Fractal scaling and crack-size effects on creep crack growth

Abstract

Scaling effects on the creep crack growth behaviour are investigated by analyzing the results of compact tension (CT) tests on different-sized notched steel specimens appearing in the literature. Creep crack growth rate data are correlated to the elastic stress-intensity factor in terms of a Paris-type law, da∕dt = C ₀ K ^q, where C ₀ turns out to be a crack-size dependent coefficient of proportionality. Considering specimens with the same loading configuration (CT) and the same thickness, the observed crack-size effect on the creep crack growth rate is discussed on the basis of self-similarity considerations, and geometrically interpreted in terms of fractal tortuosity of the crack profile. A size-independent formulation of the creep crack growth law correlating renormalized quantities is finally deduced and confirmed by the experimental results.

Keywords

Creep crack growth scaling effects incomplete self-similarity fractal geometry renormalization group theory

1. Introduction

Design and integrity assessment of structural components operating at high temperatures typically have to account for the creep behaviour under steady-state conditions, where the strain rate $\dot{{\varepsilon}}$ strongly depends on the applied stress 𝜎 according to the Norton law [15,47]: $\begin{eqnarray}\displaystyle \dot{{\varepsilon}}=A{\sigma}^{n}, & & \displaystyle\end{eqnarray}$ (1) where A and n, the latter being known as creep sensitivity [38], are temperature-dependent material parameters.

In fact, failure of structural components operating under creep conditions can occur by either creep rupture or creep crack growth. The former is most likely in components which are initially flaw-free or containing benign defects. However, the presence of inherent sharp defects, which can grow and ultimately cause brittle fracture, has necessitated characterizing the creep crack growth as well, often relegating the “flaw-free” condition to a mere idealization.

Despite the predictive character of fracture mechanics, whereby the results from laboratory specimens can be extrapolated to make predictions on full-sized structural components, and the number of studies conducted about creep, only a few of them were directly aimed at investigating size effects. In the current paper, the approach of fracture mechanics is supplemented by self-similarity and fractal geometry considerations to re-examine the observed crack-size effects on the creep crack growth behaviour for metallic materials.

2. Characterizations of creep crack growth by fracture mechanics

To properly identify a fracture parameter that is able to account for creep in the crack growth behaviour, a schematic representation of the creep deformation process in a cracked body has been considered. Immediately after loading, r ^−1∕2-type (being r the distance from the crack tip) and HRR [19,30] singular stress distributions are respectively generated in linear elastic and power-law hardening materials. In either case, if extensive creep deformation has time to increase in size before the crack propagation, neither the elastic stress-intensity factor, K, nor the J-integral [13,29] —characterizing the HRR solution— can adequately characterize the crack-tip stress field. This is due to the nonlinear and time-dependent character of creep deformation, which is not admitted in linear elastic fracture mechanics (LEFM) and elastic-plastic fracture mechanics (EPFM) formulations [16,34,46].

By analogy with the J-integral, Landes and Begley [20] and Nikbin et al. [24] defined the C*-integral as a path-independent crack-tip parameter characterizing stress and strain rate fields when creep strains dominate: $\begin{eqnarray}\displaystyle C^{\ast }=\int _{{\Gamma}}W^{\ast }\text{d}y-t_{i}(\partial \dot{u}_{i}/\partial x)\text{d}s, & & \displaystyle\end{eqnarray}$ (2) (being $W^{\ast }=\int _{0}^{\dot{{\varepsilon}}_{ij}}{\sigma}_{ij}\text{d}\dot{{\varepsilon}}_{ij}$ the strain energy density rate, 𝛤 a contour surrounding the crack tip, t the traction vector, and $\dot{u}$ the displacement rate vector) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle {\sigma}_{ij}\propto (C^{\ast }/r)^{1/(n+1)},\end{eqnarray}$ (3) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \dot{{\varepsilon}}_{ij}\propto (C^{\ast }/r)^{n/(n+1)}.\end{eqnarray}$ (4) By comparing it with the HRR solution, the analogy between creeping and power-law hardening materials is apparent.

Several studies [17,20,25,31,33] demonstrated that the creep crack growth rate (CCGR) can be described by an expression of the form: $\begin{eqnarray}\displaystyle \frac{\text{d}a}{\text{d}t}=D_{0}C^{\ast {\phi}}, & & \displaystyle\end{eqnarray}$ (5) where D ₀ is a material constant and 𝜙 ≲ 1. It can be noted that for intermediate values of the creep sensitivity n, CCGR data are correlated to the C*-integral according to Eq. (4) [16,46].

More recently, extensive work in the field of time-dependent fracture mechanics (TDFM) has been devoted to the analysis of the so-called constraint effects on CCGR. In analogy with EPFM [26,27,35], results of recent research have extended TDFM beyond the limits of single-parameter theory, where a two-parameter (C*-Q) approach has been proposed to characterize crack-tip constraint at elevated temperatures [4,6,23,35]. The Q parameter is introduced for quantifying the constraint effects during creep, in terms of deviations of the actual crack-tip stress-field—generally determined by numerical FEM analysis—from the HRR field in Eq. (3). Numerical and experimental studies reported in the literature [39,40,42,44,45,49–53] regard the dependence of these effects on loading configuration [44,49,52], as well as crack depth [39,45,51] (in-plane constraint effects) and specimen thickness (out-of-plane constraint effects) [40,42]. The general conclusion is that higher constraint levels at the crack tip provide higher creep cracking rates.

Within the single-parameter assumption, several parameters have been hystorically proposed for characterizing the creep crack growth behaviour, as reported for instance in [24,43]. It was emphasized that for creep-ductile materials, defined as those in which the creep crack growth rate is negligible compared to the creep zone expansion rate, the creep zone completely engulfs the uncracked ligament before the crack actually propagates, causing extensive stress redistribution [16,34,46]. This circumstance occurs in highly susceptible materials to creep deformation, i.e. when n ≫ 1 [38], whereby the crack-tip singularity tends to disappear according to Eq. (3a). That makes a description in terms of net section stress or reference stress 𝜎_N appropriate [18,22]: $\begin{eqnarray}\displaystyle \frac{\text{d}a}{\text{d}t}=B_{0}{\sigma}_{N}^{s}, & & \displaystyle\end{eqnarray}$ (6) where B ₀ and s are material parameters.

A relevant attempt [56–59] to consider creep as a thermally activated process led to the following Arrhenius-type relationship: $\begin{eqnarray}\displaystyle \frac{\text{d}a}{\text{d}t}\propto \exp Q^{\ast }, & & \displaystyle\end{eqnarray}$ (7) where Q* is a function dependent upon the applied stress, the absolute temperature T, the specimen size b, the gas constant R, and the activation energy ΔH _f for crack extension. With regards to the size-scale effects, the authors found a power-law dependence of the CCGR on the specimen size b. In this context, it is worth mentioning a multi-scale creep crack growth model, spanning from the micro- to the structural scale, where both microstructural (grain size) and geometric (stress tri-axiality) effects are included [55]. This model focuses on predicting fracture life of structural components under high-temperature creep conditions.

When a little creep deformation accompanies crack growth, the initial stress distribution remains virtually unaltered by creep, and K (or J) continues to characterize the stress state around the crack tip. Such creep-brittle behaviour, defined as that in which the creep crack growth rate da∕dt is comparable to the creep zone expansion rate [36,37], occurs in case of high CCGR values when the specimen always remains in small-scale creep condition. For this circumstance, occurring at short times (when the stress-strain response is predominantly elastic) or for materials designed to resist large-scale creep deformations (i.e. with creep sensitivity n ≅ 1), the creep crack growth rate is satisfactorily described in terms of stress-intensity factor K by means of a Paris-type law [12,28]: $\begin{eqnarray}\displaystyle \frac{\text{d}a}{\text{d}t}=C_{0}K^{q}, & & \displaystyle\end{eqnarray}$ (8) where C ₀ and q are material parameters.

Hereinafter, the attention will be focused on a set of CCGR data from an experimental campaign on CT steel specimens, originally correlated to both C* and K [41]. Despite the fact that CCGR is currently interpreted in terms of C* in the framework of constraint effects for most practical situations, on that occasion some success was found in correlating CCGR data to the elastic stress-intensity factor, K. In this case, the observed shift of the experimental curves is compatible with a crack-size effect, which can be interpreted in terms of self-similarity and fractal tortuosity of the crack profile, along the line of recent studies on fatigue crack growth [9–12,28].

3. Experimental characterization of creep crack growth

The creep crack growth behaviour of CT 1% CrMoV steel specimens of various sizes —standard and ultra-large specimens respectively of 50.8 mm and 254 mm in width, with thickness ranging from 6.35 to 63.5 mm— was investigated by Tabuchi et al. [41]. After introducing a fatigue pre-crack of 3 mm for standard specimens and of 15 mm for ultra-large specimens at room temperature, the authors conducted creep crack growth tests at 538 °C. The tested geometry is outlined in Fig. 1, where the markers indicate the specimens under investigation. In this paper, the analysis is restricted to specimens with the same thickness (12.7 mm), and differing in width, i.e. in the initial crack length (3 and 15 mm), in order to isolate the crack-size effects on the CCGR.

Fig. 1.

Geometry and size of CT specimens tested (adapted from [41]).

The authors discussed the experimental data both in the framework of TDFM and LEFM, correlating satisfactorily creep crack growth rate by Eqs (4) and (7), respectively. Characterization of da∕dt in terms of the C*-integral pointed out increasing creep crack growth rate with increasing specimen thickness, whereas neither specimen-width nor crack-size effects were observed. The specimen-thickness effect was already interpreted in the framework of the constraint effects, whereby the achievement of plane-strain conditions in very thick sections result in less creep ductility and higher cracking rates. Instead, the lack of crack-size (or specimen-width) effects on the creep crack growth rate was considered surprising, given that the pre-crack of ultra-large specimens was five times longer than standard ones, and worthy of further investigation.

Actually, the characterization of creep crack growth rate in terms of K by Eq. (7) did not demonstrate thickness effects for specimens with the same width, but, rather demonstrated a right- and downward shift of CCGR data with increasing specimen width, or pre-crack length. The power-law exponent q of the two fitting curves varies over the range 14.1–14.7 (the average exponent is $\bar{q}=∼14.4\pm 0.4$ ). As shown in Fig. 2, da∕dt for ultra-large specimens was about 1000 times lower than that for standard specimens for a given value of K. This may be interpreted as a crack-size effect whereby creep crack growth rate increases by decreasing the crack size, similarly to what happens with the Paris law [9–12,28]. As a straightforward consequence, short cracks under creep conditions are expected to propagate at higher rates.

Fig. 2.

Shift of experimental CCGR curves correlating da∕dt with K (adapted from [41]) for different initial crack lengths, accounting for the crack-size effect.

4. Generalized creep crack growth law

From a physical point of view, correlating creep crack growth rate da∕dt by an expression in the form of Eqs (4), (5) or (7) provides little information on the multiple factors that affect creep crack growth. In the case of the above results, where CCGR data are satisfactorily correlated to K, a functional dependence upon a series of LEFM parameters can be properly stated: $\begin{eqnarray}\displaystyle \frac{\text{d}a}{\text{d}t}={\Phi}({\sigma},K,T;{\sigma}_{u},K_{C},K_{th},X,T_{M};a). & & \displaystyle\end{eqnarray}$ (9) In particular, the governing parameters include (i) a threshold value K _th of the stress-intensity factor, below which creep crack growth does not occur [54], (ii) the thermal diffusivity X (m² s⁻¹) which controls the stress-induced migration of vacancies responsible for creep deformation, and (iii) the absolute melting temperature T _M of the material (creep is commonly found to become severe when the operating temperature exceeds 0.4 T _M). It is worth noting that the loading configuration and the specimen thickness are not included as variable parameters, since geometry (CT) and thickness (Fig. 1) are fixed in the current analysis.

Considering 𝜎_u, K _C, X, T _M as independent variables, the dimensional analysis [5] yields the following expression in terms of dimensionless parameters: $\begin{eqnarray}\displaystyle \frac{\text{d}a}{\text{d}t}=\frac{X}{a_{0}}{\Phi}_{1}\left(\frac{{\sigma}}{{\sigma}_{u}},\frac{K}{K_{C}},\frac{K_{th}}{K_{C}},\frac{T}{T_{M}},\frac{a}{a_{0}}\right), & & \displaystyle\end{eqnarray}$ (10) where a ₀ represents the fracture sensitivity of the material as a function of the fracture toughness K _C and tensile strength 𝜎_u, a ₀ = (K _C∕𝜎_u)²∕𝜋.

On the other hand, power-law relationships, such as those presented to describe creep crack growth rate, give the evidence of self-similarity, wherein a phenomenon reproduces itself over different time and space scales. It has been observed that self-similar solutions describe the intermediate asymptotic behaviour of fatigue crack growth when “the influence of the initial conditions has disappeared but the influence of the instability has not yet intruded” [1–3,32].

Accordingly, characterization of creep crack growth in terms of power laws is signature of an intermediate asymptotic regime, which is achieved in the mid-range of growth rates, i.e., when the crack has started to propagate although the material is sufficiently far from failure. The so-called incomplete self-similarity prevails at this stage, corresponding to a power-law dependence of creep crack growth rate on certain dimensionless parameters.

Let us consider a and K. Since the above experimental evidences have shown the dependence of the creep crack growth rate on a and K, incomplete self-similarity is assumed in the corresponding dimensionless parameters a∕a ₀ and K∕K _C. Thus, Eq. (9) takes the following power-law asymptotic form: $\begin{eqnarray}\displaystyle \frac{\text{d}a}{\text{d}t}=\frac{X}{a_{0}}{\Phi}_{2}\left(\frac{{\sigma}}{{\sigma}_{u}},\frac{K_{th}}{K_{C}},\frac{T}{T_{M}}\right)\left(\frac{a}{a_{0}}\right)^{{\gamma}_{2}}\left(\frac{K}{K_{C}}\right)^{{\gamma}_{1}}\equiv C_{0}K^{{\gamma}_{1}}. & & \displaystyle\end{eqnarray}$ (11) Comparing the generalized creep crack growth law of Eq. (9) with Eq. (7) gives: $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle {\gamma}_{1}=q\end{eqnarray}$ (12) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle C_{0}\equiv \frac{X}{a_{0}}{\Phi}_{2}\left(\frac{{\sigma}}{{\sigma}_{u}},\frac{K_{th}}{K_{C}},\frac{T}{T_{M}}\right)\left(\frac{a}{a_{0}}\right)^{{\gamma}_{2}}K_{C}^{-{\gamma}_{1}}.\end{eqnarray}$ (13) Equation (10) is formally identical to the classical Paris-type law in Eq. (7) for creep crack growth, but the coefficient of proportionality C ₀ is not simply a function of the material properties and temperature, but turns out to depend on the actual crack length a as well. The exponent 𝛾₂ governing such size effect cannot be obtained from the dimensional analysis, but rather using experimental data.

Deviations from the mid-range power-law behaviour, predictable within intermediate asymptotics, have been experimentally verified [54] when the conditions of non-propagating crack, for K → K _th, and Griffith-Irwin instability, for K → K _C, are approached, yielding a Paris-type curve illustrated in Fig. 3.

Fig. 3.

Schematization of CCGR diagram: power-law stage corresponds to the intermediate asymptotic behaviour.

5. Fractal approach to creep crack growth

Since the publication of the celebrated article by Mandelbrot and co-workers [21], the fractal nature of metal fracture surfaces has been widely recognized. Such a fractal character has been revealed also for metallic substructures under creep conditions [14,48]. Repeated observations at various magnifications have shown that the structure replicates itself in a self-similar way in a range of intermediate scales, falling between the micro-scale —influenced by grains, inclusions and dislocations— and the macro-scale —characterized by the size of the specimen and the notch from which the crack propagates. Accordingly, a geometric interpretation of crack-size effects on the creep crack growth rate is obtained in the context of fractal geometry [7,8], whereby the crack profile can be modelled as an invasive fractal set a*, with projected length a and topological dimension 1 + d _G (0 < d _G < 1): $\begin{eqnarray}\displaystyle a^{\ast }\simeq a^{1+d_{G}}. & & \displaystyle\end{eqnarray}$ (14) On the other hand, revisiting the Griffith energy balance in the context of fractal cracks yields the following scaling law: $\begin{eqnarray}\displaystyle K\simeq K^{\ast }a^{d_{G}/2}, & & \displaystyle\end{eqnarray}$ (15) where K* is the renormalized stress-intensity factor with anomalous physical dimensions [F][L]^{−(3+d
_G)∕2}.

Exploiting Eq. (12), an application of the derivation rule for composite functions gives: $\begin{eqnarray}\displaystyle \frac{\text{d}a}{\text{d}t}=\frac{\text{d}a}{\text{d}a^{\ast }}\frac{\text{d}a^{\ast }}{\text{d}t}=\frac{\text{d}a^{\ast }}{\text{d}t}\frac{a^{-d_{G}}}{1+d_{G}}. & & \displaystyle\end{eqnarray}$ (16) Inserting Eqs (13) and (14) into Eq. (7) yields: $\begin{eqnarray}\displaystyle \frac{\text{d}a^{\ast }}{\text{d}t}\frac{a^{-d_{G}}}{1+d_{G}}=C_{0}K^{\ast q}a^{d_{G}q/2}. & & \displaystyle\end{eqnarray}$ (17) Analogously to fractal fatigue models [9–12,28], if the following scale-invariant relationship between fractal quantities is assumed for creep crack growth: $\begin{eqnarray} \displaystyle \frac{\text{d}a^{\ast }}{\text{d}t}=C_{0}^{\ast }K^{\ast q}, & & \displaystyle \end{eqnarray}$ (18) Equation (15) reduces to: $\begin{eqnarray}\displaystyle C_{0}(a)=\frac{C_{0}^{\ast }}{(1+d_{G})}a^{-d_{G}(1+q/2)}, & & \displaystyle\end{eqnarray}$ (19) which returns by another route the crack-size dependency of the coefficient C ₀ as in Eqs (10) and (11b). Comparing Eqs (11b) and (17) gives 𝛾₂ = −d _G(1 + q∕2). Note that Eq. (17) predicts a negative scaling of C ₀ due to fractal roughness of the crack profile (i.e., for d _G > 0). Accordingly, the creep crack growth rate da∕dt in Eq. (7) is a decreasing function of the crack length as pointed out by the aforementioned experimental results. It should be noted that by combining the two scaling behaviours of da∕dt and K with respect to the crack size a —see Eqs (13) and (14)— it is possible to predict the right- and downward shift of CCGR curves.

The experimental CCGR curves (with da∕dt in mm h⁻¹ and K in MPa m^1∕2) are renormalized by a transformation of the axes K and da∕dt of the form, K → K ^∗ = Ka ^−d
_G∕2 and da∕dt → da ^∗∕dt = (1 + d _G)a ^{d
_G}da∕dt, where a is equal to 3 and 15 mm. The renormalized data collapse onto a single line (shown in Fig. 4), illustrating the fulfilment of the crack-size independent creep crack growth law given by Eq. (16). The dimensional increment d _G of the crack profile is adjusted to achieve the best data collapse, yielding d _G = 0.58 ± 0.09 and, contextually, the following scale-invariant law: $\begin{eqnarray}\displaystyle \frac{\text{d}a^{\ast }}{\text{d}t}=10^{-25.8}K^{\ast 14.4}. & & \displaystyle\end{eqnarray}$ (20)

6. Discussion and conclusions

Experimental data about creep crack propagation for specimens with a pre-existing sharp defect are re-examined, revealing a clear crack-size effect on cracking rates when correlated to the elastic stress-intensity factor, K. These findings are formalized in the framework of dimensional analysis and intermediate asymptotics, leading to a generalized formulation of the creep crack growth law, da∕dt = C ₀ K ^q, where the effects of the initial crack length as well as multiple parameters are taken into account. An application of the fractal geometry leads to the definition of a scale-invariant law, da ^∗∕dt = C ₀ ^∗ K ^∗q, able to uniquely characterize the creep crack growth behaviour of the material, regardless of the size scale. Similarly to fatigue crack propagation (where analogous scaling laws are obtained by the change of variable t → N, i.e., time → number of cycles), fractal roughness of the crack surfaces—through the dimensional increment d _G—is predicted to be responsible for the negative scaling of C ₀, whereby short cracks propagate at higher rates according to the experimental evidence.

Fig. 4.

Collapse of the renormalized CCGR data onto the fitting line represented by Eq. (18) in the fractal K* vs da*/dt diagram.

Assuming for K _th the same scaling effect as for K _C and K, $K_{th}=K_{th}^{\ast }a^{d_{G}/2}$ , the shift of the nominal CCGR curves with crack size, as represented in Fig. 5(a), is obtained. Then, the collapse of the renormalized curves onto a Paris-type scaling function (solely material and temperature dependent) of Fig. 5(b) is an obvious consequence.

Fig. 5.

Predicted nominal CCGR curves (a) and renormalized curve in the fractal diagram (b), where v _th (or $v_{th}^{\ast }$ ) and v _cr (or $v_{cr}^{\ast }$ ) denote threshold and critical CCGRs given by Eq. (14). The dashed line represents the region to be carefully investigated.

Extensive experimental work has to be conducted to confirm the present fractal approach. In particular, the investigation of a threshold limit, below which creep crack growth does not occur (see dashed branch of Paris-type curves in Fig. 5), is related to the ability to inspect smaller and smaller cracks. Further studies could also concern the application of fractal geometry concepts to interpret crack-size effects on creep crack growth rate in terms of the C*-integral.

Footnotes

Conflict of interest

None to report.

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