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A stochastic model of fatigue crack growth in heterogeneous material
Engineering Fracture Mechanics Printed in Great Britain
m3-7944/87 163.00+ 0.w Q 1987 Pergamon Journals Ltd
Vol. 26, No. 5, pp 707-728
A STOCHASTIC MODEL OF FATIGUE CRACK GROWTH IN HETEROGENEOUS MATERIAL HYNEK LAUSCHMANN Faculty of Nuclear Science and Physical Engineering, Prague, Kfemencova 10, 110 00 Praha 1, Czechoslovakia Abstract-Two ways of the phenomenoiogical modeliing of fatigue crack growth have been developed in the past - deterministic crack growth laws and stochastic crack growth models. At present time the syntheti~ng models are constitute. The object of this paper is a model describing the variability of the crack growth process under constant-cycle loading as the consequence of heterogeneous material structure. A general stochastic model with discrete time and continuous state serves as the theoretical basis. Three types of process-variability are defined and theoretically solved. Concrete form of the model in the case of power-form growth laws is presented with regard to the evaluation of experimental data and to the crack growth predictions. The model is applied to evaluate the results of crack growth experiment on the specimen from aluminium alloy.
NOMENCLATURE a
process state - crack length [mm] initial crack length [mm] process time - number of loading cycles from the origin of the process subscript denoting the i-th stage of the j-th growth curve k subscript denoting the k-th stage of the whole data set a “,,,us,,,aV,n,, (or with the subscript k) initial, middle and final crack length and time interval of the process stage determined by the subscript v crack growth rate [urn/cycle] Aa crack increment in one cycle [urn] A multiplicative factor of the growth law Au = Ag(a) conducting function of the growth law Aa = A&) g(a) mean value and variance of the A-factor E, D see A, the quantity is related to single structure area A, mean value and variance of the A,,-factor Eo, Do distribution function of the state a at time n F,,(a) probability that the state a is surpassed for the first time at a time shorter or equal to n P,,(n) 100 p% quantile of state u at time n o,“(a) 100 p% quantile of time n at which the state o is surpassed for the first time n,(a) I‘ du 00 n ij
v
=i-- ,&) model of the distribution of A-factor model of the distribution of A,-factor model .$4 with mean value x and variance y characteristic material structure dimension in the crack front direction and crack growth direction, respectively
Cmml crack front length [mm] mean crack increment in one cycle within given growth stage [pm] mean length of growth stages determined by experimental readings autocorrelation function of the quantity X in point x base function of the power-form growth Iaw Au = AM” exponent of the power-form growth law Au = AM” multiplicative factor of the dependence D = X M @ exponent of the dependence D = 2f Mm@ parameters of the deterministic power-form growth law standardized velocity V = v/(Ad Ma”)
M,,, ,1 w
the value of M on the boundary between the regions of intermediate and fine variability standardized form of q/n stationary function of n/n 707
708
HYNEK LAUSCHMANN 1. INTRODUCTION
THE PHENOMENOLOGICAL modelling of fatigue crack growth has been oriented partly towards physico-technical deterministic growth laws and partly towards stochastic models, based on mathematical abstraction. The growth laws express the dependence of the crack growth rate da/dn on stress characteristics and on the crack length, or on other factors (such as loading frequency, temperature, cycle shape, environment). Since the process history has not been respected, the validity of growth laws is practically limited to cases of constant-cycle loading regimes. The growth laws are understood as equations for the deterministic phenomenon component, the growth-process variability being put into connection with non-specified random fluctuations. The usual procedure used for evaluation of experiments includes the application of growth laws by means of regression methods. Three applications of the mean-value operator on the crack rate are involved implicitly in standard conception of the growth law: averaging along the crack front, averaging in the direction of crack propagation close to the given crack length and averaging over individual realizations of the process. The information included in the growth law is therefore rather compressed; the law is usually understood as a characteristic of the process of continuous state and time and the transition to the growth curve a(n) proceeds by integration without analytically respecting the process variability. The growth laws offer an advantage of connection with the physical description of the phenomenon, enabling the transfer of information in case of changing physical conditions. The growth laws suffer from the disadvantage of an insufficient inclusion of the process scattering properties. Stochastic crack growth models outlined especially in the works of Bogdanoff, Kozin et al.[393, Ghonem et al.[lO, 1l] and SedliEek[13-151 are based on the application of the stochastic processes theory; the model base is a general non-decreasing random process, the parameters or functions having the role of intensities are specified experimentally. The confrontation with the growth law is usually carried out by an interpretation of the growth law integral as the mean value or the median of the process or as the equation of the individual realization. The stochastic models are capable to describe precisely the random growth-process in concrete cases, without giving the possibility of predicting the growth process after the physical conditions changed. The stochastic models can be used with advantage for the description of a fatigue process in operation when the purpose is not a physical analysis of the phenomenon, but a precise compressed description of a concrete random crack growth process. Since 1981 a number of models has been published, connecting the investigation of growth process variability with the growth law ([l, 2, 12, 16-253). The scatter of the process is mostly described by a generalization of the growth law into a probabilistic form (randomization), including in several cases also the variability of loading process. It seems to be advisable to dispose of a stochastic crack growth model which would reflect the main source of process randomness in cases of constant-cycle loading, i.e. the scatter of local mechanical properties of the material due to the heterogeneous structure. Unfortunately, no practically applicable results have been attained in this way so far. The principal purpose of the present paper is a synthesis of the growth-law philosophy with a stochastic model enabling - in necessary simplification - the structural heterogeneity influence to be described. In our conception the growth law is interpreted as an equation for the crack increment in one cycle, i.e. that out of the three applications of the mean value operator in the usual conception only the averaging along the crack front has been considered. This interpretation generated a stochastic model, investigating the crack growth as the summation of random increments. 2. THEORETICAL
MODEL
Let us assume a stochastic growth process a(n) with discrete time n, n = 0,1,2,... and with continuous state a, a > 0. The course of process a(n) is defined by the increment Aa during the time interval An = 1; the increment Au depends on state a, being independent on time n and defined by the relation Au = g(u) A,
g(a) > 0, ‘4 > 0 3
(2.1)
A stochastic
model of fatiguecrack growth
709
where g(a) is a known function, A is a random factor in general, whose properties define concrete process types. The initial condition is specified in a deterministic way: at the time n = 0 the process state attains the value ao. Let us define further the inverse process n(a) as a stochastic dependence of time n, at which state a has been surpassed for the first time, on state a. The time n(a) can be understood with an error E,, < 1 as the time necessary for attaining state a. The basic characteristics of processes u(n) and n(u) are denoted as follows: F,(a) ... distribution function of state a at time n; P,(n) ... probability that the state a is for the first time surpassed at a time shorter or equal to n; u,(n) ... lOOp% quantile of the distribution of state a at time n; n,(u) ... loop% quantile of time n, at which state a is surpassed for the first time. The interconnection
of the two processes u(n) and n(u) is expressed by the relation
(2.2)
F&4 = 1 - P,(n) , from which the possibility of quantile inversion follows n = n,(u) 0
a = u&l)
.
(2.3)
The basic approach when solving this process consists in the following steps: By rearranging equation (2.1) we get
AU A
-= g(u)
.
(2.4)
Let us assume quantity 7 defined as
(2.5) where subscript k denotes individual consecutive process increments. When the increments Au are small and time n large, the sum in eq. (2.5) can be replaced by an integral in accordance with Riemann’s definition (see Fig. 1)
(2.6)
rl=
or in the case of short intervals (a,, a) in accordance with the mean value theorem by approximation a tl
--a0
a, = (a0 + a)/2 .
%T’
(2.7)
According to (2.4) it also holds that
9 =
,$,Ak’
(2.8)
i.e. the statistical properties of the quantity 17may be considered as properties of the sum of n factor A realizations. Quantity q is generally an increasing function of the process state a (see eqs (2.6) and (2.7)).
710
HYNEK LAUSCHMANN
ak-l Fig. 1. Replacing the
a
sum by an integral
It follows that the distribution function G(q) related to the growth interval (a,, a) and to time n equals
G(v)
=
F,(a)
(llo.aM
.
(2.9)
a0
Thus the statistical properties of process a(n) can be studied by using process q(n). Let us outline an approximation useful in solving some more difficult cases. It is to assess the course of the process 17(n) if it is known: (1) the distribution of A-factor does not change during the process; (2) the states of the process a(n) at times n = 0 and n = n, were a0 and a,, respectively. According to eqs (2.6) and (2.8) it can be written (2.10)
The sequence of unknown random values Ak will be.replaced by the sequence of identical values 1 = 77,/n,. The state of process q at an arbitrary time n < n, is then determined as ?/7(n)= n
3. ANALYSIS
2.
OF THE CRACK GROWTH PROCESS VARIABILITY CONSTANT-CYCLE LOADING
(2.11)
IN CASE OF
The majority of the known growth laws may be written in the form of eq. (2.1). Thus the function g(a) represents a non-random physical component of the phenomenon, whereas the factor A stands for random fluctuations. We shall limit the following analysis to cases where the variability of the crack growth has been caused only by an interaction of the crack front with the heterogeneous structure of material. The development of the plastic zone and internal stress area involves a relatively large volume of the material; therefore, it is possible to interpret the function g, including the effect of these phenomena, to be deterministic. Conversely, the individual crack increment proceeds in the elemental microvolume and may be principally affected by the local structure character. Therefore, the variability of the process is seen as related to the scattering of the individual crack increments,
711
A stochastic model of fatigue crack growth
crack front direction Fig. 2. The model structure of material in ideal crack surface. Every structural area is properties - homogeneous, characterized by a single random value A,.
as to fatigue
represented in our model by a statistical factor A. Let us characterize the distribution of the Afactor by the mean value E, variance D and model-type 9; it will be written A : L@ (6 D). To the purpose of this paper let us define the characteristic structure dimension, d, as the mean value of the distance, where the local mechanical material properties, as considered in view of their effect on the fatigue crack growth, do not change or where the change is negligible. Moreover, it will be of advantage to introduce the characteristic dimension d, in the direction of the crack front and d2 in the direction of crack growth, which is of use for the application in case of rolled materials. For general consideration the model structure will be applied instead of the real one, the former having a random distribution of structural areas of non-correlated fatigue properties, but of similar dimensions, equal to constants d, and d2 (see Fig. 2). Let us assume the growth law for the particular structural area in the form A@= g(4 Ao .
(3.1)
The growth law (3.1) determines the magnitude of the crack increment in the particular structure area -- unaffected by the adjacent structural areas - under given stress conditions. The &-factor distribution may be considered according to the growth law philosophy as a material characteristic. Let it be specified by the mean value Eo, variance f)o and model-type %sP We assume the mean value E. and the type of model bPo to be material characteristics independent of the process state. As for the variance Do we assume that it is generally dependent on the value of function g, being therefore a function of the process state a. The inconstancy of the variance characteristic expresses the fact that the effect of structural inhomogenities on crack growth rate may be sensitive to the stress level on the crack tip. 3.1 An~Iysis of rel~~~o~salong the crack front at a gisen crack length In the above interpretation the individual sample value of the A-factor represents the resultant of material properties along the actual crack front. The crack front increases or decreases by its waviness the local state of stress, responding in this way to the local higher or lower resistance of the material. The model of physical interactions along the crack front does not exist; let us, therefore, develop the most simple idea: the resulting crack increment in one cycle is the arithmetic mean of local increments by which the crack would propagate in the individual structural areas if it did not keep a continuous front. When denoting the length of crack front c, the number of the involved structural areas will be c/d,. Distribution .w(E, 0) of the A-factor is characterized by the moments E = E,,,D =
Do. dJc DO
for c 3 dj for c < d,
,
(3.2)
and by the type of model
(3.3)
712
HYNEK LAUSCHMANN
The mean value E of the A-factor is independent of the crack front length; also the type of model can be considered in the case of small differences of the crack front length as approximately constant. If variance D, and D, of the A-factor appertain to the crack front length c, and c2, respectively, then D,
CI
=
D2
~2,
~1,
c2
a
4
(3.4)
.
In case of non-linear interaction models the mean value E along with variance D are nonlinear functions of E,, D, and c. We expect that in connection with the research of interactions along the crack front the above ideas will be substantially corrected in favour of non-linear interaction models. So for instance the model representing factor A as a geometric mean of values A0 emphasizes the braking effect of barriers against crack propagation. Model-type 9 is asymptotically logarithmic-normal, the mean value E being dependent on crack front length. 3.2 Analysis of conditions in the crack growth direction The autocorrelation function of process A(a), expressing the course of the A-factor in the direction of crack propagation, is a characteristic of principal significance. With regard to a possible dependence of variance D = D(a) it is useful to apply a standardized process y(a) = (A(a) - E)/ m whose autocorrelation function under the above assumptions is q,(x) = CM_&4,
Aa + 4) =
1 - x/d2
o
for x d d2, for x > d2 .
(3.5)
The practical significance of the relation (3.5) essentially exceeds assumptions, under which it has been derived. Irrespective of the type of interaction along the crack front, the sequence of the realizations of the A-factor is conditioned by changes of structural areas affected by the crack front. Assuming a model structure with a constant d2-dimension it holds generally:
(1) q,.(O)= 1;
(2) p,.(x) = 0 for x k d2 ; (3) p,.(x) is decreasing within XE (0, d2) .
The relation (3.5) is therefore a linear approximation of function q,(x) for a general type of interaction model. The actual shape of function q,(x) differs from the theoretical dependence (3.5) partly due to a more complex type of interactions along the crack front and partly due to the random size of the structural areas. Structural areas smaller than d2 will cause the decrease of q,(x) for low xvalues; structural areas larger than d2 will cause the increase of p!(x) for large x, including also .Y> d:. Let us denote the crack growth section from the a,-length up to a-length as the growth stage and the increment a-a,, as the growth stage length. The characteristic length of the growth stage ~7is the mean value of lengths of the growth stages determined by experimental readings (we assume the lengths of these stages to be approximately constant). The mean increment G of a given stage is defined by & = (a - a,)/An, where An is the time interval of the growth stage. The essence of the experimental data evaluation problem is double sampling of the random process A(a): process A(a) with correlation properties as determined by the characteristic structure dimension dz is projected into the fatigue process a(n) by sampling with mean length z Experimental information is a result of sampling of the process u(n) with mean length & The quality of the resulting statistical properties of the investigated data will, therefore, be affected by the interrelation of dimensions d,, G and d. Three types of variability of the process within a given growth stage are defined by means of the following relations: d2 < G dz ~(6, dz>Z
. . . tine variability; 5) . . intermediate
variability;
(3.6)
. . . rough variability.
Considering that the mean increment Au varies in the course of the crack propagation
process at
A
713
stochasticmodel of fatigue crack growth
(2) (1) Fig. 3. A diagram of three types of process variability.Hsuccessive crack fronts; D experimental reading; (l)...fine variability; (2)...intermediate variability; (3)...rough variability. (The of cycles between readings is symbolic.)
constant loading cycle within several orders, a change in the type of variability may occur from the intermediate to the fine one. A chart of types of variability is given in Fig. 3. 4. STOCHASTIC
MODELS OF FATIGUE CONSTANT-CYCLE
CRACK PROPAGATION LOADING
UNDER
4.1 Model of crack growth in case offine variability In case of fine variability each crack front appears in a quite new configuration of structural inhomogenities. Conditions of crack propagation and, therefore, also individual realizations of Afactor are random without any correlation. First, let us consider the case of D = const. Assuming a sufficiently long time n the distribution of
can be expressed by using the central limit theorem as asymptotically nE and variance nD. i.e. q : JV (nE, nD) .
normal with the mean value
(4.1)
In a special case, when the distribution of the A-factor is exponential, process n(a) is the known Poisson process. This process is homogeneous for g(a) = const. and inhomogeneous if g(a) # const. To the same case a more general variant may be transformed, where the distribution of the A-factor is of gamma-type with mean E and variance D, by means of time transformation n’ = n.I?/D.
In a more general case of dependent variance D = D(a) it is appropriate to solve separately the problems of short and long growth stages. In a short growth stage the change in variance D may be assumed to be small, so that variance D may be represented - in a single growth stage - by a constant value belonging, e.g. to the midpoint of the growth stage:
The evaluation of experimental data follows the theory for D = const. In case of a long growth stage a continuous change of variance D must be taken into account. Applying the central limit theorem we obtain for the distribution of rl at sufficiently long times n n-l
V :
b/t’(nE,k;, D(d) ,
(4.2)
714
HYNEK
LAUSCHMANN
where uk are the process states at successive times k = 0, 1,2, . . . , n - 1. A correct solution encounters considerable difficulties; therefore a simple approximate solution has been derived. The sum in the term of variance may be replaced by n-multiple of the mean value D(a) along interval (ao, a), which may be found by integrating function D(a) with respect to any quantity varying linear with time n. Accepting the approximation (2.1 I), the quantity q can be used. After substitution r,r = ~(a) it can be written (4.3) Relation (4.3) is suspicious owing to 17being used for expression of variance. However, this is only a formal problem, as using relations (2.2) and (2.3) a relation for quantile n,(u) can be expressed from (4.3) (4.4) where up is 100 p% quantile of the standard normal distribution. Relation (4.4) may also serve by using (2.3) and numerical methods - as a basis for computing quantile u,(n). 4.2 A model of the crack growth in case of intermediate variability First the distribution of 17 in case of a short growth stage will be derived assuming autocorrelation function p,(x) (3.5). Within the given stage the variance of A-factor is characterized by constant D and the magnitude of individual increments of the crack by constant s. The variance of sum
is generally equal to
(4.5)
.
Only values A, belonging to --crack fronts being at a maximum distance d2 from the k-th front in both directions (i.e. distances Au, 2Aa, . . . , d*) contribute to the sum in square brackets. Neglecting diversities at peripheries of the stage we get i
j=I
COV(& A,) = 2
‘;r COv(a,,,A,,+,Kz) (4.6)
.j#k
/=I
\
u2
I
By substituting (4.6) into (4.5) and after rearranging
c+= n2 4D
(4.7)
a - a0
The distribution of v/n in case of a short growth stage may be characterized
as (4.8)
where the type of model .ti * + ,JP (E, D/n) ti*-+d
(E, D)
for Au = (a - a& for a - a0 + d2 .
+ d2 3
(4.9) (4.10)
715
A stochastic model of fatigue crack growth
Relationship (4.9) follows from the central limit theorem under conditions of a sufficiently long time and expresses a connection to the case of fine variability. When the growth stage is longer, it is no more possible to replace the mean increment z and the variance D (if varying) by a constant. Quantity r7 can be expressed by a sum of partial integrals vi over successive intervals
i
Ui-r-,Ui+r 2 2
1
of the same length r extending over n, cycles. Then the variance of q is the sum of variances of partial quantities vi, i.e. according to (4.7) 4 = T nf d2 D(uJ/r
(4.11)
.
Partial times n, are expressed by means of linear approximation
(2.11) of the process q(n) as
and the integral vi for a small value of r is expressed by approximation substituting into (4.1 l), rearranging and taking the limit r -+ 0
(2.7) as vi G r/g(aJ. After
(4.12)
The distribution
of q/n may be characterized
by notation
(4.13)
for D = D(a) or D = const., respectively. The type of model &* conforms approximately (4.9) and (4.10). According to the central limit theorem it also holds for large n &R* +
,&’ for (a-a&/d2
---t CO.
to eqs
(4.14)
Conditions for application of relations (4.13) arise also when predicting the crack propagation over long growth stages of the length a - a,, >> d2 in cases, where the results of experiment are of the type of rough variability. 4.3 A model of the crack growth in case of rough variability First the distribution of n in case of a short growth stage will be derived assuming the autocorrelation function p,,(x) (3.5). For the given stage the variance of A-factor- is characterized by constant D and the magnitude of individual crack increments by constant Au . The variance of r,~is generally determined by (4.5). It follows from the relation a - a0 < d2 that values Aj
716
HYNEK LAUSCHMANN
corresponding to all crack fronts of the given stage contribute to the term in the square brackets. Through simple rearrangement we get f Cov(A,, Ai) = 20 ,= I
(4.15)
so that the variance of 17is
(4.16)
Neglecting the term l/n’ as against one we get (4.17)
The number of structural areas affected by the crack front during one growth stage does not exceed twice the number of structural areas affected by an individual crack front. One may assume therefore that the type of model of the distribution of &z will be approximately the same as the type of model & of the distribution of A-factor, i.e. .
(4.18)
This solution relates to the limit solution in case of intermediate variability (4.10). A deviation in the variance expresses the effect of peripheries of the growth stage neglected in case of intermediate variability. If G << dZ the course of the autocorrelation characteristic of the process can be estimated from the experimental results. Details follow in Section 5.3. Predictions of the growth process in long growth stages are to be computed by means of simulation techniques dealt with in detail in Section 6. Our reasoning so far does not refer to a special case when the magnitude of the characteristic structure dimension d, is comparable to the length of the crack front or longer. Since during a short growth stage the crack front affects only a very small number of the structural areas, the distribution of q/n is close to model .%?*(E,, Do) (see also the growth equation (3.1)). Similarly, when dimension Et, exceeds the maximum possible length of the crack and simultaneously it holds D = const., the individual realization of the process occurs at an approximately constant random value A. 5, CONCRETE FORM OF THE MODEL IN THE CASE OF POWER GROWTH LAWS The large class of growth laws leads to function g in a power form g(a) = M(a,
cry, a
> 0
(5.1)
)
where M is a function of crack length a and stress characteristics of the loading cycle denoted by o, a being a material parameter. En most cases the Paris-Erdogan’s law and those of Sih are applied, where M = A&, or AS, respectively. The quantity v [see eqs (2.6) and (2.7)] is written as follows v=
-cr du
b
“M”(u)
or
rl*
a - a0 a0 + u j-+-j 3 ff, = -y--
.
(5.2)
717
A stochastic model of fatigue crack growth
In applications variance D of the A-factor often decreases with increasing M-value. Due to our limited information we suppose the functional dependence in a simple form of D(M)=&‘M-P,&‘>O,~>o.
(5.3)
A special case D = const. is obtained by substituting p = 0, D = & . According to our experience following from the application of the Paris-Erdogan’s law for steel and light alloys inequality /3 < a may be generally assumed. We assume the experimental results to be in the usual form of readings [a, n], so that the information may be arranged into triples [u,,~, au, n,il, where aoii is the initial length, au the final length and nii the time interval of i-th growth stage of j-th growth curve. If the appurtenance of a stage to a concrete growth curve is not relevant, the couple of subscripts ij may be replaced by a simple subscript k, running through the whole set of experimental results. For simplicity our further consideration assumes the lengths of the individual growth stages, as defined by experimental readings, to be approximately equal, so that it is possible to substitute for them the mean length a’.Since the loading cycle is constant, the individual growth stages may be considered as independent process realizations. 5.1 Determination of the variability-type For an a-priori determination of the variability-type let us adopt results of the current dataevaluation according to the growth law. The individual growth stages will be represented by couples CM,,, v,J, where
by a linear regression with a regression straight line log v = log A, + a, log M the deterministic parameters Ad, a, will be assessed. The growth stages are further represented by couples [a,+ V,], where the standardized velocity r/;l = v,,l(A,M$q and ~~~= (aoij + uJ2 denotes the mid-point of the growth stage. Value Vii is an estimate of quotient Au/E, where A, is the mean A-factor value in the ij-th stage: 1 au - aoi. vll = nOAd . Mgd
12-3 E nii -
-=-
E ’
(5.4)
Thus in case of rough variability the sequence of values VY, i = 1,2, . . . ; j = const., will be correlated, whilst in case of fine and intermediate variability the sequence will be uncorrelated. We shall plot the sequence of points [asrJ, Vu] in a diagram separately for each growth curve. Figure 4 represents any typical forms of diagram. 5.2 The spec$cation of the model in cases ofjine and intermediate variability The variance of the standardized velocity Vk represents an estimate of the variance of vk/ (n,E). In the intermediate variability case it is therefore approximately true (see (4.8), (5.3))
In case of fine variability (see (4.1), (5.3)) the mean increment of the given stage Au replaces characteristic structure dimension d2 in relation (5.5). For information purposes the value following from the growth law may substitute value G, i.e. Au, & A, My. We may write
4%/Iv. = AdM,“d-P
&.
(5.6)
The theoretical course of the dependence of the variance o$ on M is given in Fig. 5. When visually
718
HYNEK
LAUSCHMANN
Fig. 4. Typical realizations of the sequences of points [a,,, VJ, i = 1,2.. ; j = const. (1) fine or .’ It can be reasoned about nonintermediate variability, (2) rough variability, (3) extreme roug h varlablhty. homogenity of the set of tested bodies or of testing conditions.
evaluating the diagram of points [Mk, V,] we may immediately decide, whether it is the case of intermediate or fine variability or of a combination of both the types; in the latter case we may estimate the approximate value of the characteristic structure dimension d2 using the relation d2 = AdM;fv.
(5.7)
To estimate the exponent p we may apply the linear regression with regression equation log V, = c, - p1og Mk + c2,
(5.8)
where
log 4
” =
I
log (AJ4iq
forMk G Mtjiv , for Mk > Mdiv,
(5.9)
The estimates of the E and .#’ -parameters will be carried out using the method of moments. The transformed quantity W, defined for the k-th stage by (5.10)
where
Ok =
&
1
JZ/&
if Mk > Mdiv , if Mk < Mdiv )
(5.11)
is characterized by moments ,u,,.= 0 and o$ = A“. The above relations are a base for the estimates of the E and &‘-parameters at a given a-parameter value:
(5.12)
(5.13)
intermediate variability
Fig. 5. The theoretical
tine variability
course
of the dependence
a:(M).
A stochastic model of fatigue crack growth
719
To determine the parameter a the independence of the quantity w on the M-value may be used, expressed by the condition Cov(w, M) = 0. After substitution and rearrangement it may be written
(5.14)
By substituting (5.12) for E we obtain the solution of 8 by using numerical methods. The characteristic structure dimension d2 can be practically estimated only as the average crack increment on the boundary of the intermediate and fine variability. As far as the whole experiment proceeds in the area of the intermediate variability, we substitute V, = fi (see (5.11)); hence, as a result of estimate (5.13) combined characteristic X.d2 is obtained. In case of a pure fine variability the knowledge of dimension d, is not imperative. If the whole experiment proceeds in the zone of fine variability, we expect a normal distribution of quantity n, and, consequently, that of quantity ~1 (5.10), too. The fit of the model for short growth stages may be verified by testing the following hypothesis
w :
-4 (0, N) .
(5.15)
A similar distribution of quantity w may be assumed also in case of combination of both variabilitytypes, due to the continuity of the model close to boundary M = Mdiv. In fact, however, the type of model of the distribution of q/n in the zone of intermediate variability may vary with increasing distance from iMd+ according to the changing mean increment & (see (4.9)). We have to neglect the possible successive change of the model-type of the distribution of q/n in view of the solvability of our problem even in the case of a pure intermediate variability. We assume therefore the distribution of q/n (4.8) to be characterized by a single model through the whole experiment. The selection of the model and the test of its fit for short growth stages proceeds then in the same way as in case of rough variability. 5.3 S~~l~cu~io~ of the model in case of rough voriab~li~y From the analysis in Section 4.3 it follows that the variance of I’ may be interpreted estimate of
as an
(5.16)
By taking the logarithm of equality (5.16) we get the relation
log&=C--1ogA4,
C=const.,
(5.17)
which enables to estimate parameter /? using linear regression (under the condition of a sufficient number of realizations). The procedure used to estimate parameters E,
720
HYNEK LAUSCHMANN
and a follows from the relation (4.18), which, after substituting (5.3), may (for the k-th stage) be written as (5.18)
The distribution of the transformed quantity
is then characterized by the mean value put = 0 and by the variance 02, = .J?. Estimates of parameters E, 2and a may, therefore, be determined by using equations (5.12) through (5.14) when vk = 1 (parameter 3? is determined according to (5.13) instead of &’ ). In case of D = cons& i.e. p = 0, we select a suitable type of model &’ for the dist~bution of n/n by testing the fit of various models, whose parameters are determined by the method of moments based on the estimates of E and D. If D = D(M), it is necessary to transform quantity n/n into a stationary quantity o, the distribution of which does not change in the course of the process. Then we have to test the fit of this distribution. Table 1 is a guide for selecting test hypotheses for several current models. With regard to the application of results in the form of simulation the autocorrelation function p,,(x) of the quantity o defined above may be considered as a suitable correlation characteristic of the process. From the course of function p,(x) also follows the estimate of maximum dimension of structure inhomogenities in the direction of crack growth d2,maxas correlation length, i.e. d2,max= SW
x
(5.19)
P*Y)>0
Owing to the distortion of the autocorrelation function due to the random distribution of structural areas as to their size (see Section 3.2) it is acceptable to estimate the characteristic structure dimension d2 using regression equation (3.5) to fit empirical values of the function p,(x). The value of parameter X’ may be estimated knowing the value d2 as (5.20)
The goodness-of-fit of the model for distribution of ~/n in short growth stages, determined by experimental readings, is not a sufficient condition of the utility of the model. Slight deviations, which may be neglected in case of short stages, can attain a cumulative character. It is necessary, therefore, to verify the conformity of model predictions of crack growth under experimental conditions with growth-curves obtained experimentally. An example of the corresponding procedure is given in Sect. 6. Of substantial advantage, however, are experiments with- the constant value of M (for example A&). For an individual process realization D = const, and Aa = const., the model of distribution of n/n does not change during the process. The evaluation of the results is analogous to that of the above cases. 6. APPLICATION A best-fit model of fatigue crack growth in flat bodies from construction aluminium alloys has been obtained by application of the model described above. As an example we show the evaluation of a set of experimental results noted for an unusually large scatter. The specimen with one-side sharp notch (Fig. 6) and with preliminary fatigue crack were uniaxially loaded by constant saw-shape cycle of nominal stress range AD = 40 MPa (12 specimen) and ACJ= 60 MPa (4 specimen) at stress ratio R = 0.05, frequency f = 10 Hz and temperature T = 20°C. The propagation of the crack was observed on the outside surface in shortening time
A stochastic Table Model of the distribution Of
model
of fatigue
1. A guide for selecting
crack
test hypotheses
Quantity
Parameters
721
growth
Tested hypothesis
w
vkh
Normal 0
Logarithmicnormal
a, = Jlog(st/E2+
1) 0 : _/+“(O, 1)
pk = log E -
Weibull §
b,
> “k
412
0:
II
i5
(1)
II
0 : ZY (0, 1>tt
Gamma**
tFor
:_N ‘(0, 1)
the case of rough
variability.
$For the case of intermediate &Distribution
variability.
function
IIParameters
b,, vi are determined
by solving equations
E = v, l-(1 + I/b,),
s; = ~1’[Ql “Standard
exponential
**Distribution
YtUniform
distribution
+ 2/b,) - P(1 + I/b,)].
with distribution
function
F,,,) (CO)=
1 - em”
function
(rectangular)
distribution
on interval
(0, 1)
intervals so that the lengths of growth stages between readings were near the value of 1 mm. The growth-curves are shown in Figs 11 and 12. 6.1
The evaluation of experimental data
Under the conditions of the experiment the stress intensity factor is determined by (according to
C261)
-
t.
Fig. 6. The specimen
for fatigue
crack
growth
experiments
722
HYNEK LAUSCHMANN
AK [MPa.m”‘l 6
10
I5
+ z!5
2a
Fig. 7. The dependence of crack growth rate on the stress intensity factor range in customary form.
254.65
K(a) = a,,,
~
a
165.52
- -
- -
W
[MPa.m*] ,
(6.1)
where a[mm] is the crack length including the depth of the notch, W[mm] is the width of the specimen in the crack growth direction and cr,,, [MPa] is the nominal stress. In the range of AKE (8, 22) MPa.m* the results of test agree with the Paris-Erdogan’s growth law in the form of v = AdAIF’.
(6.2)
The estimates of parameters A, and ad specified by using linear regression are tid = 3.61, ,$, = 9.89 x 10m6(for v[p/cycle]). Dependence v(AK) in the customary form is plotted in Fig. 7. The velocity-logarithms-variance perceptibly decreases with increasing AK, so that the application of the fundamental model of linear regression is not quite correct. The variability-type has been specified by the method described in Section 5.1. The illustration
s
..
2
c= -7
I
..
+++ + ++
++ + +++++ +
+ ++++
0.5..
+ 0.2 I;
I0
.
a,@nl
IS
20
aSmm1
0.2.. -I
2s
30
s
3s
5
I0
IS
20
25
30
35
..
2
E “>
I
..
0
5..
+ + +++
++
++
+
+
+++ +
* 0.2..
++++ + 5
10
. 15
a,hml 20
25
30
35
Fig. 8. Several realizations of sequence of points [a,,, VJ, i = 1,2,...; j = const. Every diagram represents one growth-curve, i.e. one body.
A stochastic model of fatigue crack growth
723
Fig. 9. The set of a+values in normal probability paper.
of several courses of the sequence of points [Q, Vi,], i = 1,2,...; j = const., are in Fig. 8. The sequences of points are distinctly correlated, which determines the rough variability. Values of vg have been used to estimate parameter p of the dependence (5.3), which in case of Paris-Erdogan’s law obtains form D(AK)=X’AK-@, The extent of data-set being small, the application (5.16) it is valid that
X
> O,p>,O.
of relation (5.17) is not useful. According to
C$ = (V - v)” = const. AKep. After rearrangement
and substitution
(6.3)
(6.4)
of v = 1 we get (V - 1)2AK5 = const.
(6.5)
Relation (6.5) may be interpreted by the condition Cov[(V - 1)2AI?, AK] = 0, the solution of which makes for the estimate of 1 = 2.542.
Fig. 10. The autocorrelation
function p,(x)--sample
values and linear approximation.
(6.6)
HYNEK LAUSCHMANN
124
I0 n - NUMBER
0
I 00
200
300
OF CYCLES
L100
[IO31
!i0Z
600
700
Fig. 11. The comparison of model (simulated) quantile curves u,(u) to sample growth curves. Stress range Ao = 40 MPT..
B_y using relations (5.12) - (5.14) with _vk = 1 (see Sect.5.31 the estimates of parameters (x, E and H have been specified as B = 3.320, E = 2.335 x 10P5, X = 8.514 x lops, while quanity ?jk corresponding to the k-th growth stage passing from the crack length @,kto ak one is specified by
qk
(6.7)
=
The distribution of quantity v/n is logarithmic-normal with satisfactory coincidence. goodness-of-it of the model has been verified by testing the following hypothesis (see Table
where
ok=/lo+$&+
pk = log E -
The 1)
l).
0$/2.
(6.10)
The sample test-statistic for Kolmogorov-Smirnov test is D,,,, = 0.032, while the critical value at a very strict level of significance 0.2 is Dcr.O,Z= 0.067. The values of mk are plotted into the normal probability paper in Fig. 9. the goodness-of-fit of the model slightly worsens in the region of high and low quantiles. In these cases we can speculate about the influence of any factors being present with low probability and not being appreciated by the model. The distribution of q/n in the case of rough variability is very near to the model ti of Afactor distribution (see Section 4.3). Logarithmic-normal distribution of the A-factor shows to the type of interactions along the crack front, which in its consequences corresponds to the resulting crack rate as geometric mean of local rates. The acceptance of this hypothesis would have farreaching consequences on the crack growth law interpretation[ 161. A useful characteristic of the correlation properties of the process is the autocorrelation function P,(X) of the stationary process w(a) (see eq. (6.8)). The sample estimate of function p,(x) in a discrete form is shown in Fig. 10. The course of function P,(X) is construed so that every couple of growth stages belonging to the same growth-curve is arranged for the value of x obtained by rounding-off the distance between midpoints of both stages. Sample-values of p,(x) for s > 13 are not representative, being computed from a small number of couples. The linear approximation (3.5) of function p,(x) leads to the estimate of parameter d2 assessed by using the least square
125
A stochastic model of fatigue crack growth
method L& = 12.7 mm. The unexpectedly high value of the characteristic structure dimension L& is connected with the very expressive texture of material structure. The estimate 2= 8.777 x 10e8 follows from relation (5.20) for Li = 1.14 mm. 6.2 Crack growth predictions For final verification of the model-fitness the predictions of crack growth under conditions characterizing the experiment have to be compared with the sample growth-curves. The predictions
p = 0.90
p = 0.99
p = 0.50
: . . .’ *a .*
. . . . . . . . . . ...* .,...*....*......
*......,.,...............*...*.* tl-
NUMBER
p=O.lO : : : : : : : : : **=
OF CYCLES
p = 0.01 ..*“*
[IO31
s 0
100
50
200
IS0
Fig. 12. The comparison of model (simulated) quantile curves u,,(n) to sample growth Aa = 60 MPa.
curves.Stress range
of crack propagation under any other conditions - reflected in a different formula for stress intensity factor - would be construed analogically. The validity of the model is theoretically limited to cases when the crack front length is near 5 mm. The reduction of the model parameters to another crack front length requires the knowledge of the nature of interaction along the crack front. The prediction is construed by using a simulation. The stochastic process w(a) with the statedistribution (6.8) and autocorrelation function (3.5) has been simulated with step G = 1 mm, i.e. as constant in the stages of the length of 1 mm. This is obtained by simulating value w as a moving average of 13 random values with distribution &’ (0,13). The values of quantities 9, B and p are determined in advance for single gro_wth stages by using relations (6.7), (6.9) and (6.10), respectively. Parameter &’ acquires the value X = 8.546.1O-8 for a” = 1 mm. The time-interval of the growth stage is then computed as An = q/exp (wa f ,u).
(6.11)
For the purpose of practical predictions the simulation~xtent of 99 realizations suffices. Resulting from them 99 growth-curves are obtained in the form of sequences of times pertaining to the integer crack lengths {~,{Uo)= 0 , n,(uo -I” 1) , nj(GJ+ 2) Times n&z0 + h) appertaining into sequences n1.h
<
I
.
.
*
9
n,@o + r>>,“= I .
(6.12)
to the same crack length a = a0 + h are ordered as to their size
n2?h <
n3.h
<
. . . <
n99,h,
h= 1,2 ,...,
r.
(6.13)
The value ny.hmay be interpreted directly as the q-% quantile of the distribution of time necessary for attaining crack length a0 + h. Discrete representations of the fundamental process characteristics
126
+
HYNEK
LAUSCHMANN
:‘-LF 5 1.
II
+
L
+
i
3
2
+
0
I
x
0
0.5
*
+
E
0
I
q
+ n= 50000cycles
I(
0
1
x n = 100000 cvcles
D n ~200 000 cycles
Cl- CRACK
I0
Fig. 13. Sample
I5
LENGTH
[mm]
20
2s
30
35
and model (simulated) distribution function of the crack length in times n = 50,000, 100,000 and 200,000 cycles. Initial crack length a, = IO mm, stress range Au = 40 MPa.
thus follow from the matrix (2.3): For a = const.
{nq,h} either immediately
or by using linear interpolation
and relation
= a, + h
‘y;
(6.14)
q-1
for n = const.
F,,(a)
n
ao+h+
*
-
%,h
+ I
I >
n
-
II 99
nq,h
,1--L
(6.15)
100 q=,’
%.h
where %.h+
2
nq.h
;
for p = const = 4 100 uP(n) o nP(a) * Representations
~bkI0-q,h~
%
+
hl}i=
{[% + h? nq,hl}i=l
(6.16) and (6.17) differ only by the interchange
(6.16)
I ;
(6.17)
.
of co-ordinates
and by the inverse
adjudgement of probability p. The results of the simulation crack length
can be used also for predicting the crack growth from the initial into the new origin according to a0 + m, m = 1. 2, . . . . Data (6.12) transposed nj(ao+m+h)
= ni(ao+m+h)-n,{q,+m),
serve then as a basis for the above
h = 1, 2, . . .. r-m.
(6.18)
prediction.
The results of crack growth predictions are compared with sample-characteristics in Figs 11, 12, 13 and 14. The conformity of the model with initial experiment is satisfactory; the deviations are to be judged with regard to the small extent of the experiment.
7. CONCLUSIONS A stochastic model of fatigue crack growth has been developed, which is relatively from the mathematical point of view and appreciably general in the concept of randomness process. Its main contributions are: (1) The
possibility
of probabilistically
interpreted
predictions
of crack
propagation
simple of the
under
model of fatigue crack growth
A stochastic
I _
727
- +
+ a=15mm
a a-25mm n-NUMBER 0 Fig. 14. Sample
and
108
200
OF CYCLES [lo31
:
:
;
;
;
380
q00
500
600
‘700
model (simulated) distribution function of the time, in which crack length attained from initial length a, = IO mm. Stress range Acr = 40 MPa.
a = 15 and 25 mm is
conditions different from the experiment, with the model of process-variability reflecting its real causes and nature. (2) The estimation of the characteristic dimension of the material structure inhomogenities causing the dominant variability of crack growth rate under constant-cycle loading. The above results were obtained, first of all, thanks to the yet very unusual conception of the stochastic model with discrete time and continuous state, which offers an immediate analogy between the mathematical description and physical reality. In this paper the model has been presented only in the application on the fatigue crack growth under constant-cycle loading. By using a more general growth law reflecting the consequences of the non-constancy of loading cycles the process under a general loading regime can be predicted. The nature of the interaction of the local fatigue processes along the crack front is the question of principal significance. By the statistical analysis of this problem the non-integer value of the crack growth law exponent may be explained[16]. A separate paper will be presented on this theme.
REFERENCES [I] [2] [3] [4] [S] [6] [7] [8] [9] [IO] [l I] [IZ] [13] 1141 1151 [16] [17] [18] [I91 [20] [21]
F. Kozin and J. L. Bogdanoff, A critical analysis of some probabilistic models of fatigue crack growth. Engng Fracrure Mech. 14, 59-89 (1981). F. Kozin and J. L. Bogdanoff, On the probabilistic modelling of fatigue crack growth. Engng Fracrure Mech. 18, 623-632 (1983). J. L. Bogdanoff, A new cumulative damage model. Part I. J. uppl. Mech. 45, 246-250 (1978). J. L. Bogdanoff and W. Krieger, A new cumulative damage model, Part 2. J. uppl. Mech. 45, 251-258 (1978). J. L. Bogdanoff, A new cumulative damage model, Part 3. J. appl. Me& 45, 733-739 (1978). J. L. Bogdanoff and F. Kozin, A new cumulative damage model, Part 4. J. appl. Mech. 47, 40-44 (1980). J. L. Bogdanofl and F. Kozin, On nonstationary cumulative damage models. J. appl. Mech. 49, 37-42 (1982). F. Kozin and J. L. BogdanolT, On life behavior under spectrum loading. Engng Fracfure Mech. 18, 271-283 (1983). J. L. Bogdanoff and F. Kozin, Probabilistic models of fatigue crack growth - II. Engng Fracture Mech. 20, 255270 (1984). H. Ghonem. Stochastic fatigue crack initiation and propagation in polycrystalline solids. Ph.D.Thesis. McGill Universitv. Montreal 11978). H. Ghon& and J. W: Proban, Micromechanics theory of fatigue crack initiation and propagation. Engng Fracture Mech. 13, 963-977 (1980). H. Ghonem and S. Dare, Probabilistic description of fatigue crack growth in polycrystalline solids. Engng Fracture Mech. 21. 115lLl168 (1985). J. SedlLeek, The stochastic interpretation of service strength and reliability of mechanical systems. Monographs and Memorandu 6, National Research Institute of Machine Design, Prague (1968). J. SedlZek, Aditive processes with random transition to absorption state and their application on the fracture of metals (in Czech). Aplikuce matematiky 11, 2&44 (1966). J. Ntmec and J. SedlBEek, Sfatistica/ Foundations of Structure Strength (in Czech). Academia, Prague (1982). H. Lauschmann. To the probabilistic description of fatigue crack growth (in Czech). C.Sc. Dissertation. Faculty of Nuclear Science and Physical Engineering, Technical University of Prague (1985). J. N. Yang. G. C. Salivar and C. G. Anuis, Statistical modelling of fatigue-crack growth in a nickel-base superalloy. Engng Fracrure Mech. 18, 257-270 (1983). Y. K. Liu and J. N. Yang, On statistical moments of fatigue crack propagation. Engng Fracture Mech. 18, 243-256 ( 1983). R. Arone, A statistical model for fatigue fracture under constant-amplitude cyclic loading. Engng Fracture Mech. 14, 189-194 (1981). R. Arone, A stochastic model for fatigue crack growth. Proc. Fafigue 1984, Birmingham (1984). H. Ishikawa, A. Tsurui and A. Utsumi, A stochastic model of fatigue crack growth in consideration of random propagation resistance. Proc. Fafigue 1984, Birmingham (1984).
728 [22]
HYNEK
LAUSCHMANN
A. Tsurui, H. Ishikawa and A. Utsumi, Theoretical study on the distribution of fatigue crack propagation life under stationary random loading. Proc. Fatigue 1984. Birmingham (1984). 1231 F. Ellyin and C. 0. Fakinlede. Probabilistic simulation of fatigue crack growth by damage accumulation. Engng Fruciure Mech. 22, 697-712 (1985). [24] 0. Ditlevsen, Random fatigue crack growth - a first passage problem. Engng Fracture Mech. 23, 467477 (1986). [25] H. Alawi. A probabilistic model for fatigue crack growth under random loading. Engng Fracture Mech. 23, 479487 (1986). [26] P. D. Rooke and D. J. Cartwright, Compendium on stress infensifyfacfors. Her Majesty’s Stationery Office. London (1976).
m3-7944/87 163.00+ 0.w Q 1987 Pergamon Journals Ltd
Vol. 26, No. 5, pp 707-728
A STOCHASTIC MODEL OF FATIGUE CRACK GROWTH IN HETEROGENEOUS MATERIAL HYNEK LAUSCHMANN Faculty of Nuclear Science and Physical Engineering, Prague, Kfemencova 10, 110 00 Praha 1, Czechoslovakia Abstract-Two ways of the phenomenoiogical modeliing of fatigue crack growth have been developed in the past - deterministic crack growth laws and stochastic crack growth models. At present time the syntheti~ng models are constitute. The object of this paper is a model describing the variability of the crack growth process under constant-cycle loading as the consequence of heterogeneous material structure. A general stochastic model with discrete time and continuous state serves as the theoretical basis. Three types of process-variability are defined and theoretically solved. Concrete form of the model in the case of power-form growth laws is presented with regard to the evaluation of experimental data and to the crack growth predictions. The model is applied to evaluate the results of crack growth experiment on the specimen from aluminium alloy.
NOMENCLATURE a
process state - crack length [mm] initial crack length [mm] process time - number of loading cycles from the origin of the process subscript denoting the i-th stage of the j-th growth curve k subscript denoting the k-th stage of the whole data set a “,,,us,,,aV,n,, (or with the subscript k) initial, middle and final crack length and time interval of the process stage determined by the subscript v crack growth rate [urn/cycle] Aa crack increment in one cycle [urn] A multiplicative factor of the growth law Au = Ag(a) conducting function of the growth law Aa = A&) g(a) mean value and variance of the A-factor E, D see A, the quantity is related to single structure area A, mean value and variance of the A,,-factor Eo, Do distribution function of the state a at time n F,,(a) probability that the state a is surpassed for the first time at a time shorter or equal to n P,,(n) 100 p% quantile of state u at time n o,“(a) 100 p% quantile of time n at which the state o is surpassed for the first time n,(a) I‘ du 00 n ij
v
=i-- ,&) model of the distribution of A-factor model of the distribution of A,-factor model .$4 with mean value x and variance y characteristic material structure dimension in the crack front direction and crack growth direction, respectively
Cmml crack front length [mm] mean crack increment in one cycle within given growth stage [pm] mean length of growth stages determined by experimental readings autocorrelation function of the quantity X in point x base function of the power-form growth Iaw Au = AM” exponent of the power-form growth law Au = AM” multiplicative factor of the dependence D = X M @ exponent of the dependence D = 2f Mm@ parameters of the deterministic power-form growth law standardized velocity V = v/(Ad Ma”)
M,,, ,1 w
the value of M on the boundary between the regions of intermediate and fine variability standardized form of q/n stationary function of n/n 707
708
HYNEK LAUSCHMANN 1. INTRODUCTION
THE PHENOMENOLOGICAL modelling of fatigue crack growth has been oriented partly towards physico-technical deterministic growth laws and partly towards stochastic models, based on mathematical abstraction. The growth laws express the dependence of the crack growth rate da/dn on stress characteristics and on the crack length, or on other factors (such as loading frequency, temperature, cycle shape, environment). Since the process history has not been respected, the validity of growth laws is practically limited to cases of constant-cycle loading regimes. The growth laws are understood as equations for the deterministic phenomenon component, the growth-process variability being put into connection with non-specified random fluctuations. The usual procedure used for evaluation of experiments includes the application of growth laws by means of regression methods. Three applications of the mean-value operator on the crack rate are involved implicitly in standard conception of the growth law: averaging along the crack front, averaging in the direction of crack propagation close to the given crack length and averaging over individual realizations of the process. The information included in the growth law is therefore rather compressed; the law is usually understood as a characteristic of the process of continuous state and time and the transition to the growth curve a(n) proceeds by integration without analytically respecting the process variability. The growth laws offer an advantage of connection with the physical description of the phenomenon, enabling the transfer of information in case of changing physical conditions. The growth laws suffer from the disadvantage of an insufficient inclusion of the process scattering properties. Stochastic crack growth models outlined especially in the works of Bogdanoff, Kozin et al.[393, Ghonem et al.[lO, 1l] and SedliEek[13-151 are based on the application of the stochastic processes theory; the model base is a general non-decreasing random process, the parameters or functions having the role of intensities are specified experimentally. The confrontation with the growth law is usually carried out by an interpretation of the growth law integral as the mean value or the median of the process or as the equation of the individual realization. The stochastic models are capable to describe precisely the random growth-process in concrete cases, without giving the possibility of predicting the growth process after the physical conditions changed. The stochastic models can be used with advantage for the description of a fatigue process in operation when the purpose is not a physical analysis of the phenomenon, but a precise compressed description of a concrete random crack growth process. Since 1981 a number of models has been published, connecting the investigation of growth process variability with the growth law ([l, 2, 12, 16-253). The scatter of the process is mostly described by a generalization of the growth law into a probabilistic form (randomization), including in several cases also the variability of loading process. It seems to be advisable to dispose of a stochastic crack growth model which would reflect the main source of process randomness in cases of constant-cycle loading, i.e. the scatter of local mechanical properties of the material due to the heterogeneous structure. Unfortunately, no practically applicable results have been attained in this way so far. The principal purpose of the present paper is a synthesis of the growth-law philosophy with a stochastic model enabling - in necessary simplification - the structural heterogeneity influence to be described. In our conception the growth law is interpreted as an equation for the crack increment in one cycle, i.e. that out of the three applications of the mean value operator in the usual conception only the averaging along the crack front has been considered. This interpretation generated a stochastic model, investigating the crack growth as the summation of random increments. 2. THEORETICAL
MODEL
Let us assume a stochastic growth process a(n) with discrete time n, n = 0,1,2,... and with continuous state a, a > 0. The course of process a(n) is defined by the increment Aa during the time interval An = 1; the increment Au depends on state a, being independent on time n and defined by the relation Au = g(u) A,
g(a) > 0, ‘4 > 0 3
(2.1)
A stochastic
model of fatiguecrack growth
709
where g(a) is a known function, A is a random factor in general, whose properties define concrete process types. The initial condition is specified in a deterministic way: at the time n = 0 the process state attains the value ao. Let us define further the inverse process n(a) as a stochastic dependence of time n, at which state a has been surpassed for the first time, on state a. The time n(a) can be understood with an error E,, < 1 as the time necessary for attaining state a. The basic characteristics of processes u(n) and n(u) are denoted as follows: F,(a) ... distribution function of state a at time n; P,(n) ... probability that the state a is for the first time surpassed at a time shorter or equal to n; u,(n) ... lOOp% quantile of the distribution of state a at time n; n,(u) ... loop% quantile of time n, at which state a is surpassed for the first time. The interconnection
of the two processes u(n) and n(u) is expressed by the relation
(2.2)
F&4 = 1 - P,(n) , from which the possibility of quantile inversion follows n = n,(u) 0
a = u&l)
.
(2.3)
The basic approach when solving this process consists in the following steps: By rearranging equation (2.1) we get
AU A
-= g(u)
.
(2.4)
Let us assume quantity 7 defined as
(2.5) where subscript k denotes individual consecutive process increments. When the increments Au are small and time n large, the sum in eq. (2.5) can be replaced by an integral in accordance with Riemann’s definition (see Fig. 1)
(2.6)
rl=
or in the case of short intervals (a,, a) in accordance with the mean value theorem by approximation a tl
--a0
a, = (a0 + a)/2 .
%T’
(2.7)
According to (2.4) it also holds that
9 =
,$,Ak’
(2.8)
i.e. the statistical properties of the quantity 17may be considered as properties of the sum of n factor A realizations. Quantity q is generally an increasing function of the process state a (see eqs (2.6) and (2.7)).
710
HYNEK LAUSCHMANN
ak-l Fig. 1. Replacing the
a
sum by an integral
It follows that the distribution function G(q) related to the growth interval (a,, a) and to time n equals
G(v)
=
F,(a)
(llo.aM
.
(2.9)
a0
Thus the statistical properties of process a(n) can be studied by using process q(n). Let us outline an approximation useful in solving some more difficult cases. It is to assess the course of the process 17(n) if it is known: (1) the distribution of A-factor does not change during the process; (2) the states of the process a(n) at times n = 0 and n = n, were a0 and a,, respectively. According to eqs (2.6) and (2.8) it can be written (2.10)
The sequence of unknown random values Ak will be.replaced by the sequence of identical values 1 = 77,/n,. The state of process q at an arbitrary time n < n, is then determined as ?/7(n)= n
3. ANALYSIS
2.
OF THE CRACK GROWTH PROCESS VARIABILITY CONSTANT-CYCLE LOADING
(2.11)
IN CASE OF
The majority of the known growth laws may be written in the form of eq. (2.1). Thus the function g(a) represents a non-random physical component of the phenomenon, whereas the factor A stands for random fluctuations. We shall limit the following analysis to cases where the variability of the crack growth has been caused only by an interaction of the crack front with the heterogeneous structure of material. The development of the plastic zone and internal stress area involves a relatively large volume of the material; therefore, it is possible to interpret the function g, including the effect of these phenomena, to be deterministic. Conversely, the individual crack increment proceeds in the elemental microvolume and may be principally affected by the local structure character. Therefore, the variability of the process is seen as related to the scattering of the individual crack increments,
711
A stochastic model of fatigue crack growth
crack front direction Fig. 2. The model structure of material in ideal crack surface. Every structural area is properties - homogeneous, characterized by a single random value A,.
as to fatigue
represented in our model by a statistical factor A. Let us characterize the distribution of the Afactor by the mean value E, variance D and model-type 9; it will be written A : L@ (6 D). To the purpose of this paper let us define the characteristic structure dimension, d, as the mean value of the distance, where the local mechanical material properties, as considered in view of their effect on the fatigue crack growth, do not change or where the change is negligible. Moreover, it will be of advantage to introduce the characteristic dimension d, in the direction of the crack front and d2 in the direction of crack growth, which is of use for the application in case of rolled materials. For general consideration the model structure will be applied instead of the real one, the former having a random distribution of structural areas of non-correlated fatigue properties, but of similar dimensions, equal to constants d, and d2 (see Fig. 2). Let us assume the growth law for the particular structural area in the form A@= g(4 Ao .
(3.1)
The growth law (3.1) determines the magnitude of the crack increment in the particular structure area -- unaffected by the adjacent structural areas - under given stress conditions. The &-factor distribution may be considered according to the growth law philosophy as a material characteristic. Let it be specified by the mean value Eo, variance f)o and model-type %sP We assume the mean value E. and the type of model bPo to be material characteristics independent of the process state. As for the variance Do we assume that it is generally dependent on the value of function g, being therefore a function of the process state a. The inconstancy of the variance characteristic expresses the fact that the effect of structural inhomogenities on crack growth rate may be sensitive to the stress level on the crack tip. 3.1 An~Iysis of rel~~~o~salong the crack front at a gisen crack length In the above interpretation the individual sample value of the A-factor represents the resultant of material properties along the actual crack front. The crack front increases or decreases by its waviness the local state of stress, responding in this way to the local higher or lower resistance of the material. The model of physical interactions along the crack front does not exist; let us, therefore, develop the most simple idea: the resulting crack increment in one cycle is the arithmetic mean of local increments by which the crack would propagate in the individual structural areas if it did not keep a continuous front. When denoting the length of crack front c, the number of the involved structural areas will be c/d,. Distribution .w(E, 0) of the A-factor is characterized by the moments E = E,,,D =
Do. dJc DO
for c 3 dj for c < d,
,
(3.2)
and by the type of model
(3.3)
712
HYNEK LAUSCHMANN
The mean value E of the A-factor is independent of the crack front length; also the type of model can be considered in the case of small differences of the crack front length as approximately constant. If variance D, and D, of the A-factor appertain to the crack front length c, and c2, respectively, then D,
CI
=
D2
~2,
~1,
c2
a
4
(3.4)
.
In case of non-linear interaction models the mean value E along with variance D are nonlinear functions of E,, D, and c. We expect that in connection with the research of interactions along the crack front the above ideas will be substantially corrected in favour of non-linear interaction models. So for instance the model representing factor A as a geometric mean of values A0 emphasizes the braking effect of barriers against crack propagation. Model-type 9 is asymptotically logarithmic-normal, the mean value E being dependent on crack front length. 3.2 Analysis of conditions in the crack growth direction The autocorrelation function of process A(a), expressing the course of the A-factor in the direction of crack propagation, is a characteristic of principal significance. With regard to a possible dependence of variance D = D(a) it is useful to apply a standardized process y(a) = (A(a) - E)/ m whose autocorrelation function under the above assumptions is q,(x) = CM_&4,
Aa + 4) =
1 - x/d2
o
for x d d2, for x > d2 .
(3.5)
The practical significance of the relation (3.5) essentially exceeds assumptions, under which it has been derived. Irrespective of the type of interaction along the crack front, the sequence of the realizations of the A-factor is conditioned by changes of structural areas affected by the crack front. Assuming a model structure with a constant d2-dimension it holds generally:
(1) q,.(O)= 1;
(2) p,.(x) = 0 for x k d2 ; (3) p,.(x) is decreasing within XE (0, d2) .
The relation (3.5) is therefore a linear approximation of function q,(x) for a general type of interaction model. The actual shape of function q,(x) differs from the theoretical dependence (3.5) partly due to a more complex type of interactions along the crack front and partly due to the random size of the structural areas. Structural areas smaller than d2 will cause the decrease of q,(x) for low xvalues; structural areas larger than d2 will cause the increase of p!(x) for large x, including also .Y> d:. Let us denote the crack growth section from the a,-length up to a-length as the growth stage and the increment a-a,, as the growth stage length. The characteristic length of the growth stage ~7is the mean value of lengths of the growth stages determined by experimental readings (we assume the lengths of these stages to be approximately constant). The mean increment G of a given stage is defined by & = (a - a,)/An, where An is the time interval of the growth stage. The essence of the experimental data evaluation problem is double sampling of the random process A(a): process A(a) with correlation properties as determined by the characteristic structure dimension dz is projected into the fatigue process a(n) by sampling with mean length z Experimental information is a result of sampling of the process u(n) with mean length & The quality of the resulting statistical properties of the investigated data will, therefore, be affected by the interrelation of dimensions d,, G and d. Three types of variability of the process within a given growth stage are defined by means of the following relations: d2 < G dz ~(6, dz>Z
. . . tine variability; 5) . . intermediate
variability;
(3.6)
. . . rough variability.
Considering that the mean increment Au varies in the course of the crack propagation
process at
A
713
stochasticmodel of fatigue crack growth
(2) (1) Fig. 3. A diagram of three types of process variability.Hsuccessive crack fronts; D experimental reading; (l)...fine variability; (2)...intermediate variability; (3)...rough variability. (The of cycles between readings is symbolic.)
constant loading cycle within several orders, a change in the type of variability may occur from the intermediate to the fine one. A chart of types of variability is given in Fig. 3. 4. STOCHASTIC
MODELS OF FATIGUE CONSTANT-CYCLE
CRACK PROPAGATION LOADING
UNDER
4.1 Model of crack growth in case offine variability In case of fine variability each crack front appears in a quite new configuration of structural inhomogenities. Conditions of crack propagation and, therefore, also individual realizations of Afactor are random without any correlation. First, let us consider the case of D = const. Assuming a sufficiently long time n the distribution of
can be expressed by using the central limit theorem as asymptotically nE and variance nD. i.e. q : JV (nE, nD) .
normal with the mean value
(4.1)
In a special case, when the distribution of the A-factor is exponential, process n(a) is the known Poisson process. This process is homogeneous for g(a) = const. and inhomogeneous if g(a) # const. To the same case a more general variant may be transformed, where the distribution of the A-factor is of gamma-type with mean E and variance D, by means of time transformation n’ = n.I?/D.
In a more general case of dependent variance D = D(a) it is appropriate to solve separately the problems of short and long growth stages. In a short growth stage the change in variance D may be assumed to be small, so that variance D may be represented - in a single growth stage - by a constant value belonging, e.g. to the midpoint of the growth stage:
The evaluation of experimental data follows the theory for D = const. In case of a long growth stage a continuous change of variance D must be taken into account. Applying the central limit theorem we obtain for the distribution of rl at sufficiently long times n n-l
V :
b/t’(nE,k;, D(d) ,
(4.2)
714
HYNEK
LAUSCHMANN
where uk are the process states at successive times k = 0, 1,2, . . . , n - 1. A correct solution encounters considerable difficulties; therefore a simple approximate solution has been derived. The sum in the term of variance may be replaced by n-multiple of the mean value D(a) along interval (ao, a), which may be found by integrating function D(a) with respect to any quantity varying linear with time n. Accepting the approximation (2.1 I), the quantity q can be used. After substitution r,r = ~(a) it can be written (4.3) Relation (4.3) is suspicious owing to 17being used for expression of variance. However, this is only a formal problem, as using relations (2.2) and (2.3) a relation for quantile n,(u) can be expressed from (4.3) (4.4) where up is 100 p% quantile of the standard normal distribution. Relation (4.4) may also serve by using (2.3) and numerical methods - as a basis for computing quantile u,(n). 4.2 A model of the crack growth in case of intermediate variability First the distribution of 17 in case of a short growth stage will be derived assuming autocorrelation function p,(x) (3.5). Within the given stage the variance of A-factor is characterized by constant D and the magnitude of individual increments of the crack by constant s. The variance of sum
is generally equal to
(4.5)
.
Only values A, belonging to --crack fronts being at a maximum distance d2 from the k-th front in both directions (i.e. distances Au, 2Aa, . . . , d*) contribute to the sum in square brackets. Neglecting diversities at peripheries of the stage we get i
j=I
COV(& A,) = 2
‘;r COv(a,,,A,,+,Kz) (4.6)
.j#k
/=I
\
u2
I
By substituting (4.6) into (4.5) and after rearranging
c+= n2 4D
(4.7)
a - a0
The distribution of v/n in case of a short growth stage may be characterized
as (4.8)
where the type of model .ti * + ,JP (E, D/n) ti*-+d
(E, D)
for Au = (a - a& for a - a0 + d2 .
+ d2 3
(4.9) (4.10)
715
A stochastic model of fatigue crack growth
Relationship (4.9) follows from the central limit theorem under conditions of a sufficiently long time and expresses a connection to the case of fine variability. When the growth stage is longer, it is no more possible to replace the mean increment z and the variance D (if varying) by a constant. Quantity r7 can be expressed by a sum of partial integrals vi over successive intervals
i
Ui-r-,Ui+r 2 2
1
of the same length r extending over n, cycles. Then the variance of q is the sum of variances of partial quantities vi, i.e. according to (4.7) 4 = T nf d2 D(uJ/r
(4.11)
.
Partial times n, are expressed by means of linear approximation
(2.11) of the process q(n) as
and the integral vi for a small value of r is expressed by approximation substituting into (4.1 l), rearranging and taking the limit r -+ 0
(2.7) as vi G r/g(aJ. After
(4.12)
The distribution
of q/n may be characterized
by notation
(4.13)
for D = D(a) or D = const., respectively. The type of model &* conforms approximately (4.9) and (4.10). According to the central limit theorem it also holds for large n &R* +
,&’ for (a-a&/d2
---t CO.
to eqs
(4.14)
Conditions for application of relations (4.13) arise also when predicting the crack propagation over long growth stages of the length a - a,, >> d2 in cases, where the results of experiment are of the type of rough variability. 4.3 A model of the crack growth in case of rough variability First the distribution of n in case of a short growth stage will be derived assuming the autocorrelation function p,,(x) (3.5). For the given stage the variance of A-factor- is characterized by constant D and the magnitude of individual crack increments by constant Au . The variance of r,~is generally determined by (4.5). It follows from the relation a - a0 < d2 that values Aj
716
HYNEK LAUSCHMANN
corresponding to all crack fronts of the given stage contribute to the term in the square brackets. Through simple rearrangement we get f Cov(A,, Ai) = 20 ,= I
(4.15)
so that the variance of 17is
(4.16)
Neglecting the term l/n’ as against one we get (4.17)
The number of structural areas affected by the crack front during one growth stage does not exceed twice the number of structural areas affected by an individual crack front. One may assume therefore that the type of model of the distribution of &z will be approximately the same as the type of model & of the distribution of A-factor, i.e. .
(4.18)
This solution relates to the limit solution in case of intermediate variability (4.10). A deviation in the variance expresses the effect of peripheries of the growth stage neglected in case of intermediate variability. If G << dZ the course of the autocorrelation characteristic of the process can be estimated from the experimental results. Details follow in Section 5.3. Predictions of the growth process in long growth stages are to be computed by means of simulation techniques dealt with in detail in Section 6. Our reasoning so far does not refer to a special case when the magnitude of the characteristic structure dimension d, is comparable to the length of the crack front or longer. Since during a short growth stage the crack front affects only a very small number of the structural areas, the distribution of q/n is close to model .%?*(E,, Do) (see also the growth equation (3.1)). Similarly, when dimension Et, exceeds the maximum possible length of the crack and simultaneously it holds D = const., the individual realization of the process occurs at an approximately constant random value A. 5, CONCRETE FORM OF THE MODEL IN THE CASE OF POWER GROWTH LAWS The large class of growth laws leads to function g in a power form g(a) = M(a,
cry, a
> 0
(5.1)
)
where M is a function of crack length a and stress characteristics of the loading cycle denoted by o, a being a material parameter. En most cases the Paris-Erdogan’s law and those of Sih are applied, where M = A&, or AS, respectively. The quantity v [see eqs (2.6) and (2.7)] is written as follows v=
-cr du
b
“M”(u)
or
rl*
a - a0 a0 + u j-+-j 3 ff, = -y--
.
(5.2)
717
A stochastic model of fatigue crack growth
In applications variance D of the A-factor often decreases with increasing M-value. Due to our limited information we suppose the functional dependence in a simple form of D(M)=&‘M-P,&‘>O,~>o.
(5.3)
A special case D = const. is obtained by substituting p = 0, D = & . According to our experience following from the application of the Paris-Erdogan’s law for steel and light alloys inequality /3 < a may be generally assumed. We assume the experimental results to be in the usual form of readings [a, n], so that the information may be arranged into triples [u,,~, au, n,il, where aoii is the initial length, au the final length and nii the time interval of i-th growth stage of j-th growth curve. If the appurtenance of a stage to a concrete growth curve is not relevant, the couple of subscripts ij may be replaced by a simple subscript k, running through the whole set of experimental results. For simplicity our further consideration assumes the lengths of the individual growth stages, as defined by experimental readings, to be approximately equal, so that it is possible to substitute for them the mean length a’.Since the loading cycle is constant, the individual growth stages may be considered as independent process realizations. 5.1 Determination of the variability-type For an a-priori determination of the variability-type let us adopt results of the current dataevaluation according to the growth law. The individual growth stages will be represented by couples CM,,, v,J, where
by a linear regression with a regression straight line log v = log A, + a, log M the deterministic parameters Ad, a, will be assessed. The growth stages are further represented by couples [a,+ V,], where the standardized velocity r/;l = v,,l(A,M$q and ~~~= (aoij + uJ2 denotes the mid-point of the growth stage. Value Vii is an estimate of quotient Au/E, where A, is the mean A-factor value in the ij-th stage: 1 au - aoi. vll = nOAd . Mgd
12-3 E nii -
-=-
E ’
(5.4)
Thus in case of rough variability the sequence of values VY, i = 1,2, . . . ; j = const., will be correlated, whilst in case of fine and intermediate variability the sequence will be uncorrelated. We shall plot the sequence of points [asrJ, Vu] in a diagram separately for each growth curve. Figure 4 represents any typical forms of diagram. 5.2 The spec$cation of the model in cases ofjine and intermediate variability The variance of the standardized velocity Vk represents an estimate of the variance of vk/ (n,E). In the intermediate variability case it is therefore approximately true (see (4.8), (5.3))
In case of fine variability (see (4.1), (5.3)) the mean increment of the given stage Au replaces characteristic structure dimension d2 in relation (5.5). For information purposes the value following from the growth law may substitute value G, i.e. Au, & A, My. We may write
4%/Iv. = AdM,“d-P
&.
(5.6)
The theoretical course of the dependence of the variance o$ on M is given in Fig. 5. When visually
718
HYNEK
LAUSCHMANN
Fig. 4. Typical realizations of the sequences of points [a,,, VJ, i = 1,2.. ; j = const. (1) fine or .’ It can be reasoned about nonintermediate variability, (2) rough variability, (3) extreme roug h varlablhty. homogenity of the set of tested bodies or of testing conditions.
evaluating the diagram of points [Mk, V,] we may immediately decide, whether it is the case of intermediate or fine variability or of a combination of both the types; in the latter case we may estimate the approximate value of the characteristic structure dimension d2 using the relation d2 = AdM;fv.
(5.7)
To estimate the exponent p we may apply the linear regression with regression equation log V, = c, - p1og Mk + c2,
(5.8)
where
log 4
” =
I
log (AJ4iq
forMk G Mtjiv , for Mk > Mdiv,
(5.9)
The estimates of the E and .#’ -parameters will be carried out using the method of moments. The transformed quantity W, defined for the k-th stage by (5.10)
where
Ok =
&
1
JZ/&
if Mk > Mdiv , if Mk < Mdiv )
(5.11)
is characterized by moments ,u,,.= 0 and o$ = A“. The above relations are a base for the estimates of the E and &‘-parameters at a given a-parameter value:
(5.12)
(5.13)
intermediate variability
Fig. 5. The theoretical
tine variability
course
of the dependence
a:(M).
A stochastic model of fatigue crack growth
719
To determine the parameter a the independence of the quantity w on the M-value may be used, expressed by the condition Cov(w, M) = 0. After substitution and rearrangement it may be written
(5.14)
By substituting (5.12) for E we obtain the solution of 8 by using numerical methods. The characteristic structure dimension d2 can be practically estimated only as the average crack increment on the boundary of the intermediate and fine variability. As far as the whole experiment proceeds in the area of the intermediate variability, we substitute V, = fi (see (5.11)); hence, as a result of estimate (5.13) combined characteristic X.d2 is obtained. In case of a pure fine variability the knowledge of dimension d, is not imperative. If the whole experiment proceeds in the zone of fine variability, we expect a normal distribution of quantity n, and, consequently, that of quantity ~1 (5.10), too. The fit of the model for short growth stages may be verified by testing the following hypothesis
w :
-4 (0, N) .
(5.15)
A similar distribution of quantity w may be assumed also in case of combination of both variabilitytypes, due to the continuity of the model close to boundary M = Mdiv. In fact, however, the type of model of the distribution of q/n in the zone of intermediate variability may vary with increasing distance from iMd+ according to the changing mean increment & (see (4.9)). We have to neglect the possible successive change of the model-type of the distribution of q/n in view of the solvability of our problem even in the case of a pure intermediate variability. We assume therefore the distribution of q/n (4.8) to be characterized by a single model through the whole experiment. The selection of the model and the test of its fit for short growth stages proceeds then in the same way as in case of rough variability. 5.3 S~~l~cu~io~ of the model in case of rough voriab~li~y From the analysis in Section 4.3 it follows that the variance of I’ may be interpreted estimate of
as an
(5.16)
By taking the logarithm of equality (5.16) we get the relation
log&=C--1ogA4,
C=const.,
(5.17)
which enables to estimate parameter /? using linear regression (under the condition of a sufficient number of realizations). The procedure used to estimate parameters E,
720
HYNEK LAUSCHMANN
and a follows from the relation (4.18), which, after substituting (5.3), may (for the k-th stage) be written as (5.18)
The distribution of the transformed quantity
is then characterized by the mean value put = 0 and by the variance 02, = .J?. Estimates of parameters E, 2and a may, therefore, be determined by using equations (5.12) through (5.14) when vk = 1 (parameter 3? is determined according to (5.13) instead of &’ ). In case of D = cons& i.e. p = 0, we select a suitable type of model &’ for the dist~bution of n/n by testing the fit of various models, whose parameters are determined by the method of moments based on the estimates of E and D. If D = D(M), it is necessary to transform quantity n/n into a stationary quantity o, the distribution of which does not change in the course of the process. Then we have to test the fit of this distribution. Table 1 is a guide for selecting test hypotheses for several current models. With regard to the application of results in the form of simulation the autocorrelation function p,,(x) of the quantity o defined above may be considered as a suitable correlation characteristic of the process. From the course of function p,(x) also follows the estimate of maximum dimension of structure inhomogenities in the direction of crack growth d2,maxas correlation length, i.e. d2,max= SW
x
(5.19)
P*Y)>0
Owing to the distortion of the autocorrelation function due to the random distribution of structural areas as to their size (see Section 3.2) it is acceptable to estimate the characteristic structure dimension d2 using regression equation (3.5) to fit empirical values of the function p,(x). The value of parameter X’ may be estimated knowing the value d2 as (5.20)
The goodness-of-fit of the model for distribution of ~/n in short growth stages, determined by experimental readings, is not a sufficient condition of the utility of the model. Slight deviations, which may be neglected in case of short stages, can attain a cumulative character. It is necessary, therefore, to verify the conformity of model predictions of crack growth under experimental conditions with growth-curves obtained experimentally. An example of the corresponding procedure is given in Sect. 6. Of substantial advantage, however, are experiments with- the constant value of M (for example A&). For an individual process realization D = const, and Aa = const., the model of distribution of n/n does not change during the process. The evaluation of the results is analogous to that of the above cases. 6. APPLICATION A best-fit model of fatigue crack growth in flat bodies from construction aluminium alloys has been obtained by application of the model described above. As an example we show the evaluation of a set of experimental results noted for an unusually large scatter. The specimen with one-side sharp notch (Fig. 6) and with preliminary fatigue crack were uniaxially loaded by constant saw-shape cycle of nominal stress range AD = 40 MPa (12 specimen) and ACJ= 60 MPa (4 specimen) at stress ratio R = 0.05, frequency f = 10 Hz and temperature T = 20°C. The propagation of the crack was observed on the outside surface in shortening time
A stochastic Table Model of the distribution Of
model
of fatigue
1. A guide for selecting
crack
test hypotheses
Quantity
Parameters
721
growth
Tested hypothesis
w
vkh
Normal 0
Logarithmicnormal
a, = Jlog(st/E2+
1) 0 : _/+“(O, 1)
pk = log E -
Weibull §
b,
> “k
412
0:
II
i5
(1)
II
0 : ZY (0, 1>tt
Gamma**
tFor
:_N ‘(0, 1)
the case of rough
variability.
$For the case of intermediate &Distribution
variability.
function
IIParameters
b,, vi are determined
by solving equations
E = v, l-(1 + I/b,),
s; = ~1’[Ql “Standard
exponential
**Distribution
YtUniform
distribution
+ 2/b,) - P(1 + I/b,)].
with distribution
function
F,,,) (CO)=
1 - em”
function
(rectangular)
distribution
on interval
(0, 1)
intervals so that the lengths of growth stages between readings were near the value of 1 mm. The growth-curves are shown in Figs 11 and 12. 6.1
The evaluation of experimental data
Under the conditions of the experiment the stress intensity factor is determined by (according to
C261)
-
t.
Fig. 6. The specimen
for fatigue
crack
growth
experiments
722
HYNEK LAUSCHMANN
AK [MPa.m”‘l 6
10
I5
+ z!5
2a
Fig. 7. The dependence of crack growth rate on the stress intensity factor range in customary form.
254.65
K(a) = a,,,
~
a
165.52
- -
- -
W
[MPa.m*] ,
(6.1)
where a[mm] is the crack length including the depth of the notch, W[mm] is the width of the specimen in the crack growth direction and cr,,, [MPa] is the nominal stress. In the range of AKE (8, 22) MPa.m* the results of test agree with the Paris-Erdogan’s growth law in the form of v = AdAIF’.
(6.2)
The estimates of parameters A, and ad specified by using linear regression are tid = 3.61, ,$, = 9.89 x 10m6(for v[p/cycle]). Dependence v(AK) in the customary form is plotted in Fig. 7. The velocity-logarithms-variance perceptibly decreases with increasing AK, so that the application of the fundamental model of linear regression is not quite correct. The variability-type has been specified by the method described in Section 5.1. The illustration
s
..
2
c= -7
I
..
+++ + ++
++ + +++++ +
+ ++++
0.5..
+ 0.2 I;
I0
.
a,@nl
IS
20
aSmm1
0.2.. -I
2s
30
s
3s
5
I0
IS
20
25
30
35
..
2
E “>
I
..
0
5..
+ + +++
++
++
+
+
+++ +
* 0.2..
++++ + 5
10
. 15
a,hml 20
25
30
35
Fig. 8. Several realizations of sequence of points [a,,, VJ, i = 1,2,...; j = const. Every diagram represents one growth-curve, i.e. one body.
A stochastic model of fatigue crack growth
723
Fig. 9. The set of a+values in normal probability paper.
of several courses of the sequence of points [Q, Vi,], i = 1,2,...; j = const., are in Fig. 8. The sequences of points are distinctly correlated, which determines the rough variability. Values of vg have been used to estimate parameter p of the dependence (5.3), which in case of Paris-Erdogan’s law obtains form D(AK)=X’AK-@, The extent of data-set being small, the application (5.16) it is valid that
X
> O,p>,O.
of relation (5.17) is not useful. According to
C$ = (V - v)” = const. AKep. After rearrangement
and substitution
(6.3)
(6.4)
of v = 1 we get (V - 1)2AK5 = const.
(6.5)
Relation (6.5) may be interpreted by the condition Cov[(V - 1)2AI?, AK] = 0, the solution of which makes for the estimate of 1 = 2.542.
Fig. 10. The autocorrelation
function p,(x)--sample
values and linear approximation.
(6.6)
HYNEK LAUSCHMANN
124
I0 n - NUMBER
0
I 00
200
300
OF CYCLES
L100
[IO31
!i0Z
600
700
Fig. 11. The comparison of model (simulated) quantile curves u,(u) to sample growth curves. Stress range Ao = 40 MPT..
B_y using relations (5.12) - (5.14) with _vk = 1 (see Sect.5.31 the estimates of parameters (x, E and H have been specified as B = 3.320, E = 2.335 x 10P5, X = 8.514 x lops, while quanity ?jk corresponding to the k-th growth stage passing from the crack length @,kto ak one is specified by
qk
(6.7)
=
The distribution of quantity v/n is logarithmic-normal with satisfactory coincidence. goodness-of-it of the model has been verified by testing the following hypothesis (see Table
where
ok=/lo+$&+
pk = log E -
The 1)
l).
0$/2.
(6.10)
The sample test-statistic for Kolmogorov-Smirnov test is D,,,, = 0.032, while the critical value at a very strict level of significance 0.2 is Dcr.O,Z= 0.067. The values of mk are plotted into the normal probability paper in Fig. 9. the goodness-of-fit of the model slightly worsens in the region of high and low quantiles. In these cases we can speculate about the influence of any factors being present with low probability and not being appreciated by the model. The distribution of q/n in the case of rough variability is very near to the model ti of Afactor distribution (see Section 4.3). Logarithmic-normal distribution of the A-factor shows to the type of interactions along the crack front, which in its consequences corresponds to the resulting crack rate as geometric mean of local rates. The acceptance of this hypothesis would have farreaching consequences on the crack growth law interpretation[ 161. A useful characteristic of the correlation properties of the process is the autocorrelation function P,(X) of the stationary process w(a) (see eq. (6.8)). The sample estimate of function p,(x) in a discrete form is shown in Fig. 10. The course of function P,(X) is construed so that every couple of growth stages belonging to the same growth-curve is arranged for the value of x obtained by rounding-off the distance between midpoints of both stages. Sample-values of p,(x) for s > 13 are not representative, being computed from a small number of couples. The linear approximation (3.5) of function p,(x) leads to the estimate of parameter d2 assessed by using the least square
125
A stochastic model of fatigue crack growth
method L& = 12.7 mm. The unexpectedly high value of the characteristic structure dimension L& is connected with the very expressive texture of material structure. The estimate 2= 8.777 x 10e8 follows from relation (5.20) for Li = 1.14 mm. 6.2 Crack growth predictions For final verification of the model-fitness the predictions of crack growth under conditions characterizing the experiment have to be compared with the sample growth-curves. The predictions
p = 0.90
p = 0.99
p = 0.50
: . . .’ *a .*
. . . . . . . . . . ...* .,...*....*......
*......,.,...............*...*.* tl-
NUMBER
p=O.lO : : : : : : : : : **=
OF CYCLES
p = 0.01 ..*“*
[IO31
s 0
100
50
200
IS0
Fig. 12. The comparison of model (simulated) quantile curves u,,(n) to sample growth Aa = 60 MPa.
curves.Stress range
of crack propagation under any other conditions - reflected in a different formula for stress intensity factor - would be construed analogically. The validity of the model is theoretically limited to cases when the crack front length is near 5 mm. The reduction of the model parameters to another crack front length requires the knowledge of the nature of interaction along the crack front. The prediction is construed by using a simulation. The stochastic process w(a) with the statedistribution (6.8) and autocorrelation function (3.5) has been simulated with step G = 1 mm, i.e. as constant in the stages of the length of 1 mm. This is obtained by simulating value w as a moving average of 13 random values with distribution &’ (0,13). The values of quantities 9, B and p are determined in advance for single gro_wth stages by using relations (6.7), (6.9) and (6.10), respectively. Parameter &’ acquires the value X = 8.546.1O-8 for a” = 1 mm. The time-interval of the growth stage is then computed as An = q/exp (wa f ,u).
(6.11)
For the purpose of practical predictions the simulation~xtent of 99 realizations suffices. Resulting from them 99 growth-curves are obtained in the form of sequences of times pertaining to the integer crack lengths {~,{Uo)= 0 , n,(uo -I” 1) , nj(GJ+ 2) Times n&z0 + h) appertaining into sequences n1.h
<
I
.
.
*
9
n,@o + r>>,“= I .
(6.12)
to the same crack length a = a0 + h are ordered as to their size
n2?h <
n3.h
<
. . . <
n99,h,
h= 1,2 ,...,
r.
(6.13)
The value ny.hmay be interpreted directly as the q-% quantile of the distribution of time necessary for attaining crack length a0 + h. Discrete representations of the fundamental process characteristics
126
+
HYNEK
LAUSCHMANN
:‘-LF 5 1.
II
+
L
+
i
3
2
+
0
I
x
0
0.5
*
+
E
0
I
q
+ n= 50000cycles
I(
0
1
x n = 100000 cvcles
D n ~200 000 cycles
Cl- CRACK
I0
Fig. 13. Sample
I5
LENGTH
[mm]
20
2s
30
35
and model (simulated) distribution function of the crack length in times n = 50,000, 100,000 and 200,000 cycles. Initial crack length a, = IO mm, stress range Au = 40 MPa.
thus follow from the matrix (2.3): For a = const.
{nq,h} either immediately
or by using linear interpolation
and relation
= a, + h
‘y;
(6.14)
q-1
for n = const.
F,,(a)
n
ao+h+
*
-
%,h
+ I
I >
n
-
II 99
nq,h
,1--L
(6.15)
100 q=,’
%.h
where %.h+
2
nq.h
;
for p = const = 4 100 uP(n) o nP(a) * Representations
~bkI0-q,h~
%
+
hl}i=
{[% + h? nq,hl}i=l
(6.16) and (6.17) differ only by the interchange
(6.16)
I ;
(6.17)
.
of co-ordinates
and by the inverse
adjudgement of probability p. The results of the simulation crack length
can be used also for predicting the crack growth from the initial into the new origin according to a0 + m, m = 1. 2, . . . . Data (6.12) transposed nj(ao+m+h)
= ni(ao+m+h)-n,{q,+m),
serve then as a basis for the above
h = 1, 2, . . .. r-m.
(6.18)
prediction.
The results of crack growth predictions are compared with sample-characteristics in Figs 11, 12, 13 and 14. The conformity of the model with initial experiment is satisfactory; the deviations are to be judged with regard to the small extent of the experiment.
7. CONCLUSIONS A stochastic model of fatigue crack growth has been developed, which is relatively from the mathematical point of view and appreciably general in the concept of randomness process. Its main contributions are: (1) The
possibility
of probabilistically
interpreted
predictions
of crack
propagation
simple of the
under
model of fatigue crack growth
A stochastic
I _
727
- +
+ a=15mm
a a-25mm n-NUMBER 0 Fig. 14. Sample
and
108
200
OF CYCLES [lo31
:
:
;
;
;
380
q00
500
600
‘700
model (simulated) distribution function of the time, in which crack length attained from initial length a, = IO mm. Stress range Acr = 40 MPa.
a = 15 and 25 mm is
conditions different from the experiment, with the model of process-variability reflecting its real causes and nature. (2) The estimation of the characteristic dimension of the material structure inhomogenities causing the dominant variability of crack growth rate under constant-cycle loading. The above results were obtained, first of all, thanks to the yet very unusual conception of the stochastic model with discrete time and continuous state, which offers an immediate analogy between the mathematical description and physical reality. In this paper the model has been presented only in the application on the fatigue crack growth under constant-cycle loading. By using a more general growth law reflecting the consequences of the non-constancy of loading cycles the process under a general loading regime can be predicted. The nature of the interaction of the local fatigue processes along the crack front is the question of principal significance. By the statistical analysis of this problem the non-integer value of the crack growth law exponent may be explained[16]. A separate paper will be presented on this theme.
REFERENCES [I] [2] [3] [4] [S] [6] [7] [8] [9] [IO] [l I] [IZ] [13] 1141 1151 [16] [17] [18] [I91 [20] [21]
F. Kozin and J. L. Bogdanoff, A critical analysis of some probabilistic models of fatigue crack growth. Engng Fracrure Mech. 14, 59-89 (1981). F. Kozin and J. L. Bogdanoff, On the probabilistic modelling of fatigue crack growth. Engng Fracrure Mech. 18, 623-632 (1983). J. L. Bogdanoff, A new cumulative damage model. Part I. J. uppl. Mech. 45, 246-250 (1978). J. L. Bogdanoff and W. Krieger, A new cumulative damage model, Part 2. J. uppl. Mech. 45, 251-258 (1978). J. L. Bogdanoff, A new cumulative damage model, Part 3. J. appl. Me& 45, 733-739 (1978). J. L. Bogdanoff and F. Kozin, A new cumulative damage model, Part 4. J. appl. Mech. 47, 40-44 (1980). J. L. Bogdanofl and F. Kozin, On nonstationary cumulative damage models. J. appl. Mech. 49, 37-42 (1982). F. Kozin and J. L. BogdanolT, On life behavior under spectrum loading. Engng Fracfure Mech. 18, 271-283 (1983). J. L. Bogdanoff and F. Kozin, Probabilistic models of fatigue crack growth - II. Engng Fracture Mech. 20, 255270 (1984). H. Ghonem. Stochastic fatigue crack initiation and propagation in polycrystalline solids. Ph.D.Thesis. McGill Universitv. Montreal 11978). H. Ghon& and J. W: Proban, Micromechanics theory of fatigue crack initiation and propagation. Engng Fracture Mech. 13, 963-977 (1980). H. Ghonem and S. Dare, Probabilistic description of fatigue crack growth in polycrystalline solids. Engng Fracture Mech. 21. 115lLl168 (1985). J. SedlLeek, The stochastic interpretation of service strength and reliability of mechanical systems. Monographs and Memorandu 6, National Research Institute of Machine Design, Prague (1968). J. SedlZek, Aditive processes with random transition to absorption state and their application on the fracture of metals (in Czech). Aplikuce matematiky 11, 2&44 (1966). J. Ntmec and J. SedlBEek, Sfatistica/ Foundations of Structure Strength (in Czech). Academia, Prague (1982). H. Lauschmann. To the probabilistic description of fatigue crack growth (in Czech). C.Sc. Dissertation. Faculty of Nuclear Science and Physical Engineering, Technical University of Prague (1985). J. N. Yang. G. C. Salivar and C. G. Anuis, Statistical modelling of fatigue-crack growth in a nickel-base superalloy. Engng Fracrure Mech. 18, 257-270 (1983). Y. K. Liu and J. N. Yang, On statistical moments of fatigue crack propagation. Engng Fracture Mech. 18, 243-256 ( 1983). R. Arone, A statistical model for fatigue fracture under constant-amplitude cyclic loading. Engng Fracture Mech. 14, 189-194 (1981). R. Arone, A stochastic model for fatigue crack growth. Proc. Fafigue 1984, Birmingham (1984). H. Ishikawa, A. Tsurui and A. Utsumi, A stochastic model of fatigue crack growth in consideration of random propagation resistance. Proc. Fafigue 1984, Birmingham (1984).
728 [22]
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A. Tsurui, H. Ishikawa and A. Utsumi, Theoretical study on the distribution of fatigue crack propagation life under stationary random loading. Proc. Fatigue 1984. Birmingham (1984). 1231 F. Ellyin and C. 0. Fakinlede. Probabilistic simulation of fatigue crack growth by damage accumulation. Engng Fruciure Mech. 22, 697-712 (1985). [24] 0. Ditlevsen, Random fatigue crack growth - a first passage problem. Engng Fracture Mech. 23, 467477 (1986). [25] H. Alawi. A probabilistic model for fatigue crack growth under random loading. Engng Fracture Mech. 23, 479487 (1986). [26] P. D. Rooke and D. J. Cartwright, Compendium on stress infensifyfacfors. Her Majesty’s Stationery Office. London (1976).