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Modified two-parameter fracture model for bone
Engineering Fracture Mechanics 174 (2017) 44–53
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Modified two-parameter fracture model for bone Andrea Carpinteri a, Filippo Berto b, Giovanni Fortese a, Camilla Ronchei a, Daniela Scorza a, Sabrina Vantadori a,⇑ a b
Department of Civil-Environmental Engineering and Architecture, University of Parma, Parma, Italy NTNU, Department of Engineering Design and Materials, Richard Birkelands vei 2b, 7491 Trondheim, Norway
a r t i c l e
i n f o
Article history: Received 16 October 2016 Received in revised form 7 November 2016 Accepted 7 November 2016 Available online 9 November 2016 Keywords: Fracture toughness Cortical bone Quasi-brittle material Two-Parameter Model
a b s t r a c t The analysis of the bone fracture behaviour is fundamental for prevention, diagnosis and treatment of traumas. In the present paper, an experimental campaign on fracture behaviour of bovine femoral cortical bones is conducted to characterise the fracture toughness, K SIC , which is related to the structure and load-bearing capacity of bones. Firstly, K SIC is evaluated through a two-parameter model originally proposed for quasi-brittle materials. To take into account the crack deflection (kinked crack) due to osteons orientation, the twoparameter model is modified by applying the Castigliano theorem. Fracture toughness results here obtained are compared with those related to a femur of an 18-month-old bovine, available in the literature. Ó 2016 Elsevier Ltd. All rights reserved.
1. Introduction Bone is a specialised tissue which has important functions, both metabolic and mechanical [1–3]. The load-bearing capacity of bones is limited up a certain extent, beyond which they fail in a brittle manner [4,5]. The analysis of the bone fracture behaviour is fundamental for prevention, diagnosis and treatment of traumas. Basic parameters which represent the structure and functions of bone have to be measured, such as its fracture toughness [6–16]. Depending on the skeletal locations, bones may be: (i) long bones (limbs, ribs, clavicles); (ii) flat bones (skull, scapula, pelvis); (iii) short bones (vertebrae, sternum). Generally most of fractures occur in long bones due to their skeletal locations, and more precisely in the diaphysis of such bones, i.e. in their central part, which represents the longest one (Fig. 1). In the present paper, an experimental analysis of the fracture behaviour of bovine cortical bones is carried out, where specimens are extracted from diaphysises. The experimental programme is conducted to characterise the fracture toughness by employing a Two-Parameter Model (TPM), originally proposed for concrete [17–19], that is, for a quasi-brittle material showing a non-linear slow crack growth before the peak load is reached. In bones, such a behaviour is produced by mechanisms of extrinsic toughening categorised in four classes [11]: (i) constrained microcracking; (ii) crack deflection and twist; (iii) uncracked-ligament bridging; (iv) collagen-fibril bridging. The above TPM is based on experimental data obtained from three-point bending tests by using single edge-notched specimens, and employs linear elastic fracture mechanics expressions valid for Mode I loading. However, for the bone material, such a model cannot be applied in its original formulation since the crack starting from notch may deflect. In order to understand the cause of such a deflection under Mode I loading (three-point bending), the hierarchical structure of bone has to be briefly examined. More precisely, five levels can be listed [3,20,21]: (1) whole bone at the ⇑ Corresponding author. E-mail address: [email protected] (S. Vantadori). http://dx.doi.org/10.1016/j.engfracmech.2016.11.002 0013-7944/Ó 2016 Elsevier Ltd. All rights reserved.
A. Carpinteri et al. / Engineering Fracture Mechanics 174 (2017) 44–53
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Nomenclature a0 a1 ; a2 a B Ci Cu E F G
notch-depth segments of the kinked crack branch effective critical crack length specimen width linear elastic compliance or initial compliance unloading compliance elastic modulus virtual force total energy rate
K SIC
critical stress-intensity factor under Mode I (fracture toughness)
K SðIþIIÞC
critical stress-intensity factor under mixed mode i-th deflected segment along the kinked crack path specimen length peak load loading span total strain energy specimen depth relative displacement of the crack surfaces
li L Pmax S UT W DF
macrostructural level; (2) compact bone and cancellous bone block at the architecture level; (3) osteon and trabecula at the microstructural level; (4) lamella at the sub-microstructural level; (5) collagen fibril, non-collagenous and mineral components at the ultra-structural level. Osteons are more or less regular cylindrical structures, whose length ranges from 3 to 12 mm (Fig. 2). The osteons are oriented parallel to the bone axis, which consists of a vascular canal (named Haversian canal) surrounded by concentric lamellae. The interface between osteon and interstitial lamellae is called cement line (Fig. 2).
articular cartilage spongy bone
medulla medulla
marrow compact bone EPIPHYSIS
DIAPHYSIS
marrow EPIPHYSIS
Fig. 1. Schematic illustration of a long bone.
eon ost vascular canal cement line concentric lamellae Fig. 2. Cortical bone microstructural level.
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A. Carpinteri et al. / Engineering Fracture Mechanics 174 (2017) 44–53
Under three-point bending loading, when the osteons alignment is perpendicular to the loading direction (transversal specimens), the stress state is biaxial due to normal stresses produced by bending and shear stresses produced by slipping, at the cement line interface between osteon and interstitial lamellae [22]. In such a case, the crack starting from the notch is subjected to mixed mode loading (Mode I together with Mode II). On the other hand, when the osteons alignment is parallel to the loading direction (longitudinal specimens), the stress state is uniaxial. In such a case, crack starting from the notch is subjected to Mode I loading. The Two-Parameter Model can be directly applied for the latter case, whereas a modified formulation is hereafter proposed for the former case in order to take into account that Mode II loading is also present at crack tip. Firstly, the unloading compliance expression related to a singly kinked crack is determined by employing the Castigliano theorem in the way suggested by Paris [23], and the effective crack length is computed. Then, the critical stress-intensity factor at crack tip (fracture toughness K SIC ) is evaluated as a function of that for a straight crack having length equal to the projected length of the effective critical kinked crack [24,25]. The fracture toughness is evaluated using both transversal and longitudinal notched specimens obtained from a cadaveric femur diaphysis of a 24-month-old bovine. The fracture toughness results obtained herein for the above two specimen orientations are compared with those related to a femur of an 18-month-old bovine [13], experimentally determined according to the standard ASTM E399-12e3 [26]. 2. Basic composition and structure of long bones Bone is devoted to both metabolic and mechanical functions [1–3]. The former consists in regulation of calcemia during both bone mineralisation and resorption [2], and requires the participation of bone cells. The latter consists in muscles insertion, levers formation for muscles response and soft tissues protection, allowing for standing and moving [1]. As has been previously discussed, bones may be long, flat or short bones [3]. Two different regions may be distinguished in a long bone: the diaphysis, that is, the central shaft which represents the longest part, and the epiphyses at the two extremities of the long bone (Fig. 1). Articular cartilage covers the external epiphyseal surface (Fig. 1). The diaphysis consists of (see Fig. 1): (i) a thick outer layer of compact bone (also named cortical bone) made of a calcified bone matrix and soft structures (i.e. blood vessels), and (ii) a medulla, which corresponds to an axial cylinder containing marrow. The epiphyses show a thin outer compact bone (Fig. 1). The medulla is prevalent and consisting of a frame of osseous trabeculae (also named spongy bone); such trabeculae delimit irregular intercommunicating spaces containing marrow. About 90% of the diaphysis is calcified, whereas 10% consists of nonosseous cells, blood vessels, nerve fibres and bone marrow. The calcified bone matrix includes two components: (a) the organic matrix, consisting of both type I collagen fibrils (for over 90% in volume) and non-collagenous components (remaining 10% in volume), and (b) mineral substance. Collagen fibrils are formed by the assemblage of filamentous molecules arranged in a helical configuration. The molecules are stabilized by intra- and intermolecular cross-links, which are essential for the tensile strength of the fibrils. Noncollagenous components include proteins, proteoglycans, phospholipids, glycoproteins and phosphoproteins. The mineral substance is a calcium phosphate hydroxyapatite. 3. Two-parameter model According to the Two-Parameter Model [17–19], the specimens present a notch in the lower part of the middle crosssection (Fig. 3), and the following specimen sizes are suggested: width (B) depth (W) length (L) = 76.2 mm 152.4 mm 711.2 mm; loading span S = 609.6 mm; notch-depth ratio = a0 =W = 1/3; notch width <3.175 mm, and loading-span/depth ratio = S=W = 4 (Fig. 3). Note that the specimen sizes can be increased/decreased proportional to the aforementioned ones. Under three-point bending test, the specimen is monotonically loaded in Crack Mouth Opening Displacement (CMOD) control. After the peak load (Pmax) is reached, the post-peak stage follows and, when the force (P) is equal to about 95% of the peak load, the specimen is fully unloaded (Fig. 4). Then, the specimen is reloaded up to failure (Fig. 4). The linear elastic compliance C i (i.e. the initial compliance, Fig. 4) is used to determine the elastic modulus E [27]:
E¼
6Sa0 Vða0 Þ
ð1Þ
Ci W 2 B
Further, the parameter V can be expressed as follows [27]:
Vða0 Þ ¼ 0:76 2:28a0 þ 3:87a20 2:04a30 þ
0:66 1 a20
with a0 ¼
a0 W
ð2Þ
Therefore, if the crack propagates under pure Mode I, the effective critical crack length a to determine the critical stressintensity factor K SIC is obtained from the following equation by employing an iterative procedure:
E¼
6SaVðaÞ CuW 2B
ð3Þ
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47
P
a
W B
S L
a0
Fig. 3. Geometrical and testing configuration of specimen according to the Two-Parameter Model.
LOAD, P
1 0.95 (Pmax)
Ci Pmax
1
Cu CRACK MOUTH OPENING DISPLACEMENT, CMOD Fig. 4. Typical load – CMOD plot.
where C u is the unloading compliance (Fig. 4), and VðaÞ is deduced from Eq. (2) by replacing a0 with a. Finally, the critical stress-intensity factor under Mode I, K SIC , is computed by employing the measured value of the peak load, Pmax [27]:
K SIC ¼
3Pmax S pffiffiffiffiffiffi paf ðaÞ 2W 2 B
ð4Þ
where:
1 1:99 að1 aÞð2:15 3:93a þ 2:70a2 Þ f ðaÞ ¼ pffiffiffiffi p ð1 þ 2aÞð1 aÞ3=2
with a ¼
a W
ð5Þ
4. Modified two-parameter model A modified procedure is hereafter proposed when crack propagates under mixed mode loading (Mode I together with Mode II). Specimens geometry and experimental test procedure are equal to those presented in the previous Section. The elastic modulus, E, is determined through Eqs (1) and (2). If the crack propagates under mixed mode loading, the effective critical crack length is obtained by following the procedure hereafter described. Let us consider a body loaded by both a loading force, P, and a pair of virtual forces, F, in equilibrium for any distance, d (Fig. 5). The Castigliano theorem states that the displacement of any value of F (in its own direction) may be computed as follows [27]:
DF ¼
@U T @F F¼0
ð6Þ
where the total strain energy U T can be obtained by adding the strain energy due to the applied forces with no crack and that introducing the crack while holding constant the applied forces:
Z U T ¼ U NoCrack þ 0
A
@U T dA @A
being dA an infinitesimal increase of the crack area A (Fig. 5).
ð7Þ
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P F
d+
F
dA
F
P Fig. 5. Body loaded by both a loading force P and a pair of virtual forces F, in equilibrium for any distance d.
By assuming a constant loading force, P, the total energy rate, G, is equivalent to the rate of increase of the total strain energy, that is:
G¼
@U T @A
ð8Þ
and the relative displacement of crack surfaces, DF , given by the virtual forces can be computed as follows:
DF ¼
Z A @U NoCrack @G þ dA @F F¼0 @F 0 F¼0
ð9Þ
Now consider a prismatic body containing a kinked crack subjected to both three-point bending loading and a pair of virtual forces (applied in the lower part of the beam) perpendicular to the crack faces (Fig. 6). In this case, the first term on the right-hand side of Eq. (9) is equal to zero, since it corresponds to the displacement produced by the pair of virtual forces (in their own direction) in an uncracked beam, whereas the second term is a function of the Stress-Intensity Factors (SIFs) for each loading mode, which are due to both the loading force P and the virtual forces F. For such a configuration, the total energy rate G is given by:
G ¼ GI þ GII ¼
i K 2I K 2II 1 h þ ¼ ðK IP þ K IF Þ2 þ ðK IIP þ K IIF Þ2 E E E
ð10Þ
where K IP and K IIP are the Mode I and the Mode II SIFs due to the loading force, respectively, whereas K IF and K IIF are the Mode I and the Mode II SIFs due to the virtual forces, respectively. By substituting Eq. (10) in Eq. (9), we obtain:
DF ¼
1 E
Z
A
0
@ ð2K IP K IF þ 2K IIP K IIF Þ @F
dA ¼ F¼0
2 E
Z 0
A
@K IF @K IIF dA K IP þ K IIP @F F¼0 @F F¼0
ð11Þ
Each SIF term in Eq. (11) can be expressed as a function of the corresponding Mode I SIF of a straight crack having length equal to the projected length of the actual kinked crack, and the crack tip location is identified by the coordinate x shown in Fig. 6. Since the SIF expressions of K IP , K IIP , K IF , and K IIF depend on the x-coordinate, the integral over the cracked area A (Eq. (11)) can be replaced by the following three integrals along a vertical line:
Z a0 þa1 cos h @K IF;0 @K IF;1 @K IIF;1 dx dx þ K IP;1 þ K IIP;1 @F F¼0 @F F¼0 @F a0 0 F¼0 Z a0 þða1 þa2 Þ cos h @K IF;2 @K IIF;2 dx K IP;2 þ K IIP;2 þ @F F¼0 @F a0 þa1 cos h F¼0
DF ¼
2 E
Z
a0
K IP;0
P
a2a
1
W B
x
F
F
a
a0
S L Fig. 6. Geometrical and testing configuration of specimen according to the modified Two-Parameter Model.
ð12Þ
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being h the crack kinking angle, and a1 þ a2 the total length of the deflected crack, as is detailed in the following (Fig. 6). All the SIF terms in Eq. (12) can be expressed as a function of the Mode I SIFs at the crack tip of a straight crack under three-point bending (KIP,0) and under a pair of virtual forces (KIF,0) [27]:
K IP;0 ¼
3PS pffiffiffiffiffiffi 1 1:99 að1 aÞð2:15 3:93a þ 2:70a2 Þ px f ðaÞ with f ðaÞ ¼ pffiffiffiffi 2 p 2W B ð1 þ 2aÞð1 aÞ3=2
2F 0:46 þ 3:06a þ 0:84ð1 aÞ5 þ 0:66a2 ð1 aÞ2 K IF;0 ¼ pffiffiffiffiffiffi gðaÞ with gðaÞ ¼ px ð1 aÞ3=2
and a ¼
x W
ð13aÞ
ð13bÞ
The Mode I and Mode II SIFs at the crack tip of a kinked crack characterised by a short branch, under three-point bending (KIP,1, K IIP;1 ) and under a pair of virtual forces (KIF,1, K IIF;1 ) can be evaluated following the approximated procedure proposed by Cotterell and Rice for a deflected crack characterised by a short branch in a centrally cracked infinite plate [25]:
K IP;1 ¼ K IP;0 cos3 K IIP;1 ¼ K IP;0 sin
ð14aÞ
h h cos2 2 2
K IF;1 ¼ K IF;0 cos3 K IIF;1 ¼ K IF;0 sin
h 2
ð14bÞ
h 2
ð14cÞ
h h cos2 2 2
ð14dÞ
The approximation is acceptable for a1 =a0 6 0:3. The Mode I and Mode II SIFs at the crack tip of a kinked crack characterised by a long branch (a1 =a0 P 0:3), under threepoint bending (KIP,2, K IIP;2 ) and under a pair of virtual forces (KIF,2, K IIF;2 ) can be evaluated following the approximated procedure proposed by Kitagawa at al. for a deflected crack characterised by a long branch in a centrally cracked infinite plate [24]:
K IP;2 ¼ K IP;0 cos3=2 h
ð15aÞ
K IF;2 ¼ K IF;0 cos3=2 h
ð15bÞ
K IIP;2 ¼ K IP;0 sin h cos1=2 h
ð15cÞ
K IIF;2 ¼ K IF;0 sin h cos1=2 h
ð15dÞ
By substituting Eqs. (13)–(15) in Eq. (12), we obtain:
" (Z 2 2 # Z a0 þa1 cosh Z a0 a0 2 @K IF;0 h @K IF;0 @K IF;0 3h 2h DF ¼ K IP;0 dx þ cos þ sin cos K IP;0 dx K IP;0 dx E 0 2 2 2 @F F¼0 @F F¼0 @F F¼0 0 0 Z a0 þða1 þa2 Þ cosh Z a0 þa1 cosh h i @K IF;0 @K IF;0 2 2 K IP;0 dx K IP;0 dx ð16Þ þ ðcos3=2 hÞ þ ðsinhcos1=2 hÞ @F F¼0 @F F¼0 0 0
It can be noticed that all integrals are formally equal, changing only the integral extremes. An empirical solution can be employed according to that proposed by Tada et al. [27]:
2 E
Z
z
K IP;0 0
@K IF;0 6PSz dz ¼ VðaÞ @F F¼0 EW 2 B
VðaÞ ¼ 0:76 2:28a þ 3:87a2 2:04a3 þ
0:66 1 a2
with a ¼
z W
ð17Þ
being z a generic integration variable. Therefore, once each integral of Eq. (16) is empirically resolved and the unloading compliance C u (equal to DF =P) is taken into account, we can explicit the elastic modulus from Eq. (16):
a a h h a0 þ a1 cos h 0 0 2 h ða0 þ a1 cos hÞV a0 V þ cos6 þ sin cos4 a0 V 2 2 2 W W W CuW B a0 þ a1 cos h þ a2 cos h a0 þ a1 cos h 2 ða0 þ a1 cos hÞV ð18Þ þ½cos3 h þ sin h cos h ða0 þ a1 cos h þ a2 cos hÞV W W
E¼
6S
2
Now a2 can be obtained from Eq. (18) by employing an iterative procedure. Therefore, the effective critical crack length is equal to a ¼ a0 þ ða1 þ a2 Þ cos h with a1 ¼ 0:3a0 .
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As is shown in Fig. 6, the kinked crack branch consists of the two segments, named a1 and a2 . If the value of a2 obtained from Eq. (18) is negative, it means that the effective crack length is a ¼ a0 þ a1 , with a1 6 0:3a0 . Therefore, the expression of DF in such a case is given by:
DF ¼
2 E
Z
a0
K IP;0 0
Z a0 þa1 cos h @K IF;0 @K IF;1 @K IIF;1 dx dx þ K IP;1 þ K IIP;1 @F F¼0 @F F¼0 @F a0 F¼0
ð19Þ
and a1 is determined from the following equation by employing an iterative procedure:
E¼
a a a0 þ a1 cos h 0 0 2 h 6 h 4 h ða a þ sin cos V þ a cos hÞV V a þ cos 0 0 1 0 2 2 2 W W W CuW 2B 6S
ð20Þ
Finally, the critical stress-intensity factor under mixed mode, K SðIþIIÞC , is computed through Eqs. (4) and (5), by employing the above effective critical crack lengths. 5. Specimen preparation and experimental set-up The specimens tested have been extracted from a cadaveric femur diaphysis of a 24-month-old bovine, two days after slaughter. We have removed the marrow and obtained test samples by means of a low-speed saw and a milling machine. Such samples have been stored in saline solution at 3 °C for 20 h, then at room temperature for 2 h until testing. The specimens, with prismatic shape and geometrical sizes shown in Table 1, present a notch in the lower part of the middle cross-section (Fig. 6). The osteons alignment is perpendicular to the loading direction for 6 specimens (named transversal specimens) and parallel to the loading direction for 2 specimens (named longitudinal specimens). The former specimens are extracted from anterior (FA-1 and FA-2 in Table 1), posterior (FP), medial (FM-1 and FM-2) and lateral (FL) cortical bone, whereas the latter specimens from posterior cortical bone (FP-long1 and FP-long2). Specimens are tested under three-point bending. Testing is performed by means of an Instron 8862 testing machine, under crack mouth opening displacement control, employing a clip gauge at an average speed equal to 0.1 mm h1. All specimens exhibit a non-linear slow crack growth before the peak load is reached. Note that an extension of the length L has been made in some specimens by employing epoxy resins, having previously verified that such an extension would not influence the fracture behaviour of bone. The above extension is needed in order to make it possible to use the three-point flexural bend test equipment shown in Fig. 7.
Table 1 Geometrical sizes of the tested specimens. SPECIMEN no.
W [mm]
B [mm]
L [mm]
a0 [mm]
FA-1 FA-2 FP FM-1 FM-2 FL FP-long1 FP-long2
17.01 21.50 9.90 21.80 10.67 17.10 12.56 13.23
10.02 11.20 4.04 9.33 9.79 7.60 4.86 5.69
100.0 100.0 124.0 100.0 100.0 100.0 122.0 120.0
8.56 5.73 3.72 8.35 2.75 4.93 5.45 4.93
Fig. 7. Experimental testing set-up.
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A. Carpinteri et al. / Engineering Fracture Mechanics 174 (2017) 44–53
6. Results and discussion Values of fracture toughness, K SðIþIIÞC , of the above specimens are computed by employing both the proposed modified TPM and the load-CMOD curves. The initial compliance, the unloading compliance, the kinking angle and the peak load for each tested specimen are listed in Table 2. When the kinking angle value is not constant along the crack path, as is shown for example in Fig. 8 related to the specimen FA-2, the orientation of the first deflected segment such that liþ1 > li (i = 1, . . . , N, where N is a positive integer) is taken into account in the calculation of the fracture toughness. Fig. 9 shows the fracture surfaces of each specimen. Some specimens appear irregularly spray-painted since the Digital Image Correlation (DIC) technique is used to extract 2D full-field maps of displacements and strains in the mid-span zone of such specimens. Fracture toughness values K SðIþIIÞC together with the corresponding effective critical crack lengths a are listed in Table 3 (3rd and 4th columns). It can be noticed that the fracture toughness values significantly depend on the osteons alignment with respect to the loading direction. As a matter of fact, when the osteons alignment is perpendicular to the loading direction, crack grows under mixed mode (Fig. 9(a)–(f)), and a higher resistance to fracture is observed; when the osteons alignment is parallel to the loading direction, crack grows under Mode I (Fig. 9(g) and (h)), and a lower resistance to fracture is observed. Fracture toughness values computed according to the TPM, K SIC , are also listed in Table 3, together with the corresponding effective critical crack lengths a (6th column) and the elastic modulus values E. Note that, in general, the former critical SIFs (K SðIþIIÞC ) are significantly lower than the latter (K SIC ), up to about 70% for the specimen FP. pffiffiffiffiffi For transversal specimens, the average value of K SðIþIIÞC (3:87 0:15 MPa m) is in the range of the cortical bone values [28]. Such a value is then compared with the results determined by Libonati and Vergani [13] according to ASTM standards pffiffiffiffiffi m
[26], using the Linear Elastic Fracture Mechanics and considering cracks under pure Mode I loading: K SIC ¼ 5:6 0:1 MPa Table 2 Initial and unloading compliance, kinking angle and peak load for each tested specimen. SPECIMEN no.
C1 [1/MPa]
Cu [1/MPa]
h [°]
Pmax [N]
FA-1 FA-2 FP FM-1 FM-2 FL FP-long1 FP-long2
0.00032430 0.00006186 0.00068506 0.00013644 0.00014505 0.00013682 0.00067626 0.00039752
0.00038546 0.00006558 0.00094390 0.00015183 0.00014594 0.00016944 0.00079355 0.00050274
45 8 67 35 44 50 0 0
450.09 1097.66 263.22 808.91 457.73 585.96 74.95 101.07
Fig. 8. Procedure used in order to measure the kinking angle.
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A. Carpinteri et al. / Engineering Fracture Mechanics 174 (2017) 44–53
(a)
(b)
FA-1
(c)
FA-2
(d)
FM-1
FP
(e)
(f)
FM-2
(g)
FL
(h)
FP-long1
FP-long2
Fig. 9. Fracture surfaces: mixed mode for the transversal specimens (a) FA-1, (b) FA-2, (c) FP, (d) FM-1, (e) FM-2, (f) FL, and pure Mode I for the longitudinal specimens (g) FP-long1 and (h) FP-long1. Table 3 Elastic modulus, fracture toughness and critical effective crack length computed through both the modified TPM and the TPM. SPECIMEN no.
E [MPa]
K SðIþIIÞC [MPa m1/2]
a [mm]
K SIC [MPa m1/2]
a [mm]
FA-1 FA-2 FP FM-1 FM-2 FL FP-long1 FP-long2
13692.54 14393.65 14038.29 14253.00 13488.94 14042.47 12371.99 12370.47
4.09 3.68 3.74 3.92 3.85 3.91 2.14 1.99
9.25 5.97 4.91 8.88 2.77 5.92 5.82 5.52
5.01 3.70 12.78 4.47 4.82 5.03 2.14 1.99
9.08 5.96 4.32 8.80 2.76 5.62 5.82 5.52
from single edge-notched specimens under three-point bending, and K SIC ¼ 5:8 0:6 MPa
pffiffiffiffiffi m from compact specimens
under tension. The difference between such results by Libonati et al. and the average value of K SðIþIIÞC here obtained is due to the following reason: the reduction of fracture toughness for a kinked crack as compared with the straight counterpart has not been taken into account in Ref. [13]. Therefore, the present study highlights that the near-tip stress-intensity factor of a kinked crack can be considerably lower than that of a straight crack. That has to be taken into account for prevention, diagnosis and treatment of bone traumas.
7. Conclusions In the present paper, the behaviour of a compact bone has been examined in terms of fracture toughness. Fracture toughness has been experimentally evaluated through specimens obtained from the femur diaphysis of a bovine. The influence of the local biaxial stress state on the fracture behaviour has been analyzed by employing two specimen types.
A. Carpinteri et al. / Engineering Fracture Mechanics 174 (2017) 44–53
53
As a matter of fact, when the osteons alignment is perpendicular to the loading direction (transversal specimens), the stress state is biaxial due to normal stresses produced by bending and shear stresses at the cement line interface between osteons and interstitial lamellae. On the other hand, when the osteons alignment is parallel to the loading direction (longitudinal specimens), the stress state is uniaxial. Then fracture toughness values have been computed by modifying the Two-Parameter Model originally formulated for crack propagating in pure Mode I loading. Such a modified version is herein proposed for mixed mode (Mode I together with Mode II). The theoretical results obtained are compared with some data available in the literature, by highlighting that the value of the near-tip stress-intensity factor of a kinked crack can be considerably lower than that for a straight crack. Acknowledgements The authors gratefully acknowledge the financial support of the Italian Ministry of Education, University and Research (MIUR). References [1] Marks SC, Popoff SN. Bone cell biology: the regulation of development structure, and function in the skeleton. Am J Anat 1988;183:1–44. [2] Keaveny TM, Hayes WC. Mechanical properties of cortical and trabecular bone. In: Hall BK, editor. Bone. Bone growth-B, vol. 7. Boca Raton (USA): CRC Press; 1993. p. 285–344. [3] An YH, Draughn RA. Mechanical testing of bone and the bone-implant interface. Boca Raton (USA): CRC Press; 2000. [4] Li S, Abdel-Wahab A, Silberschmidt VV. Analysis of fracture processes in cortical bone tissue. Eng Fract Mech 2013;110:448–58. [5] Ural A, Mischinski S. Multiscale modeling of bone fracture using cohesive finite elements. Eng Fract Mech 2013;103:141–52. [6] Nalla RK, Kinney JH, Ritchie RO. Mechanistic fracture criteria for the failure of human cortical bone. Nat Mater 2003;2:164–8. [7] Nalla RK, Kruzic JJ, Ritchie RO. On the origin of the toughness of mineralized tissue: microcracking or crack bridging? Bone 2004;34:790–8. [8] Nalla RK, Kruzic JJ, Kinney JH, Ritchie RO. Effect of aging on the toughness of human cortical bone: evaluation by R-curves. Bone 2004;35:1240–6. [9] Nalla RK, Kruzic JJ, Kinney JH, Ritchie RO. Mechanistic aspects of fracture and R-curve behaviour in human cortical bone. Biomaterials 2005;26:217–31. [10] Nalla RK, Stölken JS, Kinney JH, Ritchie RO. Fracture in human cortical bone: local fracture criteria and toughening mechanisms. J Biomech 2005;38:1517–25. [11] Koester KJ, Ager III JW, Ritchie RO. The true toughness of human cortical bone measured with realistically short cracks. Nat Mater 2008;7:672–7. [12] Launey ME, Buehler MJ, Ritchie RO. On the mechanistic origins of toughness in bone. Annu Rev Mater Res 2010;40:25–53. [13] Libonati F, Vergani L. Bone toughness and crack propagation: an experimental study. Proc Eng 2014;74:464–7. [14] Libonati F, Colombo C, Vergani L. Design and characterization of a biomimetic composite inspired to human bone. Fatigue Fract Eng Mater Struct 2014;37:772–81. [15] Libonati F, Vergani L. Understanding the structure-property relationship in cortical bone to design a biomimetic composite. Compos Struct 2016;139:188–98. [16] Libonati F, Gu GX, Qin Z, Vergani L, Buehler MJ. Bone-inspired materials by design: toughness amplification observed using 3D printing and testing. Adv Eng Mater 2016;18:1354–63. [17] Jenq Y, Shah SP. Two parameter fracture model for concrete. J Eng Mech 1985;111:1227–41. [18] John R, Shah SP, Jenq Y. A fracture mechanics model to predict the rate sensitivity of mode I fracture for concrete. Cem Concr Res 1987;17:249–62. [19] Jenq Y, Shah SP. On the concrete fracture testing methods. In: Proceeding of the international workshop on fracture toughness and fracture energy, Sendai. p. 443–63. [20] Katz JL. The structure and biomechanics of bone. In: Vincent JFV, Currey JD, editors. The mechanical properties of biological materials (symp soc exp biol 34). Cambridge: Cambridge University Press; 1980. p. 137–68. [21] Rho JY, Kuhn-Spearing L, Zioupos P. Mechanical properties and the hierarchical structure of bone. Med Eng Phys 1998;20:92–102. [22] Martin BR, Burr DB, Sharkey NA. Skeletal tissue mechanics. New York: Springer; 1998. [23] Paris PC. The mechanics of fracture propagation and solutions to fracture arrestor problems Document D2-2195. Boeing Company; 1957. [24] Kitagawa H, Yuuki R, Ohira T. Crack-morphological aspects in fracture mechanics. Eng Fract Mech 1975;7:515–29. [25] Cotterell B, Rice JR. Slightly curved or kinked cracks. Int J Fract 1980;16:155–69. [26] ASTM E399-12e3. Standard test method for linear-elastic plane-strain fracture toughness KIc of metallic materials. [27] Tada H, Paris PC, Irwin GR. The stress analysis of cracks handbook. 3rd ed. New York: ASME Press; 2000. [28] Khanal SP, Mahfuz H, Rondinone AJ, Leventouri T. Improvement of the fracture toughness of hydroxyapatite (HAp) by incorporation of carboxyl functionalized single walled carbon nanotubes (CfSWCNTs) and nylon. Mater Sci Eng C 2016;60:204–10.
Contents lists available at ScienceDirect
Engineering Fracture Mechanics journal homepage: www.elsevier.com/locate/engfracmech
Modified two-parameter fracture model for bone Andrea Carpinteri a, Filippo Berto b, Giovanni Fortese a, Camilla Ronchei a, Daniela Scorza a, Sabrina Vantadori a,⇑ a b
Department of Civil-Environmental Engineering and Architecture, University of Parma, Parma, Italy NTNU, Department of Engineering Design and Materials, Richard Birkelands vei 2b, 7491 Trondheim, Norway
a r t i c l e
i n f o
Article history: Received 16 October 2016 Received in revised form 7 November 2016 Accepted 7 November 2016 Available online 9 November 2016 Keywords: Fracture toughness Cortical bone Quasi-brittle material Two-Parameter Model
a b s t r a c t The analysis of the bone fracture behaviour is fundamental for prevention, diagnosis and treatment of traumas. In the present paper, an experimental campaign on fracture behaviour of bovine femoral cortical bones is conducted to characterise the fracture toughness, K SIC , which is related to the structure and load-bearing capacity of bones. Firstly, K SIC is evaluated through a two-parameter model originally proposed for quasi-brittle materials. To take into account the crack deflection (kinked crack) due to osteons orientation, the twoparameter model is modified by applying the Castigliano theorem. Fracture toughness results here obtained are compared with those related to a femur of an 18-month-old bovine, available in the literature. Ó 2016 Elsevier Ltd. All rights reserved.
1. Introduction Bone is a specialised tissue which has important functions, both metabolic and mechanical [1–3]. The load-bearing capacity of bones is limited up a certain extent, beyond which they fail in a brittle manner [4,5]. The analysis of the bone fracture behaviour is fundamental for prevention, diagnosis and treatment of traumas. Basic parameters which represent the structure and functions of bone have to be measured, such as its fracture toughness [6–16]. Depending on the skeletal locations, bones may be: (i) long bones (limbs, ribs, clavicles); (ii) flat bones (skull, scapula, pelvis); (iii) short bones (vertebrae, sternum). Generally most of fractures occur in long bones due to their skeletal locations, and more precisely in the diaphysis of such bones, i.e. in their central part, which represents the longest one (Fig. 1). In the present paper, an experimental analysis of the fracture behaviour of bovine cortical bones is carried out, where specimens are extracted from diaphysises. The experimental programme is conducted to characterise the fracture toughness by employing a Two-Parameter Model (TPM), originally proposed for concrete [17–19], that is, for a quasi-brittle material showing a non-linear slow crack growth before the peak load is reached. In bones, such a behaviour is produced by mechanisms of extrinsic toughening categorised in four classes [11]: (i) constrained microcracking; (ii) crack deflection and twist; (iii) uncracked-ligament bridging; (iv) collagen-fibril bridging. The above TPM is based on experimental data obtained from three-point bending tests by using single edge-notched specimens, and employs linear elastic fracture mechanics expressions valid for Mode I loading. However, for the bone material, such a model cannot be applied in its original formulation since the crack starting from notch may deflect. In order to understand the cause of such a deflection under Mode I loading (three-point bending), the hierarchical structure of bone has to be briefly examined. More precisely, five levels can be listed [3,20,21]: (1) whole bone at the ⇑ Corresponding author. E-mail address: [email protected] (S. Vantadori). http://dx.doi.org/10.1016/j.engfracmech.2016.11.002 0013-7944/Ó 2016 Elsevier Ltd. All rights reserved.
A. Carpinteri et al. / Engineering Fracture Mechanics 174 (2017) 44–53
45
Nomenclature a0 a1 ; a2 a B Ci Cu E F G
notch-depth segments of the kinked crack branch effective critical crack length specimen width linear elastic compliance or initial compliance unloading compliance elastic modulus virtual force total energy rate
K SIC
critical stress-intensity factor under Mode I (fracture toughness)
K SðIþIIÞC
critical stress-intensity factor under mixed mode i-th deflected segment along the kinked crack path specimen length peak load loading span total strain energy specimen depth relative displacement of the crack surfaces
li L Pmax S UT W DF
macrostructural level; (2) compact bone and cancellous bone block at the architecture level; (3) osteon and trabecula at the microstructural level; (4) lamella at the sub-microstructural level; (5) collagen fibril, non-collagenous and mineral components at the ultra-structural level. Osteons are more or less regular cylindrical structures, whose length ranges from 3 to 12 mm (Fig. 2). The osteons are oriented parallel to the bone axis, which consists of a vascular canal (named Haversian canal) surrounded by concentric lamellae. The interface between osteon and interstitial lamellae is called cement line (Fig. 2).
articular cartilage spongy bone
medulla medulla
marrow compact bone EPIPHYSIS
DIAPHYSIS
marrow EPIPHYSIS
Fig. 1. Schematic illustration of a long bone.
eon ost vascular canal cement line concentric lamellae Fig. 2. Cortical bone microstructural level.
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A. Carpinteri et al. / Engineering Fracture Mechanics 174 (2017) 44–53
Under three-point bending loading, when the osteons alignment is perpendicular to the loading direction (transversal specimens), the stress state is biaxial due to normal stresses produced by bending and shear stresses produced by slipping, at the cement line interface between osteon and interstitial lamellae [22]. In such a case, the crack starting from the notch is subjected to mixed mode loading (Mode I together with Mode II). On the other hand, when the osteons alignment is parallel to the loading direction (longitudinal specimens), the stress state is uniaxial. In such a case, crack starting from the notch is subjected to Mode I loading. The Two-Parameter Model can be directly applied for the latter case, whereas a modified formulation is hereafter proposed for the former case in order to take into account that Mode II loading is also present at crack tip. Firstly, the unloading compliance expression related to a singly kinked crack is determined by employing the Castigliano theorem in the way suggested by Paris [23], and the effective crack length is computed. Then, the critical stress-intensity factor at crack tip (fracture toughness K SIC ) is evaluated as a function of that for a straight crack having length equal to the projected length of the effective critical kinked crack [24,25]. The fracture toughness is evaluated using both transversal and longitudinal notched specimens obtained from a cadaveric femur diaphysis of a 24-month-old bovine. The fracture toughness results obtained herein for the above two specimen orientations are compared with those related to a femur of an 18-month-old bovine [13], experimentally determined according to the standard ASTM E399-12e3 [26]. 2. Basic composition and structure of long bones Bone is devoted to both metabolic and mechanical functions [1–3]. The former consists in regulation of calcemia during both bone mineralisation and resorption [2], and requires the participation of bone cells. The latter consists in muscles insertion, levers formation for muscles response and soft tissues protection, allowing for standing and moving [1]. As has been previously discussed, bones may be long, flat or short bones [3]. Two different regions may be distinguished in a long bone: the diaphysis, that is, the central shaft which represents the longest part, and the epiphyses at the two extremities of the long bone (Fig. 1). Articular cartilage covers the external epiphyseal surface (Fig. 1). The diaphysis consists of (see Fig. 1): (i) a thick outer layer of compact bone (also named cortical bone) made of a calcified bone matrix and soft structures (i.e. blood vessels), and (ii) a medulla, which corresponds to an axial cylinder containing marrow. The epiphyses show a thin outer compact bone (Fig. 1). The medulla is prevalent and consisting of a frame of osseous trabeculae (also named spongy bone); such trabeculae delimit irregular intercommunicating spaces containing marrow. About 90% of the diaphysis is calcified, whereas 10% consists of nonosseous cells, blood vessels, nerve fibres and bone marrow. The calcified bone matrix includes two components: (a) the organic matrix, consisting of both type I collagen fibrils (for over 90% in volume) and non-collagenous components (remaining 10% in volume), and (b) mineral substance. Collagen fibrils are formed by the assemblage of filamentous molecules arranged in a helical configuration. The molecules are stabilized by intra- and intermolecular cross-links, which are essential for the tensile strength of the fibrils. Noncollagenous components include proteins, proteoglycans, phospholipids, glycoproteins and phosphoproteins. The mineral substance is a calcium phosphate hydroxyapatite. 3. Two-parameter model According to the Two-Parameter Model [17–19], the specimens present a notch in the lower part of the middle crosssection (Fig. 3), and the following specimen sizes are suggested: width (B) depth (W) length (L) = 76.2 mm 152.4 mm 711.2 mm; loading span S = 609.6 mm; notch-depth ratio = a0 =W = 1/3; notch width <3.175 mm, and loading-span/depth ratio = S=W = 4 (Fig. 3). Note that the specimen sizes can be increased/decreased proportional to the aforementioned ones. Under three-point bending test, the specimen is monotonically loaded in Crack Mouth Opening Displacement (CMOD) control. After the peak load (Pmax) is reached, the post-peak stage follows and, when the force (P) is equal to about 95% of the peak load, the specimen is fully unloaded (Fig. 4). Then, the specimen is reloaded up to failure (Fig. 4). The linear elastic compliance C i (i.e. the initial compliance, Fig. 4) is used to determine the elastic modulus E [27]:
E¼
6Sa0 Vða0 Þ
ð1Þ
Ci W 2 B
Further, the parameter V can be expressed as follows [27]:
Vða0 Þ ¼ 0:76 2:28a0 þ 3:87a20 2:04a30 þ
0:66 1 a20
with a0 ¼
a0 W
ð2Þ
Therefore, if the crack propagates under pure Mode I, the effective critical crack length a to determine the critical stressintensity factor K SIC is obtained from the following equation by employing an iterative procedure:
E¼
6SaVðaÞ CuW 2B
ð3Þ
A. Carpinteri et al. / Engineering Fracture Mechanics 174 (2017) 44–53
47
P
a
W B
S L
a0
Fig. 3. Geometrical and testing configuration of specimen according to the Two-Parameter Model.
LOAD, P
1 0.95 (Pmax)
Ci Pmax
1
Cu CRACK MOUTH OPENING DISPLACEMENT, CMOD Fig. 4. Typical load – CMOD plot.
where C u is the unloading compliance (Fig. 4), and VðaÞ is deduced from Eq. (2) by replacing a0 with a. Finally, the critical stress-intensity factor under Mode I, K SIC , is computed by employing the measured value of the peak load, Pmax [27]:
K SIC ¼
3Pmax S pffiffiffiffiffiffi paf ðaÞ 2W 2 B
ð4Þ
where:
1 1:99 að1 aÞð2:15 3:93a þ 2:70a2 Þ f ðaÞ ¼ pffiffiffiffi p ð1 þ 2aÞð1 aÞ3=2
with a ¼
a W
ð5Þ
4. Modified two-parameter model A modified procedure is hereafter proposed when crack propagates under mixed mode loading (Mode I together with Mode II). Specimens geometry and experimental test procedure are equal to those presented in the previous Section. The elastic modulus, E, is determined through Eqs (1) and (2). If the crack propagates under mixed mode loading, the effective critical crack length is obtained by following the procedure hereafter described. Let us consider a body loaded by both a loading force, P, and a pair of virtual forces, F, in equilibrium for any distance, d (Fig. 5). The Castigliano theorem states that the displacement of any value of F (in its own direction) may be computed as follows [27]:
DF ¼
@U T @F F¼0
ð6Þ
where the total strain energy U T can be obtained by adding the strain energy due to the applied forces with no crack and that introducing the crack while holding constant the applied forces:
Z U T ¼ U NoCrack þ 0
A
@U T dA @A
being dA an infinitesimal increase of the crack area A (Fig. 5).
ð7Þ
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A. Carpinteri et al. / Engineering Fracture Mechanics 174 (2017) 44–53
P F
d+
F
dA
F
P Fig. 5. Body loaded by both a loading force P and a pair of virtual forces F, in equilibrium for any distance d.
By assuming a constant loading force, P, the total energy rate, G, is equivalent to the rate of increase of the total strain energy, that is:
G¼
@U T @A
ð8Þ
and the relative displacement of crack surfaces, DF , given by the virtual forces can be computed as follows:
DF ¼
Z A @U NoCrack @G þ dA @F F¼0 @F 0 F¼0
ð9Þ
Now consider a prismatic body containing a kinked crack subjected to both three-point bending loading and a pair of virtual forces (applied in the lower part of the beam) perpendicular to the crack faces (Fig. 6). In this case, the first term on the right-hand side of Eq. (9) is equal to zero, since it corresponds to the displacement produced by the pair of virtual forces (in their own direction) in an uncracked beam, whereas the second term is a function of the Stress-Intensity Factors (SIFs) for each loading mode, which are due to both the loading force P and the virtual forces F. For such a configuration, the total energy rate G is given by:
G ¼ GI þ GII ¼
i K 2I K 2II 1 h þ ¼ ðK IP þ K IF Þ2 þ ðK IIP þ K IIF Þ2 E E E
ð10Þ
where K IP and K IIP are the Mode I and the Mode II SIFs due to the loading force, respectively, whereas K IF and K IIF are the Mode I and the Mode II SIFs due to the virtual forces, respectively. By substituting Eq. (10) in Eq. (9), we obtain:
DF ¼
1 E
Z
A
0
@ ð2K IP K IF þ 2K IIP K IIF Þ @F
dA ¼ F¼0
2 E
Z 0
A
@K IF @K IIF dA K IP þ K IIP @F F¼0 @F F¼0
ð11Þ
Each SIF term in Eq. (11) can be expressed as a function of the corresponding Mode I SIF of a straight crack having length equal to the projected length of the actual kinked crack, and the crack tip location is identified by the coordinate x shown in Fig. 6. Since the SIF expressions of K IP , K IIP , K IF , and K IIF depend on the x-coordinate, the integral over the cracked area A (Eq. (11)) can be replaced by the following three integrals along a vertical line:
Z a0 þa1 cos h @K IF;0 @K IF;1 @K IIF;1 dx dx þ K IP;1 þ K IIP;1 @F F¼0 @F F¼0 @F a0 0 F¼0 Z a0 þða1 þa2 Þ cos h @K IF;2 @K IIF;2 dx K IP;2 þ K IIP;2 þ @F F¼0 @F a0 þa1 cos h F¼0
DF ¼
2 E
Z
a0
K IP;0
P
a2a
1
W B
x
F
F
a
a0
S L Fig. 6. Geometrical and testing configuration of specimen according to the modified Two-Parameter Model.
ð12Þ
49
A. Carpinteri et al. / Engineering Fracture Mechanics 174 (2017) 44–53
being h the crack kinking angle, and a1 þ a2 the total length of the deflected crack, as is detailed in the following (Fig. 6). All the SIF terms in Eq. (12) can be expressed as a function of the Mode I SIFs at the crack tip of a straight crack under three-point bending (KIP,0) and under a pair of virtual forces (KIF,0) [27]:
K IP;0 ¼
3PS pffiffiffiffiffiffi 1 1:99 að1 aÞð2:15 3:93a þ 2:70a2 Þ px f ðaÞ with f ðaÞ ¼ pffiffiffiffi 2 p 2W B ð1 þ 2aÞð1 aÞ3=2
2F 0:46 þ 3:06a þ 0:84ð1 aÞ5 þ 0:66a2 ð1 aÞ2 K IF;0 ¼ pffiffiffiffiffiffi gðaÞ with gðaÞ ¼ px ð1 aÞ3=2
and a ¼
x W
ð13aÞ
ð13bÞ
The Mode I and Mode II SIFs at the crack tip of a kinked crack characterised by a short branch, under three-point bending (KIP,1, K IIP;1 ) and under a pair of virtual forces (KIF,1, K IIF;1 ) can be evaluated following the approximated procedure proposed by Cotterell and Rice for a deflected crack characterised by a short branch in a centrally cracked infinite plate [25]:
K IP;1 ¼ K IP;0 cos3 K IIP;1 ¼ K IP;0 sin
ð14aÞ
h h cos2 2 2
K IF;1 ¼ K IF;0 cos3 K IIF;1 ¼ K IF;0 sin
h 2
ð14bÞ
h 2
ð14cÞ
h h cos2 2 2
ð14dÞ
The approximation is acceptable for a1 =a0 6 0:3. The Mode I and Mode II SIFs at the crack tip of a kinked crack characterised by a long branch (a1 =a0 P 0:3), under threepoint bending (KIP,2, K IIP;2 ) and under a pair of virtual forces (KIF,2, K IIF;2 ) can be evaluated following the approximated procedure proposed by Kitagawa at al. for a deflected crack characterised by a long branch in a centrally cracked infinite plate [24]:
K IP;2 ¼ K IP;0 cos3=2 h
ð15aÞ
K IF;2 ¼ K IF;0 cos3=2 h
ð15bÞ
K IIP;2 ¼ K IP;0 sin h cos1=2 h
ð15cÞ
K IIF;2 ¼ K IF;0 sin h cos1=2 h
ð15dÞ
By substituting Eqs. (13)–(15) in Eq. (12), we obtain:
" (Z 2 2 # Z a0 þa1 cosh Z a0 a0 2 @K IF;0 h @K IF;0 @K IF;0 3h 2h DF ¼ K IP;0 dx þ cos þ sin cos K IP;0 dx K IP;0 dx E 0 2 2 2 @F F¼0 @F F¼0 @F F¼0 0 0 Z a0 þða1 þa2 Þ cosh Z a0 þa1 cosh h i @K IF;0 @K IF;0 2 2 K IP;0 dx K IP;0 dx ð16Þ þ ðcos3=2 hÞ þ ðsinhcos1=2 hÞ @F F¼0 @F F¼0 0 0
It can be noticed that all integrals are formally equal, changing only the integral extremes. An empirical solution can be employed according to that proposed by Tada et al. [27]:
2 E
Z
z
K IP;0 0
@K IF;0 6PSz dz ¼ VðaÞ @F F¼0 EW 2 B
VðaÞ ¼ 0:76 2:28a þ 3:87a2 2:04a3 þ
0:66 1 a2
with a ¼
z W
ð17Þ
being z a generic integration variable. Therefore, once each integral of Eq. (16) is empirically resolved and the unloading compliance C u (equal to DF =P) is taken into account, we can explicit the elastic modulus from Eq. (16):
a a h h a0 þ a1 cos h 0 0 2 h ða0 þ a1 cos hÞV a0 V þ cos6 þ sin cos4 a0 V 2 2 2 W W W CuW B a0 þ a1 cos h þ a2 cos h a0 þ a1 cos h 2 ða0 þ a1 cos hÞV ð18Þ þ½cos3 h þ sin h cos h ða0 þ a1 cos h þ a2 cos hÞV W W
E¼
6S
2
Now a2 can be obtained from Eq. (18) by employing an iterative procedure. Therefore, the effective critical crack length is equal to a ¼ a0 þ ða1 þ a2 Þ cos h with a1 ¼ 0:3a0 .
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A. Carpinteri et al. / Engineering Fracture Mechanics 174 (2017) 44–53
As is shown in Fig. 6, the kinked crack branch consists of the two segments, named a1 and a2 . If the value of a2 obtained from Eq. (18) is negative, it means that the effective crack length is a ¼ a0 þ a1 , with a1 6 0:3a0 . Therefore, the expression of DF in such a case is given by:
DF ¼
2 E
Z
a0
K IP;0 0
Z a0 þa1 cos h @K IF;0 @K IF;1 @K IIF;1 dx dx þ K IP;1 þ K IIP;1 @F F¼0 @F F¼0 @F a0 F¼0
ð19Þ
and a1 is determined from the following equation by employing an iterative procedure:
E¼
a a a0 þ a1 cos h 0 0 2 h 6 h 4 h ða a þ sin cos V þ a cos hÞV V a þ cos 0 0 1 0 2 2 2 W W W CuW 2B 6S
ð20Þ
Finally, the critical stress-intensity factor under mixed mode, K SðIþIIÞC , is computed through Eqs. (4) and (5), by employing the above effective critical crack lengths. 5. Specimen preparation and experimental set-up The specimens tested have been extracted from a cadaveric femur diaphysis of a 24-month-old bovine, two days after slaughter. We have removed the marrow and obtained test samples by means of a low-speed saw and a milling machine. Such samples have been stored in saline solution at 3 °C for 20 h, then at room temperature for 2 h until testing. The specimens, with prismatic shape and geometrical sizes shown in Table 1, present a notch in the lower part of the middle cross-section (Fig. 6). The osteons alignment is perpendicular to the loading direction for 6 specimens (named transversal specimens) and parallel to the loading direction for 2 specimens (named longitudinal specimens). The former specimens are extracted from anterior (FA-1 and FA-2 in Table 1), posterior (FP), medial (FM-1 and FM-2) and lateral (FL) cortical bone, whereas the latter specimens from posterior cortical bone (FP-long1 and FP-long2). Specimens are tested under three-point bending. Testing is performed by means of an Instron 8862 testing machine, under crack mouth opening displacement control, employing a clip gauge at an average speed equal to 0.1 mm h1. All specimens exhibit a non-linear slow crack growth before the peak load is reached. Note that an extension of the length L has been made in some specimens by employing epoxy resins, having previously verified that such an extension would not influence the fracture behaviour of bone. The above extension is needed in order to make it possible to use the three-point flexural bend test equipment shown in Fig. 7.
Table 1 Geometrical sizes of the tested specimens. SPECIMEN no.
W [mm]
B [mm]
L [mm]
a0 [mm]
FA-1 FA-2 FP FM-1 FM-2 FL FP-long1 FP-long2
17.01 21.50 9.90 21.80 10.67 17.10 12.56 13.23
10.02 11.20 4.04 9.33 9.79 7.60 4.86 5.69
100.0 100.0 124.0 100.0 100.0 100.0 122.0 120.0
8.56 5.73 3.72 8.35 2.75 4.93 5.45 4.93
Fig. 7. Experimental testing set-up.
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A. Carpinteri et al. / Engineering Fracture Mechanics 174 (2017) 44–53
6. Results and discussion Values of fracture toughness, K SðIþIIÞC , of the above specimens are computed by employing both the proposed modified TPM and the load-CMOD curves. The initial compliance, the unloading compliance, the kinking angle and the peak load for each tested specimen are listed in Table 2. When the kinking angle value is not constant along the crack path, as is shown for example in Fig. 8 related to the specimen FA-2, the orientation of the first deflected segment such that liþ1 > li (i = 1, . . . , N, where N is a positive integer) is taken into account in the calculation of the fracture toughness. Fig. 9 shows the fracture surfaces of each specimen. Some specimens appear irregularly spray-painted since the Digital Image Correlation (DIC) technique is used to extract 2D full-field maps of displacements and strains in the mid-span zone of such specimens. Fracture toughness values K SðIþIIÞC together with the corresponding effective critical crack lengths a are listed in Table 3 (3rd and 4th columns). It can be noticed that the fracture toughness values significantly depend on the osteons alignment with respect to the loading direction. As a matter of fact, when the osteons alignment is perpendicular to the loading direction, crack grows under mixed mode (Fig. 9(a)–(f)), and a higher resistance to fracture is observed; when the osteons alignment is parallel to the loading direction, crack grows under Mode I (Fig. 9(g) and (h)), and a lower resistance to fracture is observed. Fracture toughness values computed according to the TPM, K SIC , are also listed in Table 3, together with the corresponding effective critical crack lengths a (6th column) and the elastic modulus values E. Note that, in general, the former critical SIFs (K SðIþIIÞC ) are significantly lower than the latter (K SIC ), up to about 70% for the specimen FP. pffiffiffiffiffi For transversal specimens, the average value of K SðIþIIÞC (3:87 0:15 MPa m) is in the range of the cortical bone values [28]. Such a value is then compared with the results determined by Libonati and Vergani [13] according to ASTM standards pffiffiffiffiffi m
[26], using the Linear Elastic Fracture Mechanics and considering cracks under pure Mode I loading: K SIC ¼ 5:6 0:1 MPa Table 2 Initial and unloading compliance, kinking angle and peak load for each tested specimen. SPECIMEN no.
C1 [1/MPa]
Cu [1/MPa]
h [°]
Pmax [N]
FA-1 FA-2 FP FM-1 FM-2 FL FP-long1 FP-long2
0.00032430 0.00006186 0.00068506 0.00013644 0.00014505 0.00013682 0.00067626 0.00039752
0.00038546 0.00006558 0.00094390 0.00015183 0.00014594 0.00016944 0.00079355 0.00050274
45 8 67 35 44 50 0 0
450.09 1097.66 263.22 808.91 457.73 585.96 74.95 101.07
Fig. 8. Procedure used in order to measure the kinking angle.
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A. Carpinteri et al. / Engineering Fracture Mechanics 174 (2017) 44–53
(a)
(b)
FA-1
(c)
FA-2
(d)
FM-1
FP
(e)
(f)
FM-2
(g)
FL
(h)
FP-long1
FP-long2
Fig. 9. Fracture surfaces: mixed mode for the transversal specimens (a) FA-1, (b) FA-2, (c) FP, (d) FM-1, (e) FM-2, (f) FL, and pure Mode I for the longitudinal specimens (g) FP-long1 and (h) FP-long1. Table 3 Elastic modulus, fracture toughness and critical effective crack length computed through both the modified TPM and the TPM. SPECIMEN no.
E [MPa]
K SðIþIIÞC [MPa m1/2]
a [mm]
K SIC [MPa m1/2]
a [mm]
FA-1 FA-2 FP FM-1 FM-2 FL FP-long1 FP-long2
13692.54 14393.65 14038.29 14253.00 13488.94 14042.47 12371.99 12370.47
4.09 3.68 3.74 3.92 3.85 3.91 2.14 1.99
9.25 5.97 4.91 8.88 2.77 5.92 5.82 5.52
5.01 3.70 12.78 4.47 4.82 5.03 2.14 1.99
9.08 5.96 4.32 8.80 2.76 5.62 5.82 5.52
from single edge-notched specimens under three-point bending, and K SIC ¼ 5:8 0:6 MPa
pffiffiffiffiffi m from compact specimens
under tension. The difference between such results by Libonati et al. and the average value of K SðIþIIÞC here obtained is due to the following reason: the reduction of fracture toughness for a kinked crack as compared with the straight counterpart has not been taken into account in Ref. [13]. Therefore, the present study highlights that the near-tip stress-intensity factor of a kinked crack can be considerably lower than that of a straight crack. That has to be taken into account for prevention, diagnosis and treatment of bone traumas.
7. Conclusions In the present paper, the behaviour of a compact bone has been examined in terms of fracture toughness. Fracture toughness has been experimentally evaluated through specimens obtained from the femur diaphysis of a bovine. The influence of the local biaxial stress state on the fracture behaviour has been analyzed by employing two specimen types.
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As a matter of fact, when the osteons alignment is perpendicular to the loading direction (transversal specimens), the stress state is biaxial due to normal stresses produced by bending and shear stresses at the cement line interface between osteons and interstitial lamellae. On the other hand, when the osteons alignment is parallel to the loading direction (longitudinal specimens), the stress state is uniaxial. Then fracture toughness values have been computed by modifying the Two-Parameter Model originally formulated for crack propagating in pure Mode I loading. Such a modified version is herein proposed for mixed mode (Mode I together with Mode II). The theoretical results obtained are compared with some data available in the literature, by highlighting that the value of the near-tip stress-intensity factor of a kinked crack can be considerably lower than that for a straight crack. 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