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A multifractal analysis of fatigue crack growth and its application to concrete
Engineering Fracture Mechanics 77 (2010) 974–984
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A multifractal analysis of fatigue crack growth and its application to concrete Andrea Carpinteri *, Andrea Spagnoli, Sabrina Vantadori Department of Civil and Environmental Engineering & Architecture, University of Parma, Viale Usberti 181/A, 43100 Parma, Italy
a r t i c l e
i n f o
Article history: Received 31 August 2009 Received in revised form 18 December 2009 Accepted 31 January 2010 Available online 12 February 2010 Keywords: Size effect Fatigue and fracture Crack growth Fractal geometry Concrete Crack growth rate Fatigue crack propagation
a b s t r a c t As is well-known, strength of materials is influenced by the specimen or structure size. In particular, several experimental campaigns have shown a decrease of the material strength under static or fatigue loading with increasing structure size, and some theoretical arguments have been proposed to interpret such a phenomenon. As far as fatigue crack growth is concerned, limited information on size effect is available in the literature, particularly for so-called quasi-brittle materials like concrete. In the present paper, by exploiting concepts of fractal geometry, some definitions of fracture energy and stress intensity factor based on physical dimensions different from the classical ones are discussed. A multifractal sizedependent fatigue crack growth law (expressing crack growth rate against stress intensity factor range) is proposed and used to interpret relevant experimental data related to concrete. Ó 2010 Elsevier Ltd. All rights reserved.
1. Introduction The nominal strength and fracture toughness of a structural material have experimentally been observed to change by scaling the specimen size. Therefore, such mechanical properties cease to be material constants, and this phenomenon is called size effect. In particular, tensile strength [1] and fatigue strength [2] usually decrease by increasing the specimen (or structure) size. Theoretical scaling laws describing the variation of material strength against a characteristic structure size have been proposed. Some of the scaling law parameters are normally determined by the fitting of experimental data. Scale ranges in laboratory tests are generally small and, therefore, it is crucial for a scaling law to correctly extrapolate the strength of large structures for which the scale jump from the laboratory specimens may span several orders of magnitude. Size effect on material strength under static loading has extensively been analysed. For example, Weibull [3] proposed the statistical concept of the weakest link in a chain, that is, by increasing the structure volume, the probability of failure increases owing to the higher probability of finding a critical microcrack provoking macroscopic fracture. More recently, a theoretical approach presented in Refs. [4–6] considers the fractal nature of the material microstructure [7,8] and the renormalization group theory [9–11] to explain size effect on both tensile strength and fracture toughness. The reacting cross section (ligament) of a given structure shows a self-similar weakening due to the material heterogeneity, cracks, defects, etc. [12]. Such a surface may be modelled through a lacunar fractal set, which presents a dimension lower than that of the Euclidean domain where it is contained. Then, a new definition of tensile strength with physical dimensions depending on the fractal dimension of the ligament (renormalization procedure) has been proposed [4]. On the other hand, * Corresponding author. Fax: +39 0521 905924. E-mail address: [email protected] (A. Carpinteri). 0013-7944/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.engfracmech.2010.01.019
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Nomenclature a, a0 an a C C1 d D E f G G* lch KI KIC K I K IC m M Mn M(a) N Nn R We Ws
a b1 dn k n
Euclidean (projected) crack length or semi-length crack length or semi-length at the nth scale of observation renormalized crack length or semi-length material constant of the Paris–Erdogan law (see Eq. (18)) coefficient of the size-dependent fatigue crack growth law (see Eqs. (22b) and (26)), function of the characteristic structure size D fractal dimensional increment with respect to the Euclidean space, 0 6 d 6 1 characteristic structure size Young’s modulus parameter of the proposed multifractal relationship (see Eqs. (25) and (26)) nominal fracture energy renormalized fracture energy parameter of the proposed multifractal relationship (see Eqs. (23), (25), (26)) stress intensity factor for Mode I critical stress intensity factor (fracture toughness) for Mode I renormalized stress intensity factor for Mode I renormalized critical stress intensity factor (fracture toughness) for Mode I material constant of the Paris–Erdogan law (see Eq. (18)) measurement at the scale dn , for dn ? 0 measurement at the scale dn a-dimensional Hausdorff measure number of fatigue cycles number of segments of length dn correlation coefficient of best-fitting elastic strain energy release energy dissipated at the crack surface dimension of a fractal set in a two-dimensional analysis (a = 1 ± d), 0 6 a 6 2 exponent of the size-dependent fatigue crack growth law (see Eq. (21b)), function of the fractal dimensional increment d scale of observation of the fractal set scale factor relative crack depth
by treating a fracture surface as an invasive fractal set (i.e. with a dimension higher than that of the Euclidean domain where it is contained), a renormalized fracture toughness has been determined [4,5]. Mandelbrot [13] pointed out a non-uniform (multifractal) scaling of the natural fractals, different from the uniform one of the mathematical fractals, i.e. in the physical reality a transition occurs from a fractal regime for small structures to a Euclidean one for structures large enough with respect to a characteristic material length. In other words, the effect of the microstructural disorder (heterogeneity and/or micromechanical damage) of a given material on the macroscopic mechanical behaviour gradually vanishes by increasing the structure size [6]. As is well-known, experimental data in fatigue are typically expressed in terms of S–N curves. Therefore, size effect is described by the scaling of either finite-life fatigue strength or fatigue limit. Fatigue limit of materials decreases with increasing specimen size, although the testing method adopted (reversed direct stress, reversed plane bending, rotating bending, etc.) significantly influences such a decrease. Two theoretical arguments currently accepted to interpret size effect in fatigue are as follows: statistical theory of random strength [3] and stress gradient [14–16]. The present authors have recently discussed a theoretical explanation of size effect on fatigue limit based on fractal concepts, i.e. modelling the ligament of a structure through a lacunar fractal set, and a monofractal scaling law and a multifractal scaling law have been proposed [17]. For a crack subjected to fatigue loading, size effect during its growth as well as at its threshold condition of fatigue growth can be observed. Such size effect-related phenomena have been analysed using concepts of fractal geometry in Refs. [18,19] and Ref. [20] for fatigue crack growth and fatigue threshold condition, respectively. Furthermore, by applying dimensional analysis and similarity concepts, an interpretation of size effect on fatigue crack growth of metals has been proposed in Refs. [21–23]. In the present paper, crack surfaces are modelled through fractal geometry concepts, that is, size effect in fatigue is explained by considering the fractal nature of the material microstructure, and non-classical definitions of both fracture energy and stress intensity factor are discussed. Then, after reviewing a monofractal size-dependent crack propagation law (see Refs. [18,19]), a new multifractal size-dependent crack propagation law is proposed. Finally, such a law is used to theoretically interpret some relevant experimental crack growth data related to concrete [24–26].
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2. Fracture energy and stress intensity factor for self-similar fractal cracks Now some basic concepts of fractal geometry, extensively discussed in Refs. [7,8], are shortly reviewed. A Euclidean domain presents an integer dimension (for instance, [L]1 for a curve, [L]2 for a surface, [L]3 for a solid), while a fractal domain is a mathematical set with a non-integer (or fractal) dimension a. To describe the meaning of such a fractal dimension a, let us consider a plane curve, the length of which might approximately be measured by means of a number Nn of linear segments of length dn. The corresponding measured length (the so-called measurement at the scale dn) is Mn = Nn dn. Such a measure Mn is independent of dn for a straight line, while Mn asymptotically converges towards a positive finite value, M, for dn ? 0 in the case of a generic Euclidean curve. On the contrary, when a fractal curve is considered, the following law holds [7,8]:
Mn / dd n
ð1Þ
where d = a 1 is the fractal dimensional increment or decrement with respect to the Euclidean space. Note that for a > 1 (corresponding to so-called invasive fractals) Mn ? 1 for dn ? 0, while for a < 1 (corresponding to so-called lacunar fractals) Mn ? 0 for dn ? 0. According to the Euclidean geometry, the measure M of a smooth curve is equal to its length, the measure of a plane surface is given by its area, whereas the volume represents the measure of a non-spongy three-dimensional domain. On magnification of a Euclidean domain V by a scale factor k, the length of a curve, the area of a surface and the volume of a threedimensional solid are multiplied by k, k2 , k3 , respectively, that is Mðk VÞ ¼ kk MðVÞ, where k ¼ 1; 2; 3. On the other hand, for a fractal set V having a dimension a, the following scale invariance property applies:
MðaÞ ðk VÞ ¼ ka M ðaÞ ðVÞ
ð2Þ
where M(a)(V) is the a-dimensional Hausdorff measure of V [8,9]. As an example of a fractal object, the well-known invasive von Koch curve V (Fig. 1), obtained by replacing, at each step, the middle-third with the other two sides of an equilateral triangle based on the replaced middle-third, consists of four subsets geometrically similar to V but scaled by a factor k = 1/3. Consequently, the fractal dimension a of V is obtained as follows:
MðaÞ ðVÞ ¼ 4ðð1=3Þa MðaÞ ðVÞÞ that is,
1 ¼ 4ð1=3Þa ! a ¼ ln 4= ln 3 ¼ 1:262
ð3Þ
Therefore, the dimensional increment d due to its densifying nature is equal to 0.262. Now examine a two-dimensional crack problem, by pointing out that generalisation to three dimensional cases does not affect the ensuing arguments provided that a fractal dimension of the crack surface can be defined. From a macroscopic point of view, the crack being considered resembles a straight cut of length a (Euclidean or projected size). By treating the crack as a deterministic self-similar invasive fractal curve, such as the von Koch curve in Fig. 1, its morphology appears to be similar at different scales of observation (or, in other words, at different steps in the fractal generation procedure). At the nth scale of
E0
E1
E2
E3
En Fig. 1. Mathematical self-similar fractal domain: the invasive von Koch curve.
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observation, the crack length is equal to an, so that the length of the fractal crack at the 0th scale of observation is equal to its Euclidean size, i.e. a0 = a. Obviously, the applicability of a self-similar fractal model is confined to the so-called mesoscopic scale range of the material, defined by an upper limit, corresponding to the geometrical dimensions of the cracked body, and a lower limit, related to the microstructure of the medium (e.g. aggregate size for concrete, grain size for metals) [6]. At the nth scale of observation of the above fractal crack, the total energy Ws dissipated at the surface of the crack in a plate of unit thickness is given by:
W s ¼ Gn a n
ð4Þ
where Gn is the fracture energy at the nth scale of observation. A new definition of fracture energy has been proposed [4] in order to obtain a material property, the so-called scale-invariant (renormalized) fracture energy G, which is independent of the scale of observation. Accordingly, the total energy dissipated at the surface of the crack can be written as a function of G:
W s ¼ G a
ð5Þ a
1+d
where a = a is the fractal length of the crack, having physical dimensions equal to [L] , with a = 1 + d. By applying the dimensional analysis to Eq. (5), the following non-integer physical dimensions of the scale-invariant (renormalized) fracture energy G can be determined:
½G ¼ ½F½Lð1þdÞ
ð6Þ
Note that, for the limit case of d = 0 (Euclidean cracks), the physical dimensions of G are the conventional ones ([G] = [F]][L]1), while G has physical dimensions ½G ¼ ½F½L2 for the limit case of d = 1. Quasi-brittle materials present an intermediate behaviour characterised by 0 < d < 1. By analysing geometrically similar bodies containing a fractal crack through a self-similar fractal approach (also called monofractal approach, i.e. the fractal dimension a is constant at any scale of observation), a power scaling law for fracture energy has been derived [4]. For instance, let us consider the two cracked bodies of unit thickness in Fig. 2, made up of the same material, with characteristic structure sizes DA = 1 and DB = D, and Euclidean crack lengths aA = n1 and aB = nD, respectively, where n is the relative crack depth. Since the renormalized fracture energy G is a material constant, we can write:
G ¼
W ðAÞ s 1þd
ðn1Þ
¼
W ðBÞ s
ð7Þ
ðnDÞ1þd
ðBÞ where W ðAÞ s and W s are the energies dissipated at the surface of the crack in the two bodies A and B, respectively. The nominal fracture energies for the two bodies are equal to:
GðAÞ ¼
W ðAÞ s n1
ð8aÞ
GðBÞ ¼
W ðBÞ s nD
ð8bÞ
Using Eq. (7), Eqs. (8) yield the following scaling law [4]:
GðBÞ ¼ GðAÞ Dd
ð9Þ
Now consider the well-known Griffith’s energetic approach to the problem of an infinite plate of unit thickness, containing a crack of Euclidean length 2a and subjected to a remote tensile stress r (Fig. 3) [4]. The elastic strain energy release caused by the crack depends on its Euclidean length, that is:
We ¼
pr2 a2
ð10Þ
E
aA = ξ
DA = 1
L
DB = D aB = ξ D
(D/1)L Fig. 2. Two-dimensional geometrically similar bodies containing a crack: the typical case of three-point bend specimens.
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Fig. 3. Griffith’s problem for fractal cracks (in shadow is the zone of elastic strain energy release).
On the other hand, the energy dissipated at the surface of the fractal crack can be expressed using the renormalized fracture energy G (see Eq. (5)):
W s ¼ 2G a
ð11Þ
Consequently, the variation dWe of elastic strain energy release is defined with respect to the Euclidean crack length (2a), while the variation dWs of energy dissipation refers to the fractal crack length (2a). Differentiating Eqs. (10) and (11) against a and a, respectively, we obtain:
2pr2 a da E dW s ¼ 2G da ¼ 2ð1 þ dÞG ad da
dW e ¼
ð12aÞ ð12bÞ
The energy balance dWe = dWs yields :
r2 pa1d ¼ ð1 þ dÞG E
ð13Þ
An extension to fractal cracks of the classical definition of stress intensity factor (SIF) KI of Westergaard [27] for the probpffiffiffiffiffiffi lem under pffiffiffiffiffiffi consideration (K I ¼ r pa), and that of critical SIF (fracture toughness) KIC of Irwin [28] for plane stress states (K IC ¼ GE) has been proposed in Ref. [4], and the following renormalized quantities have been defined: d
K I ¼ rðpa1d Þ1=2 ¼ K I a2
ð14aÞ
K IC ¼ ½ð1 þ dÞG E1=2
ð14bÞ
so that Eq. (13) reads: 2 K 2 I ¼ K IC
ð15Þ
where the renormalized
½K I ¼ ½F½L
K I
and
K IC
have the following physical dimensions:
3þd 2
ð16Þ
For a general crack problem, Eq. (14a) should be written as follows:
K I ¼ Y rðpa1d Þ1=2
ð17Þ
where Y is a geometrical and loading dimensionless factor, and r is the nominal tensile stress perpendicular to the crack. Accordingly, the stress field ahead of the crack tip is proportional to K I (instead of KI). 3. A size-dependent fatigue crack growth law within the framework of monofractal geometry The well-known Paris–Erdogan law [29] describes the kinetics of crack propagation in the intermediate range of DKI:
da ¼ CðDK I Þm dN
ð18Þ
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where C and m are material constants, N is the number of fatigue loading cycles; da/dN and DKI are the crack propagation rate and the SIF range, respectively. The crack growth rate has experimentally been observed to be dependent on the size of specimens or structural components. Therefore, exploiting the renormalized quantities related to fractal cracks (whose surfaces can be modelled as invasive fractals, see previous Sections), the following size-dependent fatigue crack growth law has been conceived [18,19]:
da ¼ CðDK I Þm dN
ð19Þ
Since Eq. (19) is expressed through the renormalized quantities a and K I , it turns out to be scale-invariant. A scaling law can be obtained from Eq. (19) by rewriting such an equation in terms of the nominal crack propagation rate da/dN and the nominal SIF range DKI . As a matter of fact, the crack propagation rate da/dN of the fractal crack can be computed, by also recalling that a = a1+d:
da da da da ¼ ð1 þ dÞad ¼ dN dN da dN
ð20Þ
By employing Eqs. (14a) and (20), Eq. (19) becomes:
da C m ¼ ab1 DK m I ¼ C 1 ðaÞðDK I Þ dN 1 þ d
ð21aÞ
where
b1 ¼ dð1 þ
m Þ 2
ð21bÞ
For geometrically similar cracked bodies (e.g. see Fig. 2), a is proportional to D (i.e. a = nD). Consequently, Eq. (21a) becomes:
da ¼ C 1 ðDÞðDK I Þm dN
ð22aÞ
where
C 1 ðDÞ ¼
m C ðnDÞdð1þ 2 Þ 1þd
ð22bÞ
is a function of the structure size D. Therefore, Eq. (22a) represents a size-dependent fatigue crack growth law. Such an equation is formally similar to the classical Paris–Erdogan law (Eq. (18)), but the coefficient C1 depends on D, whereas C in Eq. (18) is assumed to be a material constant. Eq. (22a) represents a monofractal approach to size effect on fatigue crack growth. Consequently, it can appropriately be applied to a limited range of fatigue crack lengths (in the long crack fatigue growth regime). On the other hand, a multifractal approach would be more suitable to also model the propagation of fatigue cracks long with respect to the microstructural size (these cracks seem to follow the classical Paris–Erdogan law which is independent of the structure size D), since such an approach could describe a transition from a fractal regime (dependent on D) to a Euclidean one (independent of D) [6]. Therefore, a multifractal approach is analysed in the following Section. 4. A size-dependent fatigue crack growth law within the framework of multifractal geometry In the physical reality, a transition occurs from a fractal regime for small structures to a Euclidean one for structures large enough with respect to a characteristic material length, as was pointed out by Mandelbrot [13]. In other words, the effect of the microstructural disorder (heterogeneity and/or micromechanical damage) of a given material on the macroscopic mechanical behaviour gradually vanishes by increasing the structure size [6]. In line with a multifractal hypothesis, the fractal increment d can be assumed to vary with the scale of observation of the f with f = material fractal. In the following the renormalized crack length is assumed to be equal to a = a1+d, where d ¼ f þa parameter. In this way, the fractal increment d is a function of the crack length a and varies between 1 (for a ? 0) and 0 (for a ? 1). Moreover, the SIF range to be introduced in Eq. (19) can be written as follows:
DK I ¼ DK I ð1 þ
lch 1=2 Þ a
ð23Þ
where lch is a material parameter. The definition of Eq. (23) is in line with the multifractal expression for fracture energy proposed in Ref. [6]. Moreover, such an equation is reminiscent of other equations dealing with non-propagating fatigue cracks which include a material parameter in the SIF expression (e.g. see Ref. [30]). In order to derive a multifractal size-dependent fatigue crack growth law, the same thread of logic of the previous Section is followed. The chain rule for the derivation of a with respect to N now yields: f af ln a þ ðf þ aÞð2f þ aÞ da da da da ¼ af þa ¼ dN dN da dN ðf þ aÞ2
ð24Þ
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(a) Log da/dN 1 m m
C1 (D A ) 1 C1 (D B ) DB > DA
Log ΔKI
(b)
Log C1 multifractal
1
|β1| monofractal
Log D Fig. 4. Size effect in fatigue crack growth: (a) da/dN–DKI curves (see Eq. (22a)) and (b) C1–D curves (see Eqs. (22b) and (26)).
Thus, by employing Eqs. (23) and (24), Eq. (19) becomes:
da ¼ C 1 ðaÞðDK I Þm dN
ð25aÞ
where
C 1 ðaÞ ¼ C
m=2 lch 1þ a a ðaf ln a þ ðf þ aÞð2f þ aÞÞ ðf þ aÞ2
f f þa
ð25bÞ
For geometrically similar cracked bodies (e.g. see Fig. 2), a is proportional to D (i.e. a = nD). Consequently, Eq. (25a) can formally be written as a function of the structure size D (see Eq. (22a)). In this way a multifractal size-dependent fatigue crack growth law is obtained, where the coefficient C1 depends on D according to the following relationship:
C 1 ðDÞ ¼ C
m=2 lch 1 þ f nD ðnDÞf þnD ðnDf ln nD þ ðf þ nDÞð2f þ nDÞÞ ðf þ nDÞ2
ð26Þ
Note that, in comparison with Eq. (22b), Eq. (26) depends on two further material parameters, f and lch. Eq. (22a) demonstrates a size effect on fatigue crack propagation. In more details, since C1 depends on the structure size D (see Eqs. (22b) and (26) for monofractal and multifractal approach, respectively), the crack growth rate da/dN in turn depends on D. Typically the crack growth rate da/dN decreases with increasing D. If we assume that the relative crack depth n in Eqs. (22b) and (26) is approximately constant during crack propagation, Eq. (22a) describes a linear relationship in the bilogarithmic plane da/dN–DKI (Fig. 4a). In Fig. 4b two sample curves of the monofractal (Eq. (22b)) and multifractal (Eq. (26)) relationships are reported in the bilogarithmic plane C1–D. It can be noted that the curve of Eq. (26) is non-linear in such a plane. 5. Theoretical interpretation of some experimental fatigue data Some experimental data, related to fatigue crack propagation in normal strength (NS) concrete [24,25] and high strength (HS) concrete [26], are here employed to show an application of the proposed scaling laws (see Eqs. (22b) and (26)). Two series of three-point bend NS plain-concrete specimens were tested under fatigue loading [24,25]. Each series consisted of three two-dimensional geometrically similar cracked specimens (see Fig. 2). The beam height (characteristic structure size) was equal to DA = 38 mm, DB = 76 mm, DC = 152 mm, respectively. The span L of the beams was 2.5 times the height D, the initial length of the crack was equal to D/6, and the beam thickness was equal to 38 mm. The ratio of cement:sand:aggregate:water in the concrete mix was 1:2:2:0.6 by weight. The aggregate was crushed limestone of maximum size 12.7 mm. The sand was siliceous river sand passing through 5-mm sieve. Type I Portland cement with no admixtures was used. After 28-day curing, the mean compression strength of cylindrical specimens was equal to 32.8 MPa.
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1E-002
DA = 38 mm DB = 76 mm
da /dN (m/cycle)
1E-003
DC = 152 mm
1E-004
1E-005
1E-006
0.4
0.5
0.6 0.7 0.8 0.9
2
1
ΔKI (MPa√m) Fig. 5. Crack growth da/dN–DKI data for NS concrete and different specimen sizes [25]: dashed regression lines refer to best-fit slopes (mA = 12.184, = 10.395). mB = 9.975, mC = 9.025), while continuous regression lines refer to an average slope (m
(a) C1 [m/(MPa√m)10.395]
1E-002 1E-003 1E-004
⏐ 1⏐
1E-005 1E-006 1E-002
C1 [m/(MPa√m)8.155]
(b)
1E-003 1E-004
⏐ 1⏐
1E-005 1E-006 20
40
60
80
100
200
400
D (mm) Fig. 6. Best-fitting of C1(D) to determine b1 (Eq. 22b) and the fractal increment d (Eq. 21b): (a) b1 = 4.697 and d = 0.758 for NS concrete [25] and (b) b1 = 1.358 and d = 0.267 for HS concrete [26].
The crack growth rate against SIF range data (corresponding to 37, 36 and 29 experimental points for DA, DB and DC, respectively) are presented in Fig. 5 along with regression lines based on Eq. (22a) (the experimental data are taken from Ref. [25]). Dashed lines refer to best-fit parameters m (mA = 12.184, mB = 9.975, mC = 9.025) and C1 (Log C1(DA) = 1.603, Log C1(DB) = 3.666, Log C1(DC) = 4.818). The correlation coefficient R is equal to 0.865, 0.909 and 0.912 for DA, DB and DC, respectively. = 10.395) is conSince the variation of the best-fit slope m with respect to the structure size is small, an average slope (m sidered in the following, so that the new best-fitting yields Log C1(DA) = 2.058 (R = 0.846), Log C1(DB) = 3.632 (R = 0.907), Log C1(DC) = 4.886 (R = 0.891) (see continuous lines in Fig. 5). Therefore, according to the power law expressed in Eq. (22b), the exponent b1 can be determined by plotting C1 against D and assuming the relative crack depth constant and equal to its initial value D/6 [18], as is shown in Fig. 6a (b1 = 4.697, R = 0.996). Consequently, the corresponding value of the fractal = 10.395 and b1 = 4.697). increment d results to be equal to 0.758 from Eq. (21b) (for m ¼ m
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1E-002
da /dN (m/cycle)
1E-003
1E-004
1E-005
DA = 38 mm DB =108 mm DC = 304 mm
1E-006 0.8 0.9
1
2
3
ΔKI (MPa√m)
Fig. 7. Crack growth da=dN DK I data for HS concrete and different specimen sizes [26]: dashed regression lines refer to best-fit slopes (mA = 8.444, = 8.155). mB = 8.603, mC = 7.417), while continuous regression lines refer to an average slope (m
C1 [m/(MPa√m)10.395]
(a)
1E-001
monofractal
1E-002
multifractal
1E-003 1E-004 1E-005 1E-006 1E-007
C1 [m/(MPa√m)8.155]
(b)
1E-001
monofractal 1E-002
multifractal
1E-003 1E-004 1E-005 1E-006 1E-007 30
50
70 90 100
200
400 600 800 1000
D (mm) Fig. 8. Interpolation of experimental C 1 data using the multifractal relationship in Eq. (26) for: (a) NS concrete [25] and (b) HS concrete [26]. The monofractal relationship in Eq. (22b) is also reported for comparison.
Furthermore, one series of three-point bend HS plain-concrete specimens was tested [26], consisting of three two-dimensional geometrically similar cracked specimens (see Fig. 2). The beam height was equal to DA = 38 mm, DB = 108 mm and DC = 304 mm, respectively. The span and the thickness of the beams and the relative initial crack length were equal to those for NS concrete tests, previously discussed. The ratio of the mix components to cement by weight was as follows: 1.00 Portland cement, 0.316 water, 0.132 fly ash, 0.0507 silica fume, 2.18 maximum-diameter crushed aggregate, 1.51 siliceous sand, 0.0019 retarder, 0.00951 superplasticizer. The maximum size of the aggregate was equal to 9.5 mm. At the beginning of the fatigue tests, the mean compression strength of cylindrical specimens was equal to 90.3 MPa. The crack growth rate against SIF range data [26] are reported in Fig. 7 along with regression lines based on Eq. (22a) (17, 16 and 12 experimental points for DA, DB and DC, respectively). Dashed lines refer to best-fit parameters m (mA = 8.444, mB = 8.603, mC = 7.417) and C1 (Log C1(DA) = 4.719, Log C1(DB) = 5.561, Log C1(DC) = 5.720). The correlation coefficient
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R is equal to 0.965, 0.905 and 0.761 for DA, DB and DC, respectively. Similar to NS concrete data of Refs. [24,25], a small variation of the best-fit slope m with respect to the structure size is observed also for HS concrete, and an average slope = 8.155) can be considered, so that the new best-fitting yields Log C1(DA) = 4.690 (R = 0.964), Log C1(DB) = 5.481 (m (R = 0.902), Log C1(DC) = 5.916 (R = 0.753) (see continuous lines in Fig. 7). Then, the exponent b1 (in Eq. (22b)) can be determined by plotting C1 against D and again by assuming the relative crack depth constant and equal to its initial value D/6 [18], as is shown in Fig. 6b (b1 = 1.358, R = 0.974). Finally, from Eq. (21b), the fractal increment d results to be equal to 0.267 (for = 8.155 and b1 = 1.358). m¼m By analysing the obtained values of the fractal increment d for NS and HS concrete mixes, it seems that d tends to increase with increasing maximum size dmax of aggregate (NS concrete: dmax = 12.7 mm, d = 0.758; HS concrete: dmax = 9.5 mm, d = 0.267) [18,19]. Such a result is consistent from a physical point of view, that is to say, the invasive properties (d) are connected to the microstructural properties (e.g. dmax). In other words, a concrete with a coarse microstructure (i.e. large value of dmax) presents a fracture surface dimension much greater than 2 (i.e. d is large), while a fine concrete (i.e. small value of dmax) shows a fracture surface which is almost flat (i.e. d is small). Experimentally, this trend of behaviour has been observed by performing a digital scanning and analysis of the fracture surfaces for different concrete mixes under static loading [31]. Now, within the framework of a multifractal approach, Eq. (26) is used to describe the experimental trend of behaviour of C1(D). Assuming the relative crack depth constant and equal to its initial value D/6, the three parameters C, f and lch appearing in Eq. (26) can be computed through a non-linear best-fitting procedure. As a matter of fact, three experimental values C1(D) are available for each (NS and HS) concrete series (see the C1(D) values previously mentioned in the present Section), and therefore C, f and lch can actually be worked out by solving a 3 3 system of non-linear equations (see Eq. (26)). The obtained pffiffiffiffiffi results are as follows: for NS concrete C = 2.415 107 (for DKI expressed in MPa m and da/dN in m/cycle), f = 0.0259 m pffiffiffiffiffi 7 (for DKI expressed in MPa m and da/dN in m/cycle), and lch = 0.0199 m (Fig. 8a); for HS concrete C = 9.743 10 f = 0.0188 m and lch = 0.000396 m (Fig. 8b). It is worth remarking that, since the system under consideration is non-linear, the solution is not unique: for each concrete mix, the results reported above correspond to a solution of the system for positive values of C, f and lch. It can be noted that, according to the multifractal approach, the size effect in the crack growth rate tends, conversely to the trend described by the monofractal approach (see Fig. 8, where the monofractal C1–D curves are also reported for comparison), to reduce as the structure size increases beyond a certain threshold (the C1–D curve approaches an horizontal asymptote for D ? 1). Furthermore, it can be noted that lch increases with increasing the characteristic length of material microstructure (e.g. the maximum size of aggregate, which is larger in the NS concrete as compared to that in the HS concrete). A similar trend of lch against a characteristic length of material microstructure is reported in Ref. [6]. 6. Conclusions By modelling a crack surface as an invasive fractal set, a new definition of fracture energy is required to obtain a parameter (renormalized fracture energy) which is independent of the scale of observation of the fractal set and can be considered as a material constant. Then, by investigating the Griffith’s problem within the framework of fractal cracks, a renormalized SIF based on physical dimensions different from the classical ones can be obtained. In the present paper, such a SIF is firstly introduced in the classical Paris–Erdogan law for fatigue crack propagation. This leads to size effects whose physical source is hence related to the fact that, on one side, the energy consumption during crack propagation occurs on the crack surface which is reasonably well described by a fractal domain whereas, on the other side, the energy release occurs in the bulk volume of the material, that is, in a classical Euclidean domain. A so-called monofractal size-dependent fatigue crack growth law, which is characterised by a power law (with the exponent dependent on the fractal dimensional increment d) of the structure size D, is proposed. Then, by applying some concepts of multifractal geometry (the fractal dimension is made to vary with the scale of observation of the fractal), a new definition of renormalized crack length and SIF is put forward. This yields a so-called multifractal size-dependent fatigue crack growth law. Both the presented fatigue crack growth laws are herein used to theoretically interpret some experimental data related to concrete. Acknowledgements The authors gratefully acknowledge the research support for this work provided by the Italian Ministry for University and Technological and Scientific Research (MIUR). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]
Griffith AA. The phenomenon of rupture and flow in solids. Philos Trans Roy Soc 1921;A221:163–98. Peterson RE. Model testing as applied to strength of materials. J Appl Mech 1933;1:79–85. Weibull W. A statistical theory for the strength of materials. Stockholm: Swedish Royal Institute for Engineering Research; 1939. Carpinteri Al. Scaling laws and renormalization groups for strength and toughness of disordered materials. Int J Solids Struct 1994;31:291–302. Carpinteri Al, Chiaia B. Crack-resistance behavior as a consequence of self-similar fracture topologies. Int J Fract 1996;76:327–40. Carpinteri Al, Chiaia B. Multifractal scaling laws in the breaking behaviour of disordered materials. Chaos Soliton Fract 1997;8:135–50. Mandelbrot BB. The fractal geometry of nature. New York: W.H. Freeman and Company; 1982. Falconer K. Fractal geometry: mathematical foundations and applications. Chichester: Wiley; 1990. Wilson KG. Renormalization group and critical phenomena. Phys Rev 1971;B4:3174–205. Barenblatt GI. Similarity, self-similarity and intermediate asymptotics. New York: Consultants Bureau; 1979.
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A. Carpinteri et al. / Engineering Fracture Mechanics 77 (2010) 974–984 Herrmann HJ, Roux S, editors. Statistical models for the fracture of disordered media. Amsterdam: North-Holland; 1990. Mandelbrot BB, Passoja DE, Paullay AJ. Fractal character of fracture surfaces of metals. Nature 1984;308:721–2. Mandelbrot BB. Self-affine fractals and fractal dimension. Phys Scr 1985;32:257–60. Neuber H. Theory of notch stresses. Berling: Springer-Verlag; 1958. Peterson RE. Notch sensitivity. In: Sines G, Waisman JL, editors. Metal fatigue. New York: McGraw Hill; 1959. p. 293–306. Findley WN. An explanation of size effect in fatigue of metals. J Mech Engng Sci 1972;14:424–5. Carpinteri A, Spagnoli A, Vantadori S. An approach to size effect in fatigue of metals using fractal theories. Fatigue Fract Engng Mater Struct 2002;25:619–27. Carpinteri A, Spagnoli A. A fractal analysis of size effect on fatigue crack growth. Int J Fatigue 2004;26:125–33. Carpinteri A, Spagnoli A, Vantadori S, Viappiani D. Influence of the crack morphology on the fatigue crack growth rate: a continuously-kinked crack model based on fractals. Engng Fract Mech 2008;75:579–89. Spagnoli A. Fractality in the threshold condition of fatigue crack growth: an interpretation of the Kitagawa diagram. Chaos Soliton Fract 2004;22:589–98. Barenblatt GI, Botvina LR. Incomplete self-similarity of fatigue in the linear range of crack growth. Fatigue Fract Engng Mater Struct 1980;3:193–202. Spagnoli A. Self-similarity and fractals in the Paris range of fatigue crack growth. Mech Mater 2005;37:519–29. Carpinteri Al, Paggi M. Self-similarity and crack growth instability in the correlation between the Paris’ constants. Engng Fract Mech 2007;74:1041–53. Bazant ZP, Size KXU. Size effect in fatigue fracture of concrete. ACI Mater J 1991;88:390–9. Bazant ZP, Planas J. Fracture and size effect in concrete and other quasibrittle materials. New York: CRC Press; 1998. Bazant ZP, Shell WF. Fatigue fracture of high-strength concrete and size effect. ACI Mater J 1993;90:472–8. Westergaard HM. Bearing pressure and cracks. J Appl Mech 1939;6:49–53. Irwin GR. Analysis of stresses and strains near the end of a crack traversing a plate. J Appl Mech 1957;24:109–14. Paris PC, Erdogan F. A critical analysis of crack propagation laws. J Basic Engng 1963;85D:528–34. El Haddad MH, Topper TH, Smith KN, Prediction KN. Prediction of non-propagating cracks. Engng Fract Mech 1979;11:573–84. Carpinteri A, Chiaia B, Invernizzi S. Three-dimensional fractal analysis of concrete fracture at the meso-level. Theor Appl Fract Mech 1999;31:163–72.
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Engineering Fracture Mechanics journal homepage: www.elsevier.com/locate/engfracmech
A multifractal analysis of fatigue crack growth and its application to concrete Andrea Carpinteri *, Andrea Spagnoli, Sabrina Vantadori Department of Civil and Environmental Engineering & Architecture, University of Parma, Viale Usberti 181/A, 43100 Parma, Italy
a r t i c l e
i n f o
Article history: Received 31 August 2009 Received in revised form 18 December 2009 Accepted 31 January 2010 Available online 12 February 2010 Keywords: Size effect Fatigue and fracture Crack growth Fractal geometry Concrete Crack growth rate Fatigue crack propagation
a b s t r a c t As is well-known, strength of materials is influenced by the specimen or structure size. In particular, several experimental campaigns have shown a decrease of the material strength under static or fatigue loading with increasing structure size, and some theoretical arguments have been proposed to interpret such a phenomenon. As far as fatigue crack growth is concerned, limited information on size effect is available in the literature, particularly for so-called quasi-brittle materials like concrete. In the present paper, by exploiting concepts of fractal geometry, some definitions of fracture energy and stress intensity factor based on physical dimensions different from the classical ones are discussed. A multifractal sizedependent fatigue crack growth law (expressing crack growth rate against stress intensity factor range) is proposed and used to interpret relevant experimental data related to concrete. Ó 2010 Elsevier Ltd. All rights reserved.
1. Introduction The nominal strength and fracture toughness of a structural material have experimentally been observed to change by scaling the specimen size. Therefore, such mechanical properties cease to be material constants, and this phenomenon is called size effect. In particular, tensile strength [1] and fatigue strength [2] usually decrease by increasing the specimen (or structure) size. Theoretical scaling laws describing the variation of material strength against a characteristic structure size have been proposed. Some of the scaling law parameters are normally determined by the fitting of experimental data. Scale ranges in laboratory tests are generally small and, therefore, it is crucial for a scaling law to correctly extrapolate the strength of large structures for which the scale jump from the laboratory specimens may span several orders of magnitude. Size effect on material strength under static loading has extensively been analysed. For example, Weibull [3] proposed the statistical concept of the weakest link in a chain, that is, by increasing the structure volume, the probability of failure increases owing to the higher probability of finding a critical microcrack provoking macroscopic fracture. More recently, a theoretical approach presented in Refs. [4–6] considers the fractal nature of the material microstructure [7,8] and the renormalization group theory [9–11] to explain size effect on both tensile strength and fracture toughness. The reacting cross section (ligament) of a given structure shows a self-similar weakening due to the material heterogeneity, cracks, defects, etc. [12]. Such a surface may be modelled through a lacunar fractal set, which presents a dimension lower than that of the Euclidean domain where it is contained. Then, a new definition of tensile strength with physical dimensions depending on the fractal dimension of the ligament (renormalization procedure) has been proposed [4]. On the other hand, * Corresponding author. Fax: +39 0521 905924. E-mail address: [email protected] (A. Carpinteri). 0013-7944/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.engfracmech.2010.01.019
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Nomenclature a, a0 an a C C1 d D E f G G* lch KI KIC K I K IC m M Mn M(a) N Nn R We Ws
a b1 dn k n
Euclidean (projected) crack length or semi-length crack length or semi-length at the nth scale of observation renormalized crack length or semi-length material constant of the Paris–Erdogan law (see Eq. (18)) coefficient of the size-dependent fatigue crack growth law (see Eqs. (22b) and (26)), function of the characteristic structure size D fractal dimensional increment with respect to the Euclidean space, 0 6 d 6 1 characteristic structure size Young’s modulus parameter of the proposed multifractal relationship (see Eqs. (25) and (26)) nominal fracture energy renormalized fracture energy parameter of the proposed multifractal relationship (see Eqs. (23), (25), (26)) stress intensity factor for Mode I critical stress intensity factor (fracture toughness) for Mode I renormalized stress intensity factor for Mode I renormalized critical stress intensity factor (fracture toughness) for Mode I material constant of the Paris–Erdogan law (see Eq. (18)) measurement at the scale dn , for dn ? 0 measurement at the scale dn a-dimensional Hausdorff measure number of fatigue cycles number of segments of length dn correlation coefficient of best-fitting elastic strain energy release energy dissipated at the crack surface dimension of a fractal set in a two-dimensional analysis (a = 1 ± d), 0 6 a 6 2 exponent of the size-dependent fatigue crack growth law (see Eq. (21b)), function of the fractal dimensional increment d scale of observation of the fractal set scale factor relative crack depth
by treating a fracture surface as an invasive fractal set (i.e. with a dimension higher than that of the Euclidean domain where it is contained), a renormalized fracture toughness has been determined [4,5]. Mandelbrot [13] pointed out a non-uniform (multifractal) scaling of the natural fractals, different from the uniform one of the mathematical fractals, i.e. in the physical reality a transition occurs from a fractal regime for small structures to a Euclidean one for structures large enough with respect to a characteristic material length. In other words, the effect of the microstructural disorder (heterogeneity and/or micromechanical damage) of a given material on the macroscopic mechanical behaviour gradually vanishes by increasing the structure size [6]. As is well-known, experimental data in fatigue are typically expressed in terms of S–N curves. Therefore, size effect is described by the scaling of either finite-life fatigue strength or fatigue limit. Fatigue limit of materials decreases with increasing specimen size, although the testing method adopted (reversed direct stress, reversed plane bending, rotating bending, etc.) significantly influences such a decrease. Two theoretical arguments currently accepted to interpret size effect in fatigue are as follows: statistical theory of random strength [3] and stress gradient [14–16]. The present authors have recently discussed a theoretical explanation of size effect on fatigue limit based on fractal concepts, i.e. modelling the ligament of a structure through a lacunar fractal set, and a monofractal scaling law and a multifractal scaling law have been proposed [17]. For a crack subjected to fatigue loading, size effect during its growth as well as at its threshold condition of fatigue growth can be observed. Such size effect-related phenomena have been analysed using concepts of fractal geometry in Refs. [18,19] and Ref. [20] for fatigue crack growth and fatigue threshold condition, respectively. Furthermore, by applying dimensional analysis and similarity concepts, an interpretation of size effect on fatigue crack growth of metals has been proposed in Refs. [21–23]. In the present paper, crack surfaces are modelled through fractal geometry concepts, that is, size effect in fatigue is explained by considering the fractal nature of the material microstructure, and non-classical definitions of both fracture energy and stress intensity factor are discussed. Then, after reviewing a monofractal size-dependent crack propagation law (see Refs. [18,19]), a new multifractal size-dependent crack propagation law is proposed. Finally, such a law is used to theoretically interpret some relevant experimental crack growth data related to concrete [24–26].
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2. Fracture energy and stress intensity factor for self-similar fractal cracks Now some basic concepts of fractal geometry, extensively discussed in Refs. [7,8], are shortly reviewed. A Euclidean domain presents an integer dimension (for instance, [L]1 for a curve, [L]2 for a surface, [L]3 for a solid), while a fractal domain is a mathematical set with a non-integer (or fractal) dimension a. To describe the meaning of such a fractal dimension a, let us consider a plane curve, the length of which might approximately be measured by means of a number Nn of linear segments of length dn. The corresponding measured length (the so-called measurement at the scale dn) is Mn = Nn dn. Such a measure Mn is independent of dn for a straight line, while Mn asymptotically converges towards a positive finite value, M, for dn ? 0 in the case of a generic Euclidean curve. On the contrary, when a fractal curve is considered, the following law holds [7,8]:
Mn / dd n
ð1Þ
where d = a 1 is the fractal dimensional increment or decrement with respect to the Euclidean space. Note that for a > 1 (corresponding to so-called invasive fractals) Mn ? 1 for dn ? 0, while for a < 1 (corresponding to so-called lacunar fractals) Mn ? 0 for dn ? 0. According to the Euclidean geometry, the measure M of a smooth curve is equal to its length, the measure of a plane surface is given by its area, whereas the volume represents the measure of a non-spongy three-dimensional domain. On magnification of a Euclidean domain V by a scale factor k, the length of a curve, the area of a surface and the volume of a threedimensional solid are multiplied by k, k2 , k3 , respectively, that is Mðk VÞ ¼ kk MðVÞ, where k ¼ 1; 2; 3. On the other hand, for a fractal set V having a dimension a, the following scale invariance property applies:
MðaÞ ðk VÞ ¼ ka M ðaÞ ðVÞ
ð2Þ
where M(a)(V) is the a-dimensional Hausdorff measure of V [8,9]. As an example of a fractal object, the well-known invasive von Koch curve V (Fig. 1), obtained by replacing, at each step, the middle-third with the other two sides of an equilateral triangle based on the replaced middle-third, consists of four subsets geometrically similar to V but scaled by a factor k = 1/3. Consequently, the fractal dimension a of V is obtained as follows:
MðaÞ ðVÞ ¼ 4ðð1=3Þa MðaÞ ðVÞÞ that is,
1 ¼ 4ð1=3Þa ! a ¼ ln 4= ln 3 ¼ 1:262
ð3Þ
Therefore, the dimensional increment d due to its densifying nature is equal to 0.262. Now examine a two-dimensional crack problem, by pointing out that generalisation to three dimensional cases does not affect the ensuing arguments provided that a fractal dimension of the crack surface can be defined. From a macroscopic point of view, the crack being considered resembles a straight cut of length a (Euclidean or projected size). By treating the crack as a deterministic self-similar invasive fractal curve, such as the von Koch curve in Fig. 1, its morphology appears to be similar at different scales of observation (or, in other words, at different steps in the fractal generation procedure). At the nth scale of
E0
E1
E2
E3
En Fig. 1. Mathematical self-similar fractal domain: the invasive von Koch curve.
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observation, the crack length is equal to an, so that the length of the fractal crack at the 0th scale of observation is equal to its Euclidean size, i.e. a0 = a. Obviously, the applicability of a self-similar fractal model is confined to the so-called mesoscopic scale range of the material, defined by an upper limit, corresponding to the geometrical dimensions of the cracked body, and a lower limit, related to the microstructure of the medium (e.g. aggregate size for concrete, grain size for metals) [6]. At the nth scale of observation of the above fractal crack, the total energy Ws dissipated at the surface of the crack in a plate of unit thickness is given by:
W s ¼ Gn a n
ð4Þ
where Gn is the fracture energy at the nth scale of observation. A new definition of fracture energy has been proposed [4] in order to obtain a material property, the so-called scale-invariant (renormalized) fracture energy G, which is independent of the scale of observation. Accordingly, the total energy dissipated at the surface of the crack can be written as a function of G:
W s ¼ G a
ð5Þ a
1+d
where a = a is the fractal length of the crack, having physical dimensions equal to [L] , with a = 1 + d. By applying the dimensional analysis to Eq. (5), the following non-integer physical dimensions of the scale-invariant (renormalized) fracture energy G can be determined:
½G ¼ ½F½Lð1þdÞ
ð6Þ
Note that, for the limit case of d = 0 (Euclidean cracks), the physical dimensions of G are the conventional ones ([G] = [F]][L]1), while G has physical dimensions ½G ¼ ½F½L2 for the limit case of d = 1. Quasi-brittle materials present an intermediate behaviour characterised by 0 < d < 1. By analysing geometrically similar bodies containing a fractal crack through a self-similar fractal approach (also called monofractal approach, i.e. the fractal dimension a is constant at any scale of observation), a power scaling law for fracture energy has been derived [4]. For instance, let us consider the two cracked bodies of unit thickness in Fig. 2, made up of the same material, with characteristic structure sizes DA = 1 and DB = D, and Euclidean crack lengths aA = n1 and aB = nD, respectively, where n is the relative crack depth. Since the renormalized fracture energy G is a material constant, we can write:
G ¼
W ðAÞ s 1þd
ðn1Þ
¼
W ðBÞ s
ð7Þ
ðnDÞ1þd
ðBÞ where W ðAÞ s and W s are the energies dissipated at the surface of the crack in the two bodies A and B, respectively. The nominal fracture energies for the two bodies are equal to:
GðAÞ ¼
W ðAÞ s n1
ð8aÞ
GðBÞ ¼
W ðBÞ s nD
ð8bÞ
Using Eq. (7), Eqs. (8) yield the following scaling law [4]:
GðBÞ ¼ GðAÞ Dd
ð9Þ
Now consider the well-known Griffith’s energetic approach to the problem of an infinite plate of unit thickness, containing a crack of Euclidean length 2a and subjected to a remote tensile stress r (Fig. 3) [4]. The elastic strain energy release caused by the crack depends on its Euclidean length, that is:
We ¼
pr2 a2
ð10Þ
E
aA = ξ
DA = 1
L
DB = D aB = ξ D
(D/1)L Fig. 2. Two-dimensional geometrically similar bodies containing a crack: the typical case of three-point bend specimens.
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Fig. 3. Griffith’s problem for fractal cracks (in shadow is the zone of elastic strain energy release).
On the other hand, the energy dissipated at the surface of the fractal crack can be expressed using the renormalized fracture energy G (see Eq. (5)):
W s ¼ 2G a
ð11Þ
Consequently, the variation dWe of elastic strain energy release is defined with respect to the Euclidean crack length (2a), while the variation dWs of energy dissipation refers to the fractal crack length (2a). Differentiating Eqs. (10) and (11) against a and a, respectively, we obtain:
2pr2 a da E dW s ¼ 2G da ¼ 2ð1 þ dÞG ad da
dW e ¼
ð12aÞ ð12bÞ
The energy balance dWe = dWs yields :
r2 pa1d ¼ ð1 þ dÞG E
ð13Þ
An extension to fractal cracks of the classical definition of stress intensity factor (SIF) KI of Westergaard [27] for the probpffiffiffiffiffiffi lem under pffiffiffiffiffiffi consideration (K I ¼ r pa), and that of critical SIF (fracture toughness) KIC of Irwin [28] for plane stress states (K IC ¼ GE) has been proposed in Ref. [4], and the following renormalized quantities have been defined: d
K I ¼ rðpa1d Þ1=2 ¼ K I a2
ð14aÞ
K IC ¼ ½ð1 þ dÞG E1=2
ð14bÞ
so that Eq. (13) reads: 2 K 2 I ¼ K IC
ð15Þ
where the renormalized
½K I ¼ ½F½L
K I
and
K IC
have the following physical dimensions:
3þd 2
ð16Þ
For a general crack problem, Eq. (14a) should be written as follows:
K I ¼ Y rðpa1d Þ1=2
ð17Þ
where Y is a geometrical and loading dimensionless factor, and r is the nominal tensile stress perpendicular to the crack. Accordingly, the stress field ahead of the crack tip is proportional to K I (instead of KI). 3. A size-dependent fatigue crack growth law within the framework of monofractal geometry The well-known Paris–Erdogan law [29] describes the kinetics of crack propagation in the intermediate range of DKI:
da ¼ CðDK I Þm dN
ð18Þ
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where C and m are material constants, N is the number of fatigue loading cycles; da/dN and DKI are the crack propagation rate and the SIF range, respectively. The crack growth rate has experimentally been observed to be dependent on the size of specimens or structural components. Therefore, exploiting the renormalized quantities related to fractal cracks (whose surfaces can be modelled as invasive fractals, see previous Sections), the following size-dependent fatigue crack growth law has been conceived [18,19]:
da ¼ CðDK I Þm dN
ð19Þ
Since Eq. (19) is expressed through the renormalized quantities a and K I , it turns out to be scale-invariant. A scaling law can be obtained from Eq. (19) by rewriting such an equation in terms of the nominal crack propagation rate da/dN and the nominal SIF range DKI . As a matter of fact, the crack propagation rate da/dN of the fractal crack can be computed, by also recalling that a = a1+d:
da da da da ¼ ð1 þ dÞad ¼ dN dN da dN
ð20Þ
By employing Eqs. (14a) and (20), Eq. (19) becomes:
da C m ¼ ab1 DK m I ¼ C 1 ðaÞðDK I Þ dN 1 þ d
ð21aÞ
where
b1 ¼ dð1 þ
m Þ 2
ð21bÞ
For geometrically similar cracked bodies (e.g. see Fig. 2), a is proportional to D (i.e. a = nD). Consequently, Eq. (21a) becomes:
da ¼ C 1 ðDÞðDK I Þm dN
ð22aÞ
where
C 1 ðDÞ ¼
m C ðnDÞdð1þ 2 Þ 1þd
ð22bÞ
is a function of the structure size D. Therefore, Eq. (22a) represents a size-dependent fatigue crack growth law. Such an equation is formally similar to the classical Paris–Erdogan law (Eq. (18)), but the coefficient C1 depends on D, whereas C in Eq. (18) is assumed to be a material constant. Eq. (22a) represents a monofractal approach to size effect on fatigue crack growth. Consequently, it can appropriately be applied to a limited range of fatigue crack lengths (in the long crack fatigue growth regime). On the other hand, a multifractal approach would be more suitable to also model the propagation of fatigue cracks long with respect to the microstructural size (these cracks seem to follow the classical Paris–Erdogan law which is independent of the structure size D), since such an approach could describe a transition from a fractal regime (dependent on D) to a Euclidean one (independent of D) [6]. Therefore, a multifractal approach is analysed in the following Section. 4. A size-dependent fatigue crack growth law within the framework of multifractal geometry In the physical reality, a transition occurs from a fractal regime for small structures to a Euclidean one for structures large enough with respect to a characteristic material length, as was pointed out by Mandelbrot [13]. In other words, the effect of the microstructural disorder (heterogeneity and/or micromechanical damage) of a given material on the macroscopic mechanical behaviour gradually vanishes by increasing the structure size [6]. In line with a multifractal hypothesis, the fractal increment d can be assumed to vary with the scale of observation of the f with f = material fractal. In the following the renormalized crack length is assumed to be equal to a = a1+d, where d ¼ f þa parameter. In this way, the fractal increment d is a function of the crack length a and varies between 1 (for a ? 0) and 0 (for a ? 1). Moreover, the SIF range to be introduced in Eq. (19) can be written as follows:
DK I ¼ DK I ð1 þ
lch 1=2 Þ a
ð23Þ
where lch is a material parameter. The definition of Eq. (23) is in line with the multifractal expression for fracture energy proposed in Ref. [6]. Moreover, such an equation is reminiscent of other equations dealing with non-propagating fatigue cracks which include a material parameter in the SIF expression (e.g. see Ref. [30]). In order to derive a multifractal size-dependent fatigue crack growth law, the same thread of logic of the previous Section is followed. The chain rule for the derivation of a with respect to N now yields: f af ln a þ ðf þ aÞð2f þ aÞ da da da da ¼ af þa ¼ dN dN da dN ðf þ aÞ2
ð24Þ
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(a) Log da/dN 1 m m
C1 (D A ) 1 C1 (D B ) DB > DA
Log ΔKI
(b)
Log C1 multifractal
1
|β1| monofractal
Log D Fig. 4. Size effect in fatigue crack growth: (a) da/dN–DKI curves (see Eq. (22a)) and (b) C1–D curves (see Eqs. (22b) and (26)).
Thus, by employing Eqs. (23) and (24), Eq. (19) becomes:
da ¼ C 1 ðaÞðDK I Þm dN
ð25aÞ
where
C 1 ðaÞ ¼ C
m=2 lch 1þ a a ðaf ln a þ ðf þ aÞð2f þ aÞÞ ðf þ aÞ2
f f þa
ð25bÞ
For geometrically similar cracked bodies (e.g. see Fig. 2), a is proportional to D (i.e. a = nD). Consequently, Eq. (25a) can formally be written as a function of the structure size D (see Eq. (22a)). In this way a multifractal size-dependent fatigue crack growth law is obtained, where the coefficient C1 depends on D according to the following relationship:
C 1 ðDÞ ¼ C
m=2 lch 1 þ f nD ðnDÞf þnD ðnDf ln nD þ ðf þ nDÞð2f þ nDÞÞ ðf þ nDÞ2
ð26Þ
Note that, in comparison with Eq. (22b), Eq. (26) depends on two further material parameters, f and lch. Eq. (22a) demonstrates a size effect on fatigue crack propagation. In more details, since C1 depends on the structure size D (see Eqs. (22b) and (26) for monofractal and multifractal approach, respectively), the crack growth rate da/dN in turn depends on D. Typically the crack growth rate da/dN decreases with increasing D. If we assume that the relative crack depth n in Eqs. (22b) and (26) is approximately constant during crack propagation, Eq. (22a) describes a linear relationship in the bilogarithmic plane da/dN–DKI (Fig. 4a). In Fig. 4b two sample curves of the monofractal (Eq. (22b)) and multifractal (Eq. (26)) relationships are reported in the bilogarithmic plane C1–D. It can be noted that the curve of Eq. (26) is non-linear in such a plane. 5. Theoretical interpretation of some experimental fatigue data Some experimental data, related to fatigue crack propagation in normal strength (NS) concrete [24,25] and high strength (HS) concrete [26], are here employed to show an application of the proposed scaling laws (see Eqs. (22b) and (26)). Two series of three-point bend NS plain-concrete specimens were tested under fatigue loading [24,25]. Each series consisted of three two-dimensional geometrically similar cracked specimens (see Fig. 2). The beam height (characteristic structure size) was equal to DA = 38 mm, DB = 76 mm, DC = 152 mm, respectively. The span L of the beams was 2.5 times the height D, the initial length of the crack was equal to D/6, and the beam thickness was equal to 38 mm. The ratio of cement:sand:aggregate:water in the concrete mix was 1:2:2:0.6 by weight. The aggregate was crushed limestone of maximum size 12.7 mm. The sand was siliceous river sand passing through 5-mm sieve. Type I Portland cement with no admixtures was used. After 28-day curing, the mean compression strength of cylindrical specimens was equal to 32.8 MPa.
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1E-002
DA = 38 mm DB = 76 mm
da /dN (m/cycle)
1E-003
DC = 152 mm
1E-004
1E-005
1E-006
0.4
0.5
0.6 0.7 0.8 0.9
2
1
ΔKI (MPa√m) Fig. 5. Crack growth da/dN–DKI data for NS concrete and different specimen sizes [25]: dashed regression lines refer to best-fit slopes (mA = 12.184, = 10.395). mB = 9.975, mC = 9.025), while continuous regression lines refer to an average slope (m
(a) C1 [m/(MPa√m)10.395]
1E-002 1E-003 1E-004
⏐ 1⏐
1E-005 1E-006 1E-002
C1 [m/(MPa√m)8.155]
(b)
1E-003 1E-004
⏐ 1⏐
1E-005 1E-006 20
40
60
80
100
200
400
D (mm) Fig. 6. Best-fitting of C1(D) to determine b1 (Eq. 22b) and the fractal increment d (Eq. 21b): (a) b1 = 4.697 and d = 0.758 for NS concrete [25] and (b) b1 = 1.358 and d = 0.267 for HS concrete [26].
The crack growth rate against SIF range data (corresponding to 37, 36 and 29 experimental points for DA, DB and DC, respectively) are presented in Fig. 5 along with regression lines based on Eq. (22a) (the experimental data are taken from Ref. [25]). Dashed lines refer to best-fit parameters m (mA = 12.184, mB = 9.975, mC = 9.025) and C1 (Log C1(DA) = 1.603, Log C1(DB) = 3.666, Log C1(DC) = 4.818). The correlation coefficient R is equal to 0.865, 0.909 and 0.912 for DA, DB and DC, respectively. = 10.395) is conSince the variation of the best-fit slope m with respect to the structure size is small, an average slope (m sidered in the following, so that the new best-fitting yields Log C1(DA) = 2.058 (R = 0.846), Log C1(DB) = 3.632 (R = 0.907), Log C1(DC) = 4.886 (R = 0.891) (see continuous lines in Fig. 5). Therefore, according to the power law expressed in Eq. (22b), the exponent b1 can be determined by plotting C1 against D and assuming the relative crack depth constant and equal to its initial value D/6 [18], as is shown in Fig. 6a (b1 = 4.697, R = 0.996). Consequently, the corresponding value of the fractal = 10.395 and b1 = 4.697). increment d results to be equal to 0.758 from Eq. (21b) (for m ¼ m
982
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1E-002
da /dN (m/cycle)
1E-003
1E-004
1E-005
DA = 38 mm DB =108 mm DC = 304 mm
1E-006 0.8 0.9
1
2
3
ΔKI (MPa√m)
Fig. 7. Crack growth da=dN DK I data for HS concrete and different specimen sizes [26]: dashed regression lines refer to best-fit slopes (mA = 8.444, = 8.155). mB = 8.603, mC = 7.417), while continuous regression lines refer to an average slope (m
C1 [m/(MPa√m)10.395]
(a)
1E-001
monofractal
1E-002
multifractal
1E-003 1E-004 1E-005 1E-006 1E-007
C1 [m/(MPa√m)8.155]
(b)
1E-001
monofractal 1E-002
multifractal
1E-003 1E-004 1E-005 1E-006 1E-007 30
50
70 90 100
200
400 600 800 1000
D (mm) Fig. 8. Interpolation of experimental C 1 data using the multifractal relationship in Eq. (26) for: (a) NS concrete [25] and (b) HS concrete [26]. The monofractal relationship in Eq. (22b) is also reported for comparison.
Furthermore, one series of three-point bend HS plain-concrete specimens was tested [26], consisting of three two-dimensional geometrically similar cracked specimens (see Fig. 2). The beam height was equal to DA = 38 mm, DB = 108 mm and DC = 304 mm, respectively. The span and the thickness of the beams and the relative initial crack length were equal to those for NS concrete tests, previously discussed. The ratio of the mix components to cement by weight was as follows: 1.00 Portland cement, 0.316 water, 0.132 fly ash, 0.0507 silica fume, 2.18 maximum-diameter crushed aggregate, 1.51 siliceous sand, 0.0019 retarder, 0.00951 superplasticizer. The maximum size of the aggregate was equal to 9.5 mm. At the beginning of the fatigue tests, the mean compression strength of cylindrical specimens was equal to 90.3 MPa. The crack growth rate against SIF range data [26] are reported in Fig. 7 along with regression lines based on Eq. (22a) (17, 16 and 12 experimental points for DA, DB and DC, respectively). Dashed lines refer to best-fit parameters m (mA = 8.444, mB = 8.603, mC = 7.417) and C1 (Log C1(DA) = 4.719, Log C1(DB) = 5.561, Log C1(DC) = 5.720). The correlation coefficient
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R is equal to 0.965, 0.905 and 0.761 for DA, DB and DC, respectively. Similar to NS concrete data of Refs. [24,25], a small variation of the best-fit slope m with respect to the structure size is observed also for HS concrete, and an average slope = 8.155) can be considered, so that the new best-fitting yields Log C1(DA) = 4.690 (R = 0.964), Log C1(DB) = 5.481 (m (R = 0.902), Log C1(DC) = 5.916 (R = 0.753) (see continuous lines in Fig. 7). Then, the exponent b1 (in Eq. (22b)) can be determined by plotting C1 against D and again by assuming the relative crack depth constant and equal to its initial value D/6 [18], as is shown in Fig. 6b (b1 = 1.358, R = 0.974). Finally, from Eq. (21b), the fractal increment d results to be equal to 0.267 (for = 8.155 and b1 = 1.358). m¼m By analysing the obtained values of the fractal increment d for NS and HS concrete mixes, it seems that d tends to increase with increasing maximum size dmax of aggregate (NS concrete: dmax = 12.7 mm, d = 0.758; HS concrete: dmax = 9.5 mm, d = 0.267) [18,19]. Such a result is consistent from a physical point of view, that is to say, the invasive properties (d) are connected to the microstructural properties (e.g. dmax). In other words, a concrete with a coarse microstructure (i.e. large value of dmax) presents a fracture surface dimension much greater than 2 (i.e. d is large), while a fine concrete (i.e. small value of dmax) shows a fracture surface which is almost flat (i.e. d is small). Experimentally, this trend of behaviour has been observed by performing a digital scanning and analysis of the fracture surfaces for different concrete mixes under static loading [31]. Now, within the framework of a multifractal approach, Eq. (26) is used to describe the experimental trend of behaviour of C1(D). Assuming the relative crack depth constant and equal to its initial value D/6, the three parameters C, f and lch appearing in Eq. (26) can be computed through a non-linear best-fitting procedure. As a matter of fact, three experimental values C1(D) are available for each (NS and HS) concrete series (see the C1(D) values previously mentioned in the present Section), and therefore C, f and lch can actually be worked out by solving a 3 3 system of non-linear equations (see Eq. (26)). The obtained pffiffiffiffiffi results are as follows: for NS concrete C = 2.415 107 (for DKI expressed in MPa m and da/dN in m/cycle), f = 0.0259 m pffiffiffiffiffi 7 (for DKI expressed in MPa m and da/dN in m/cycle), and lch = 0.0199 m (Fig. 8a); for HS concrete C = 9.743 10 f = 0.0188 m and lch = 0.000396 m (Fig. 8b). It is worth remarking that, since the system under consideration is non-linear, the solution is not unique: for each concrete mix, the results reported above correspond to a solution of the system for positive values of C, f and lch. It can be noted that, according to the multifractal approach, the size effect in the crack growth rate tends, conversely to the trend described by the monofractal approach (see Fig. 8, where the monofractal C1–D curves are also reported for comparison), to reduce as the structure size increases beyond a certain threshold (the C1–D curve approaches an horizontal asymptote for D ? 1). Furthermore, it can be noted that lch increases with increasing the characteristic length of material microstructure (e.g. the maximum size of aggregate, which is larger in the NS concrete as compared to that in the HS concrete). A similar trend of lch against a characteristic length of material microstructure is reported in Ref. [6]. 6. Conclusions By modelling a crack surface as an invasive fractal set, a new definition of fracture energy is required to obtain a parameter (renormalized fracture energy) which is independent of the scale of observation of the fractal set and can be considered as a material constant. Then, by investigating the Griffith’s problem within the framework of fractal cracks, a renormalized SIF based on physical dimensions different from the classical ones can be obtained. In the present paper, such a SIF is firstly introduced in the classical Paris–Erdogan law for fatigue crack propagation. This leads to size effects whose physical source is hence related to the fact that, on one side, the energy consumption during crack propagation occurs on the crack surface which is reasonably well described by a fractal domain whereas, on the other side, the energy release occurs in the bulk volume of the material, that is, in a classical Euclidean domain. A so-called monofractal size-dependent fatigue crack growth law, which is characterised by a power law (with the exponent dependent on the fractal dimensional increment d) of the structure size D, is proposed. Then, by applying some concepts of multifractal geometry (the fractal dimension is made to vary with the scale of observation of the fractal), a new definition of renormalized crack length and SIF is put forward. This yields a so-called multifractal size-dependent fatigue crack growth law. Both the presented fatigue crack growth laws are herein used to theoretically interpret some experimental data related to concrete. Acknowledgements The authors gratefully acknowledge the research support for this work provided by the Italian Ministry for University and Technological and Scientific Research (MIUR). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]
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