solution of Eqns (35) and (36) with allowance for E q n (34) determines the j-stage of loading on which fracture takes place, so that the total lifetime tRZ under an arbitrary n u m b e r of loading stages is determined by the relation tax=t1
Here ta,
t2, ...,
+ t2 +
... + t R ( 6 j ) [ l
are loading durations on each stage.
-- ~ O ) - l ] l+m+q(~j)
(37)
1148
V.P. Golub
In the specific case of two stepwise loading (i = 2), for the total time to rupture t~z we will obtain the following from Eqn (37) tRX = t I + t 2
l
tR(0"1)J
(38)
or after some simple transformations for the creep residual lifetime t* one obtains
V t* l :'(~)
tt
t.(0-,~+ L ~ J
=
i,
(39)
where 7(0") is a loading history parameter. For additional loading (0"t < 0"2) and partial unloading (61 > 0-2) one has, respectively,
1 + m + q(0-1)
7(0-) =
1 + m + q(0"2)
< 1 and 7(0")=
1 + m + q((rl) 1 + m + q(0"2)
(40)
> 1,
because q(O'l) < q(O'2) at at < 0-2 and q(0-1) > q(a2) at 0-1 > 0"2" As an example, Fig. 5(a) compares experimental data (points) with calculated lifetime curves (broken lines) for E1437B alloy under additional (0-t = 300 MPa, 0"z = 400 MPa) loading (o) and partial (ai = 400 MPa, 0"2 = 300 MPa) unloading (©) at 75OC. In the case of cyclic loading, integrating Eqn (33) with the initial condition coj = 0 at n = 0 and using the procedure mentioned above for the total number of cycles to failure nRz, under two stepwise
(b)
(a) 1"° ~..x~ ~ .
o"\
[ '
0.5 -
I.O ,,.,..
\.
\\
"'.
\
• 0.0
\
\,,o
~ \\ 0.5
•
~.
\
\\N
0.5
N\
.\
\
~
*\
\\
°\\
\~ 1.0
0.0
/
I
"-..o~ _
0.5
1.0
n/np.(eq)
t/tR(~O
(c)
(d)
1.0[~- -~..~
1.0
~
I o.,_
0.5
[A,A ~ / V M''-"
~''~.
\
°), I
0.0
0.5 t/tR(tr)
1.0
0.0
0.5
! .0
n/nR(o')
Fig. 5. Comparison of experimental data (points) with calculated life-time curves (broken lines) under stepped static (a) and cyclic (b) loading and under creel~fatigue (c) and fatigue~reep (d) alternation.
Non-linear one-dimensionalcontinuum damage theory
1149
loadings we will obtain the following nRZ = nt + n2
1
nR(at}
(41)
or after some simple transformations for the residual fatigue lifetime n~ one obtains nR(a 1) + [ _ ~ j
= 1,
(42)
where r/(a) is a loading history parameter, too. For additional loading and partial unloading we have, respectively, tl(a)-
1 +k+fl(a,) 1 +k+fi(ax) > 1 and ~/(a)< 1, 1 + k + fl(a2) 1 + k + fl(O'2)
(43)
because fl(al) > fi(a2) and ol < o2 and fl(a0 < fi(a2) at al > a2. The resulting comparison between experimental data (points) and predicted lifetime curves (broken lines) for E1437B alloy under additional (a~ = 200 MPa, a2 = 300 MPa) loading (e) and partial (at = 300 MPa, a2 = 200 MPa) unloading ((2)) at 8 0 0 C is shown in Fig. 5(b). 4.3. Lifetime under creep-fatigue alternation Now consider the conditions when static and cyclic loads act consistently. In this case, the damage kinetics has been determined by the simultaneous development of creep and fatigue processes. Using the superposition principle, the initial differential damage equation may be written as follows dcoz d l d d~- - d t (am, O)E) -1- --V ~ (O'a' ")~)'
(44)
where cox = cot + o)f is the value of the total cumulative damage. In Eqn (44) it is also assumed that, the creep damage kinetics has been given by the static component am of the load and the fatigue damage kinetics by its cyclic component era. We will consider two types of creep and fatigue processes alternation. In the first case, a portion of life tc is spent in creep and the remaining life t* being calculated under fatigue. Following integration of Eqn (44) with the initial condition ~o~ = 0 at t = 0 one obtains the following expression for the total damage co* cumulated to the moment of failure co~ = 1 -
1
6
| i = ~
t~'
tR~m)~
;l+,n+~l,°.,
/R(Ga))
,
(45)
from which assuming that, the fracture occurs at the damage value o)~ = I for the total lifetime tRXwe have the relation
7 1-k+fiqao) tRX = tc + t* = tc + /Rf
1
to [ c ; ~
(46)
tRcJ
where tRC = tR(O'm) and tRf = tR(G) are the time to rupture under pure creep and pure fatigue, respectively. In the second case, a portion of life tf is spent in fatigue and the remaining life t* being calculated under creep. So by analogy with the previous case one obtains the following for the total damage co~' cumulated to the moment of failure co~ = 1 --
1
tf
1+ .... ql~) _ _ _
tR~a)
t + k-/~/~,
(47)
tR(O'm)
and for the total lifetime tRX one obtains the relation
I tRX =
~f -]- t~ = tf -v tRC
where tRC = tR(am) and tRf = fR(O'a) as above. MS 38-10-H
~1 .... + ql~mi 1 -- tt- ~-; ~ , tRfJ
(48)
1150
V.P. Golub
The resulting c o m p a r i s o n between experimental data (points) a n d predicted lifetime curves (broken lines) for E1867 alloy u n d e r creel~fatigue a n d fatigue-creep a l t e r n a t i o n at 900°C a n d am = 300 M P a , aa = 200 M P a is s h o w n in Figs. 5(c) a n d 5(d), respectively. 5. CONCLUSIONS The n o n - l i n e a r d a m a g e c u m u l a t i v e theory has been developed for the uniaxial static a n d cyclic loading. The d a m a g e n o n - l i n e a r i t y c o n d i t i o n s are e n u n c i a t e d o n the basis of the "separability principle" giving the d a m a g e kinetics d e p e n d i n g on the stresses in the n o r m a l i z e d time scale. The constitutive e q u a t i o n s of the theory have been c o n s t r u c t e d a n d a technique is p r o p o s e d for the d e t e r m i n a t i o n of its coefficients for two structures of initial d a m a g e e v o l u t i o n a r y equations. W i t h i n the framework of the theory suggested, the p r o b l e m s of the total a n d residual lifetime calculation u n d e r stepped l o a d i n g a n d u n d e r creep fatigue a l t e r n a t i o n c o n d i t i o n s are solved. T h e o r y predictions are in good agreement with experimental results o b t a i n e d for some heat-resistant alloys. It must be emphasized that, all predictions have been m a d e using material coefficients which have been d e t e r m i n e d only from pure creep or (and) pure fatigue tests a n d stress-strain diagrams. REFERENCES 1. 2. 3. 4. 5. 6. 7.
L. M. Kachanov, On time to rupture under creep conditions (in Russian), lzv. AN SSSR, OTH, 8, 26 (1958). Yu. N. Rabotnov, Creep problems in structural members (Russian translation). North-Holland, Amsterdam (1969). J. Janson and J. Hult, Fracture mechanics and damage mechanics--a combined approach. J. Mech. Appl. 1, 69 (1977). F. A. Leckie and D. R. Hayhurst, Creep rupture of structures, Proc. R. Soc., Lond. A240, 323 (1974). J. Hult, Creep in Continua and Structures, Topics in Applied Continuum Mechanics. Springer, Vienna (1974). J. Lemaitre, How to use damage mechanics, Nucl. Engng Design 80, 233 (1984). J.-L. Chaboche, Continuous damage mechanics- a tool to describe phenomena before crack initiation. Nucl. Engng Design 64, 233 (1981). 8. J. Lemaitre and J.-k Chaboche, A non-linear model of creep-fatigue damage cumulation and interaction. In Mechanics of Visco Plastic Media and Bodies, p. 291. Springer, Berlin (1975). 9. R. Billardon and J. Lemaitre, Numerical prediction of crack growth under mixed-mode loading by continuum damage theory. Proc. of AUM Congress, Lyon (1983). 10. K. C. To, A phenomenological theory of subcritical creep crack growth under constant loading in an inert enviroment. Int. J. Fract. 11(4), (19751. 11. D. R. Hayhurst, P. R. Brown and C. J. Marrison, The role of continuum damage in creep crack growth. Phil. Trans. R. Soc. Lond. A311, 131 (1984). 12. D. R. Hayhurst, P. R. Dimmer and C. J. Morrison, Development of continuum damage in the creep rupture of notched bars. Phil. Trans. R. Soc. Lond. A311, 103 (1984). 13. S. Murakami, M. Kawai and H. Rong, Finite element analysis of creep crack growth by a local approach. Int. J. Mech. Sci. 30(7), 491 (1988). 14. D. R. Hayhurst, Creep rupture under multiaxial states of stress. J. Mech. Phys. Solids 20, 381 (1972). 15. S. Murakami and N. Ohno, A continuum theory of creep and creep damage. Proc. of3rd I U T A M Syrup. Creep in Struetures, p. 422. Springer, Berlin (1981). 16. V. P. Golub, A. D. Pogrebnjak and A. V. Romanov, The applicability of the linear damage summation hypothesis in creep and fatigue problems (in Russian). Prohl. Proch. 10, 20 (1993). 17. V. P. Golub and A. V. Romanov, Kinetics of damage in isotropic materials under creep conditions. Soy. Appl. Mech. 25(12), 1264 (1989).