d e n t a l m a t e r i a l s 3 0 ( 2 0 1 4 ) 1224–1233
Available online at www.sciencedirect.com
ScienceDirect journal homepage: www.intl.elsevierhealth.com/journals/dema
On the possibility of estimating the fracture toughness of enamel Miguel Ángel Garrido a,∗ , Isabel Giráldez b , Laura Ceballos b , Jesús Rodríguez a a
Departamento de Tecnología Mecánica, Escuela Superior de Ciencias Experimentales y Tecnología, Universidad Rey Juan Carlos, C/ Tulipán s/n, Móstoles, Madrid, Spain b Departamento de Estomatología, Facultad de Ciencias de la Salud, Universidad Rey Juan Carlos, Avda. Atenas s/n, Alcorcón, Madrid, Spain
a r t i c l e
i n f o
a b s t r a c t
Article history:
Objectives. There are many works that have attempted to estimate the fracture toughness
Received 9 July 2013
of enamel by indentation techniques using equations whose success in determining the
Received in revised form
actual value of fracture toughness, rely on a particular three-dimensional pattern consisting
15 July 2014
of cracks growing from the edges of the indentation. Recently, an alternative methodology
Accepted 8 August 2014
based on an energetic approach has been developed to estimate the fracture toughness of coatings by depth sensing indentation that is not less affected by the cracks pattern generated. In this work, the energetic approach to indentation fracture toughness of bovine
Keywords:
enamel is presented and compared with those toughness values obtained using the tradi-
Enamel
tional expressions reported in the literature.
Indentation fracture toughness
Methods. Indentation tests were carried out using a diamond Berkovich indenter onto the
Energetic approach
enamel surface of eight incisors from bovines of two years old. A continuous stiffness measurement methodology was used with a frequency of 45 Hz and displacement amplitude of 2 nm up to a maximum penetration depth of 2000 nm. Results. The results showed that some modifications in the energetic methodology should be performed in order to apply it successfully. Significance. The fracture toughness values obtained using the traditional equation and applying the energetic methodology, were significantly different, although the values were within the range obtained by other authors. © 2014 Academy of Dental Materials. Published by Elsevier Ltd. All rights reserved.
1.
Introduction
Enamel is the hardest and stiffest tissue of mammals. Its microstructure consists of rods encapsulated by thin protein rich sheaths that are arranged parallel in a direction
perpendicular to the dentino-enamel junction (DEJ) from dentin to the outer enamel surface. The enamel microstructure of all mammals appears to be very similar on a histochemical and anatomic basis [1–5]. Numerous methods have been employed to experimentally measure the fracture toughness (KC ) of the enamel. The
∗ Corresponding author at: Escuela Superior de Ciencias Experimentales y Tecnología, Departamento de Tecnología Mecánica, Universidad Rey Juan Carlos, C/ Tulipán s/n, E 28933 Móstoles, Madrid, Spain. Tel.: +34 914887177; fax: +34 914888150. E-mail address: [email protected] (M.Á. Garrido). http://dx.doi.org/10.1016/j.dental.2014.08.364 0109-5641/© 2014 Academy of Dental Materials. Published by Elsevier Ltd. All rights reserved.
1225
d e n t a l m a t e r i a l s 3 0 ( 2 0 1 4 ) 1224–1233
determination of KC by indentation techniques is based on measuring the size of cracks induced in a material during indentation [6]. Several expressions are available to determine KC by this technique, depending on the indenter geometry and crack morphology [7–10]. In 1987, Laugier [10] determined that:
KC = xV
a 1/2 E 2/3 P l
H
(1)
c3/2
where a is the half-diagonal of the indentation impression, l is the crack length measured from the indentation imprint edge, and c is the crack length measured from the center of the indentation imprint. Additionally, Laugier [10] showed that the radial and half-penny models make similar predictions when xv = 0.015. The Eq. (1) was developed for ceramic materials and for the symmetrical Vickers indentations. Some efforts have been made to obtain similar equation but properly modified for Berkovich indentations. Ouchterlony [11] investigated the nature of the radial cracking and determined a modification factor for stress intensity factor to account the number of radial cracks, n, formed.
k1 =
n/2 1 + (n/2) sin(2/n)
(2)
The ratio of k1 values for n = 4 (Vickers indenter) and n = 3 (Berkovich indenter) is 1.073. Introducing this ratio in Eq. (1), a modified Laugier model adapted to Berkovich indentations can be obtained, according to Eq. (3).
KC = 1.073 · xV
a 1/2 E 2/3 P l
H
c3/2
(3)
However, these equations provide an actual value of the fracture toughness by indentation only if a particular pattern of cracks running from the indentation imprint vertices is obtained. This work has been developed to answer the question about what happens if cracks generated by indentation do not show that characteristic pattern. Concerning this point, there could be an alternative methodology based on the energy released during cracking process, as a measurement of the fracture toughness [12]. This energetic method calculates the fracture toughness comparing the area limiting by the load-displacement curve with crack generation and the hypothetical one if cracking does not occur [12]. For a thin film, Li et al. [12] determined the critical stressintensity factor assuming Mode I:
KIC =
E (1 − v2 )2CR
1/2 U ·
t
(4)
where E is the elastic modulus and is the Poisson’s ratio, KIC is the indentation fracture toughness of the material, U is a measure of the fracture energy, t is the film thickness and
2CR is the crack length. The product of 2CR and t gives the cracked area, Acrack , therefore, Eq. (4) can be written as:
KIC =
E (1 − 2 )
1/2 U ·
(5)
Acrack
Eq. (5) allows obtaining fracture toughness values from indentation tests. Under this testing method, the strain energy, U, can be obtained from the area between the hypothetical loading curve if no cracking exists and the experimental one. This equation can be used even when a particular cracks pattern is not obtained, although the critical point of Eq. (5) is to determine again the cracked area. Hereafter, Eq. (3) is referred in this paper as the traditional equation to distinguish the equation using the energetic methodology, Eq. (5). There is some scatter in the literature about the actual fracture toughness of enamel [13]. Hassan et al. [14] have reported values of human tooth enamel, using a Vickers indenter, in the range of 0.7–1.37 MPa m1/2 . Xu et al. [15] reported fracture toughness values of 0.84 MPa m1/2 for labial human enamel also using a Vickers indenter in the same range of those determined by Padmanabhan et al. [16]. Bajaj and Arola [17] reported fracture toughness values for human enamel that ranged from 1.79 MPa m1/2 to 2.37 MPa m1/2 obtained from Rcurve analysis, and Baldassarri et al. [18] obtained values of 0.5 MPa m1/2 and 1.3 MPa m1/2 for transversal and midsagital enamel orientation, respectively, using a Vickers indenter on rat tooth. There are various reasons that may cause the large variation in reported fracture toughness values. The biological nature of the enamel suggests that the compositional variations among specimens could impact to the toughness values [19,20]. Additionally, the microstructure orientation of enamel could also contribute to this variability [21]. Therefore, the aim of this study was to compare the fracture toughness values obtained from the energetic methodology using depth sensing indentation technique with those obtained applying the traditional equation based in a specific pattern of cracks.
2.
Materials and methods
2.1.
Preparation of specimens
Eight incisors were extracted from bovines of two years old. Teeth were cleaned and stored in artificial saliva (Table 1) prior to the tests. The labial surfaces of the specimens were polished
Table 1 – The composition of artificial saliva. Composition Sodium carboxymethyl cellulose Sorbitol Sodium chloride Potassium phosphate Calcium chloride, dihydrate Magnesium chloride, hexahydrate Dibasic sodium diphosphate Purified water
g/100 g of solution 1.000 3.000 0.084 0.120 0.015 0.005 0.034 bal
1226
d e n t a l m a t e r i a l s 3 0 ( 2 0 1 4 ) 1224–1233
in two steps, using a mechanical grinder (Labopol-5, Struers, Copenhagen, Denmark) with polishing cloths of alumina suspension slurry of 3 m and 0.020 m, respectively. This polish sequence provided an average roughness lower than 20 nm. This value was checked by atomic force microscopy measurements. Afterwards, the specimens were kept fully hydrated in artificial saliva at room temperature before the indentation tests. The composition of saliva is detailed in Table 1.
2.2.
Indentation tests
A preliminary study of Vickers indentations on the bovine enamel was carried out, following the ASTM standard requirements. Two different maximum loads were applied: 5 N, 20 N. The time to load and the dwell time was 15 s. Additionally, hardness and elastic modulus of bovine enamel surfaces were studied by depth sensing indentation (Nanoindenter XP-MTS System Corporation) using a Berkovich diamond indenter with a tip radius of 50 nm. A Continuous Stiffness Measurement module (CSM) [22,23] was chosen. A maximum penetration depth of 300 nm was fixed for all indentation tests. Three rows of 20 indentations were performed on each sample. Each indentation was separated by 100 m from each other. During the loading branch, continuous loading–unloading cycles with amplitude of 2 nm and a frequency of 45 Hz were superimposed. The Oliver–Pharr methodology [22] was applied on each of these partial unloading cycles, providing values of elastic modulus (E), and hardness (H), as a continuous function of load or penetration depth. Before each batch tests, the Berkovich indenter was calibrated on a standard fused silica.
2.3.
Fracture toughness analysis
Additionally, 5 indentations were done on each sample using the CSM module up to a maximum penetration depth of 2000 nm. This penetration depth was sufficient to generate a characteristic pattern of cracks, allowing us to apply Eqs. (3) and (5). After the indentation tests, the residual imprints were observed by scanning electron microscopy (SEM) (Hitachi S-3400N) under low vacuum condition. The fracture toughness of the enamel was determined using the traditional methodology and the energetic one. The application of each methodology has difficulties. To estimate the fracture toughness by the traditional methodology implies to obtain a characteristic pattern of cracks. This problem was overcome choosing adequate indentation test conditions. To determine the fracture toughness by the energetic methodology, it is necessary to calculate the energy released during the cracking process, U. The cracking area, Acrack , has to be estimated for both methodologies.
2.3.1.
Determination of U
The proposed procedure to calculate the energy released during the cracking process, is based on the contact stiffness variation due to cracking. Besides the cracking process, other phenomena may affect the contact stiffness during indentation, as quasi-plastic deformation phenomena associated with the movement of the water and protein phase. However,
given the conditions under which the indentation tests were made, with very high strain rates, the contribution of these phenomena to the variation of the contact stiffness, measured from the elastic unloading branch of each cycle, can be considered negligible. Additionally, it is well known that using a pyramidal indenter, the ratio of the contact area to the depth of the indentation, remains constant for an increasing indenter load, i.e. the Berkovich indentation exhibits geometrical similarity. Consequently, the strain within the material is constant, independent of the penetration depth, except for very small values, where the apex radius effect is dominant. Therefore, the plasticity effects should be equally important at low or high penetrations. If at low penetration depths, a linear tendency of the contact stiffness versus square root of the contact area is obtained, any different trend observed at high depths should not be a consequence of the plastic deformation of the enamel during the indentation process. Therefore, when the contact stiffness versus square root of the projected contact area does not show a linear tendency, it is, most probably, a consequence of a cracking process developed during the indentation process. It is well known [24–28] that the area under the loaddisplacement curve during the unloading process is the corresponding elastic work. However, if cracking occurs, part of the elastic energy stored during the indentation test will be released to create new crack surfaces. This will be reflected as a change in the contact stiffness and, consequently, the relationship between contact stiffness, S, and the squared root of the contact area, A, will not be linear, as the contact theory predicts [22,29]. Therefore, the linear tendency of contact stiffness versus squared root of the contact area will only remain before cracking occurs. Then, it is possible to identify in the S–A1/2 curve, a critical point associated with the onset of cracking (S* , A1/2* ). According to the Oliver–Pharr methodology [22], it is possible to calculate the load and penetration depth values corresponding to the critical point associated with the cracking initiation, P* and h* . For values (P, h) lower than the critical point, the indentation process will take place without cracking. For higher values, the indentation process will be characterized by the nucleation and propagation of cracks. The load-penetration depth curve for values lower than the critical one can be extrapolated according to the Kick’s law [27], P = E · h2 , to the maximum penetration depth or indentation load, obtaining a hypothetical curve characteristic of a non cracking process.
2.3.2.
Determination of Acrack
Following the assumption of Laugier to develop the indentation fracture toughness model for Berkovich indentation [10], the radial crack geometry inside the enamel was postulated in this work (see Fig. 1a). Under this assumption, the authors propose two possible ways to estimate the cracked area: using the crack lengths directly; and through the contact stiffness variation. The cracked area estimation through the crack length was determined assuming that the crack from the edge of the indentation imprint, run through the protein interlayer (Fig. 1b). This phenomenon was previously observed by other authors [14,30–33]. Under this assumption, the crack path around each prism was approached as a half of a
d e n t a l m a t e r i a l s 3 0 ( 2 0 1 4 ) 1224–1233
1227
where N corresponding to the number of cracks, a and c are cracks lengths defined in Fig. 1. Considering that the experimental conditions of the indentations were carefully selected to obtain a characteristic pattern composed of three cracks, each one running from each indenter Berkovich corner, N = 3. The cracking area estimation from the contact stiffness can be expressed as a function of the elastic modulus of the enamel according to the Oliver and Pharr methodology [22]. Therefore, the variation of the contact stiffness observed from the critical point (P* , h* ), is associated with a progressive decrease in the corresponding elastic modulus. The decrease of this property can be used to estimate the cracked area, mainly following the results of Budiansky and O’Connell [35], Case and Kim [36–38], Fan et al. [39], and others authors [40–42]. According to these studies, the elastic modulus, E, of the microcracked body can be expressed by: E = ES (1 − fε)
(10)
where Es refers to the non-microcraked elastic modulus. The function f depends on the spatial orientation of the microcracks [35,37]. The crack damage parameter ε can be expressed [35,40–42] as:
ε= Fig. 1 – (a) Crack geometry idealization along the load direction; (b) schematic of the sinuous path followed by the crack through the rods interfaces.
circumference (Fig. 1b). Additionally, this sinuous crack propagation, was considered effective from the end of the indentation imprint, a, to the end of the crack, c (Fig. 1b). Therefore, the length of each crack, Lc , could be written as: Lc =
(c − a) + a 2
(6)
In order to obtain the cracked area, an elliptical geometry of the crack shape in the depth direction was expected (Fig. 1a). The major axis was equal to the crack length, Lc , and the minor one was assumed equal to c/2 (Fig. 1a). Similar postulation was previously pointed out by Hayashi-Sakai et al. [34]. Therefore, the area of each crack, Ac , was written as: Ac =
Lc c 2 2
(7)
If N cracks were generated during the indentation process, the total cracked area, Acrack could be written as: Acrack = N · Ac
(8)
Substituting Eqs. (6) and (7) into Eq. (8), the total cracked area result:
Acrack = N ·
2 2 (2 − ) c + a·c 8 8
2
A2 P
· NV
(11)
where Nv is the volume number density of microcracks, A2 is the mean of the square of the cracked area and P is the mean crack perimeter. The volume number of density is defined by the relation between the number of cracks, N, and a representative volume, V [43]: NV =
N V
(12)
In this study and based on the cylindrical cavity model for cracked brittle solids subjected to indentation tests [44], a cylindrical representative volume of radius equal to the mean crack length, c, and total depth of c/2 was considered (Fig. 1). Therefore, the volume of material affected by the indentation process could be estimated by Eq. (13): V = · c2 ·
c 2
(13)
According to Fig. 1:
A=
2
c 1 2 2
P=
c 2
(14)
(15)
Substituting Eqs. (14) and (15) into Eq. (11):
(9)
ε=
c3 · NV 16
(16)
1228
d e n t a l m a t e r i a l s 3 0 ( 2 0 1 4 ) 1224–1233
Assuming radial crack geometry, as defined Laugier [10], with aligned crack orientation [25]: f =
16 (1 − 2 ) 3
(17)
where Q is a constant bounded between two limits. Substituting Eq. (22) into Eq. (24), this constant was equal to: Q1 =
2 34 64
(26)
where is the Poisson’s ratio of the non-microcracked body. Substituting Eqs. (16) and (17) into Eq. (10):
Similarly, if Eq. (23) is substituted into Eq. (24), Q constant was equal to:
1 − 2 3 N ES − E = c ES 3 V
Q2 =
(18)
The first term of the Eq. (18) represents the elastic modulus variation regarding to the value of the non-microcraked solid. If E represents the elastic modulus evolution, Eq. (18) could be written as: E 1 − 2 3 N = c ES 3 V
(19)
Using the Eq. (14), the total cracked area, Acrack could be estimated by Eq. (20):
c 2 1
Acrack = N 2 2
(20)
where N represents the number of cracks generated by the indentation process. Substituting Eq. (19) into Eq. (20), the total cracked area could estimate as: Acrack =
3 V E 8 (1 − 2 )c ES
(21)
Considering the sinuous path of the crack, through the protein interlayer (Figs. 1 and 2), the c parameter in Eqs. (13) and (21), changed to (/2)·c. However, concerning the volume affected by the indentation process, V (Eq. (13)), two limiting situations could be identified. A first circumstance in which the longitudinal axes of the prisms were oriented perpendicularly to the surface; and another situation where the longitudinal axes were oriented parallel to the surface. Considering the first extreme situation, the volume affected, V1 , can be written as: V1 = ·
2 c 2
c
2
=
3 2
c
(22)
For the second extreme situation, the volume, V2 , was: V2 = ·
2 c 2
c
22
=
2
3 2
c
(23)
Similarly, the cracked area (Eq. (21)) could be written as: Acrack =
3 V E 8 (1 − 2 )((/2)c) ES
(24)
Substituting Eq. (22) or (23) into Eq. (24), the total cracked area could be written as: Acrack = Q
E c2 (1 − 2 ) ES
(25)
34 64
(27)
Eq. (25) represented the total cracked area depending on the elastic modulus decrease with the penetration depth, recorded during the indentation process. Eqs. (9) and (25) represented the total cracked area obtained from different approximations. Now, it is possible to apply the energetic methodology through Eq. (2).
3.
Results
Fig. 2a and b shows two optical images of representative residual imprints obtained from Vickers indentations. For both loads, it is observed that the cracks progressed randomly from the vertices and faces of the indentation. Besides, the prisms geometry is clearly revealed, which suggests that the cracks predominantly run surrounding the hydroxyapatite rods, i.e. through protein interfaces. Fig. 2c shows a detail of a crack path, observing a preferential propagation along the protein interlayer. Similar phenomenon has been observed by other researchers [14,30–33]. Fig. 3 shows two SEM images of typical residual imprints from depth sensing indentations with Berkovich tip at different maximum penetration depths. Their corresponding curves of the contact stiffness versus squared root of the contact area for 300 nm (Fig. 3a) and 2000 nm (Fig. 3b) penetrations depths were also included. Indentations at maximum penetration depth of 300 nm were characterized by the absence of cracks at the edges of the residual imprint (Fig. 3a). The contact stiffness, S, showed a linear variation with the square root of the contact area, A, between the indenter and the material surface (Fig. 3a); then, the indentation process was carried out without cracking. By contrast, the indentations obtained at higher penetration depths of 2000 nm were characterized by the presence of cracks extended from the edges of the residual imprints (Fig. 3b). In addition, it was also observed how the cracks deflected during their propagation, and the contact stiffness lost its linear variation with the square root of contact area for values above a threshold one (Fig. 3b). Fig. 3a shows a SEM image of a representative Berkovich indentation up to maximum penetration depth of 300 nm. The size of the residual imprint ranged from 1.5 to 2 m, however, the zone affected by the indentation process is higher than the indentation imprint [45]. Consequently several rods participate in the indentation process. Therefore, it can be assumed that the mechanical properties obtained through the indentation test, were average values. Similar reasoning could be made for indentations up to 2000 nm of penetration depth.
d e n t a l m a t e r i a l s 3 0 ( 2 0 1 4 ) 1224–1233
1229
Fig. 2 – Typical cracks pattern obtained from Vickers indentations on enamel: (a) 5 N; (b) 20 N; (c) detail of a crack propagating along the protein interlayer.
Table 2 summarizes the hardness and elastic modulus of bovine enamel obtained from indentation tests at maximum penetration depth of 300 nm, where the loading process did not cause any cracking. The methodology followed to obtain
both properties was described by Oliver and Pharr [22]. The elastic modulus and hardness values remained roughly constant regardless of the penetration depth (Fig. 4a). Therefore, indentation size effect could be considered negligible under
Fig. 3 – Typical pattern of cracks obtained from nanoindentations carried out at different maximum penetration depths: (a) 300 nm; (b) 2000 nm. Their corresponding indentation contact stiffness versus squared root of the contact area plots were put over each image.
1230
d e n t a l m a t e r i a l s 3 0 ( 2 0 1 4 ) 1224–1233
Table 2 – Experimental results of elastic modulus, E, hardness, H and indentation fracture toughness, KC . Mean values and standard deviations (in brackets). The methodologies and equations followed to obtain the each property are included in the last column. Property
Value
Equation/Reference
Elastic modulus, E (GPa) Hardness, H (GPa) Crack length, c (m) Imprint size, a (m) Indentation fracture √ toughness, KC (MPa m)
92 (6) 4.7 (0.3) 12 (3) 7 (2) 0.7 (0.2)
Oliver–Pharr [22] Oliver–Pharr [22] – – Eq. (3) [10,11]
these test conditions. Additionally, the force-penetration curves fit a quadratic law, as predicted by the Kick’s law [27] (Fig. 4b). Table 2 includes fracture toughness data of enamel provided by the traditional equation, Eq. (3), obtained from the results reported by Laugier [10] and Ouchterlony [11].
Fig. 4 – (a) Elastic modulus and hardness values versus penetration depth of a representative indentation test on bovine enamel following the continuous stiffness measurement methodology up to maximum penetration depth of 300 nm; (b) load, P, versus squared penetration depth, h2 , curve for a representative indentation test on enamel up to 300 nm of penetration depth.
Fig. 5 – Elastic modulus evolution with the penetration depth during a representative Berkovich indentation test up to maximum penetration depth of 2500 nm.
Fig. 5 shows a representative variation of the elastic modulus versus penetration depth, corresponding to the Berkovich indentation on the enamel up to maximum penetration depth of 2500 nm. This figure revealed that the elastic modulus remained constant with the indentation depth up to a critical point was reached. The elastic modulus gradually decreased from this penetration depth. Consequently, two regions could be identified concerning to the elastic modulus evolution with the penetration depth. Both sections were fitted by linear tendencies. The critical point could be experimentally determined as the intersection between the linear fits of the E constant and E decreasing regions. This point corresponded to a penetration depth of 1000 nm (Fig. 5). The average value of the elastic modulus variation, E, is included in Table 3. Fig. 6a shows a typical curve of contact stiffness versus squared root of the contact area. The linearity only remained up to a squared root of the contact area equivalent to a maximum penetration depth of 1000 nm, roughly. The critical point of cracking onset (S* , A1/2* ) was calculated for each test. Following the methodology described in the Fracture toughness analysis section, the critical point associated with the cracking initiation, P* and h* , was determined on each test (Fig. 6b). The load-penetration depth curve for values lower than the critical one was extrapolated to the maximum penetration depth of 2000 nm, obtaining the hypothetical curve without associated cracking process. Fig. 6b shows a comparative example between the experimental indentation curve, with cracking, and the hypothetical indentation curve, without cracking. The difference in area between the two force–displacement curves gave the energy released during the cracking process, U. The average values and standard deviations are included in Table 3. After that, the cracked area was determined using both possibilities: through the equation based measurement of crack length, c, Eq. (9); and the equation based on the crack length, c, and the elastic modulus variation due to the cracking process, Eq. (25). Table 3 summarizes the average cracked area value calculated by both equations and the fracture toughness
1231
d e n t a l m a t e r i a l s 3 0 ( 2 0 1 4 ) 1224–1233
Table 3 – Experimental results of fracture toughness of the enamel using the alternative energetic methodology. Average values of the energy released to create new crack surfaces, U; the elastic modulus variation, E; and the cracked area Acrack were included. The corresponding standard deviations have been also included (in brackets). Property U (mN nm) E (GPa) Acrack (m2 ) Acrack (m2 ) Crack length, c (m) √ Indentation fracture toughness, KC (MPa m) by crack measurement √ Indentation fracture toughness, KC (MPa m) by E measurement
via the alternative energetic methodology using both expressions for the cracked area.
4.
Discussion
According to the results of Vickers indentations, in which a random patter of cracks was obtained, the applicability of traditional equations (Eq. (3)) is compromised. Consequently, the possibility of using an alternative methodology, that would not have to depend on the generation of a particular pattern of cracks, has been explored. In that sense, it seems that energetic methodology fulfills this condition (Eq. (5)). To check its reliability, an experimental campaign of Berkovich indentations at high penetration depths was designed, so that a characteristic pattern of cracks was obtained allowing the application of both methodologies. Both methodologies need to know the elastic modulus of the indented material; therefore, additional indentation tests at small penetration depths, under free induced cracks conditions, were made on the enamel. Concerning elastic modulus and hardness obtained from indentations up to maximum penetration depth of 300 nm, the average values included in Table 2 were very similar to
Value 2710 (126) 16 (3) 152 (10) 86 (3) 135 (3) 12 (3) 1.34 (0.14) 1.77 (0.23) 1.41 (0.23)
Equation/Reference – – According Eq. (9) Eq. (25) using Q1 Eq. (25) using Q2 – Using Eq. (5) and Acrack from Eq. (9) Using Eq. (5) and Acrack from Eq. (25) with Q1 Using Eq. (5) and Acrack from Eq. (25) with Q2
those reported in previous works [15,46–49]. Furthermore, the contact stiffness evolution versus squared root of the contact area was linear (Fig. 3a). Both circumstances let to assume that the indentation process designed to obtain the elastic modulus and hardness, was developed under free cracks conditions, or at least, was developed so that the cracks inherent to the enamel microstructure did not have a significant effect. Figs. 2 and 3b show how the cracks appeared zigzagged. Similar mechanism was also observed by other researchers [14,32,33], suggesting that it is more difficult for cracks to cut through the enamel rods that to go around them. The fracture toughness value obtained through the traditional methodology (Table 2) was in the order of 0.7 MPa*m1/2 . This value is in the range of the results reported by other authors [14]. When the fracture toughness values obtained by both methodologies, traditional (Table 2) and energetic (Table 3) were compared, a considerable difference could be observed. The toughness obtained by following the energetic methodology was, at least, two times the value obtained by the traditional one. These values were similar to those reported by other authors who have used fracture mechanics tests to evaluate the fracture toughness of enamel. Bechtle et al. [30] used a three-point bending test to estimate the fracture
Fig. 6 – (a) Contact stiffness versus squared root of the contact area for a representative indentation test; (b) indentation load versus penetration depth of an experimental and hypothetical indentation curve.
1232
d e n t a l m a t e r i a l s 3 0 ( 2 0 1 4 ) 1224–1233
toughness of enamel. They reported values that range from 0.8 to 1.5 MPa*m1/2 at the beginning of crack propagation up to 4.4 MPa*m1/2 for 500 m crack extension. In a recently published work, the fracture toughness values derived from CT samples consisting of human enamel cubes mounted in epoxy resin ranged from 0.5 to 0.8 MPa*m1/2 at the beginning of crack propagation up to 2.4 MPa*m1/2 at 1.5 mm crack extension [17].
5.
Conclusions
In this work the suitability of the equation traditionally used for Berkovich indentation to determine the indentation toughness of the enamel has been analysed. The difficulty of obtaining an appropriate crack pattern requires the investigation of alternative ways to estimate the fracture toughness. In this work, an alternative energetic methodology, adapted to the depth sensing indentation results, has been developed. The method is based on the contact stiffness variation due to the cracking process. The values obtained using this methodology, were clearly different than those reported by other authors that use traditional indentation equations.
references
[1] Habelitz S, Marshall SJ, Marshall GW, Balooch M. Mechanical properties of human dental enamel on the nanometre scale. Arch Oral Biol 2001;46:173–83. [2] Poole DFG, Brooks AW. The arrangement of crystallites in enamel prisms. Arch Oral Biol 1961;5:14–27. [3] Ge J, Cui FZ, Wang XM, Feng HL. Property variations in the prism and the organic sheath within enamel by nanoindentation. Biomaterials 2005;26:3333–9. [4] Oesterle LJ, Shellhart WC, Belanger GK. The use of bovine enamel in bonding studies. Am J Orthod Dentofac Orthop 1998;114:514–20. [5] Sanches RP, Otani C, Damiao AJ, Miyakawa W. AFM characterization of bovine enamel and dentine after acid etching. Micron 2009;40:502–6. [6] Kruzic JJ, Kim DK, Koester KJ, Ritchie RO. Indentation techniques for evaluating the fracture toughness of biomaterials and hard tissues. J Mech Behav Biomed 2009;2:384–95. [7] Lawn BR, Evans AG, Marshall DB. Elastic/plastic indentation damage in ceramics: the median/radial crack system. J Am Ceram Soc 1980;63:574–81. [8] Anstis GR, Chantikul P, Lawn BR, Marshall DB. A critical evaluation of indentation techniques for measuring fracture toughness: I. Direct crack measurements. J Am Ceram Soc 1981;64:533–8. [9] Niihara K, Morena R, Hasselmann DPH. A fracture mechanics analysis of indentation-induced Palmqvist crack in ceramics. J Mater Sci Lett 1983;2:221–3. [10] Laugier MT. Palmqvist indentation toughness in WC–Co composites. J Mater Sci Lett 1987;6:897–900. [11] Ouchterlony F. Stress intensity factors for the expansion loaded star crack. Eng Fract Mech 1976;8:447–8. [12] Li X, Diao D, Bhushan B. Fracture mechanisms of thin amorphous carbon films in nanoindentation. Acta Mater 1997;45:4453–61. [13] Soderholm KJ. Review of the fracture toughness approach. Dent Mater 2010;26:63–77. [14] Hassan R, Caputo AA, Bunshash RF. Fracture toughness of human enamel. J Dent Res 1981;60:820–7.
[15] Xu HHK, Smith DT, Jahanmir S, Romberg E, Kelly JR, Thompson VP, et al. Indentation damage and mechanical properties of human enamel and dentin. J Dent Res 1998;77:472–80. [16] Padmanabhan SK, Balakrishnan A, Chu MC, Kim TN, Cho SJ. Micro-indentation fracture behavior of human enamel. Dent Mater 2010;26:100–4. [17] Bajaj D, Arola DD. On the R-curve behavior of human tooth enamel. Biomaterials 2009;30:4037–46. [18] Baldassarri M, Margolis HC, Beniash E. Compositional determinants of mechanical properties of enamel. J Dent Res 2008;87: 645–9. [19] Van der Linden AHIM, Booij M, ten Bosch JJ, Arends T. Protein loss of bovine dental enamel during in-vitro subsurface demineralization. Arch Oral Biol 1985;30:645–50. [20] Robinson C, Lowe NR, Weatherell JA. Amino acid composition, distribution and origin of “tuft” protein in human and bovine dental enamel. Arch Oral Biol 1975;20:29–42. [21] Rasmussen ST, Patchin RE. Fracture properties of human enamel and dentin in an aqueous environment. J Dent Res 1984;63:1362–8. [22] Oliver WC, Pharr GM. An improved technique for determining hardness and elastic modulus using load and displacement sensing indentation experiments. J Mater Res 1992;7:1564–83. [23] Pethica JB, Oliver WC. Mechanical properties of nanometer volumes of material: use of the elastic response of small area indentations. Materials Research Society Symposium Proceedings. Materials Research Society, Pittsburgh, PA 1989;130:13–23. [24] Shorshorov MKh, Bulychev SI, Alekhin VO. Work of plastic and elastic deformation during indenter indentation. Sov Phys Dokl 1981;26:769–71. [25] Cheng Y-T, Cheng C-M. Relationships between hardness, elastic modulus, and the work of indentation. Appl Phys Lett 1998;73:614–6. [26] Oliver WC, Pharr GM. Review. Measurement of hardness and elastic modulus by instrumented indentation: advances in understanding and refinements to methodology. J Mater Res 2004;19:3–20. [27] Cheng Y-T, Cheng C-M. Scaling, dimensional analysis, and indentation measurements. Mater Sci Eng R 2004;44: 91–149. [28] Jha KK, Zhang S, Suksawang N, Wang T-L, Agarwal A. Work-of-indentation as a means to characterize indenter geometry and load–displacement response of a material. J Phys D: Appl Phys 2013;46:1–11. [29] Pharr GM. Measurement of mechanical properties by ultra-low load indentation. Mater Sci Eng A 1998;253:151–9. [30] Bechtle S, Habelitz S, Klocke A, Fett T, Schneider GA. The fracture behaviour of dental enamel. Biomaterials 2010;31:375–84. [31] Kunjalukkal S, Balakrishnan A, Chu M-C, Nam T, Jai S. Micro-indentation fracture behavior of human enamel. Dent Mater 2010;26:100–4. [32] Rasmussen ST, Patchin RE, Scott DB, Heuer AH. Fracture properties of human enamel and dentin. J Dent Res 1976;55:154–64. [33] Boyde A. Microstructure of enamel. In: Chadwick DJ, Cardew G, editors. Ciba Foundation Symposium, 205: dental enamel. New York: Wiley; 1997. p. 18–31. [34] Hayashi-Sakai S, Sakai J, Sakamoto M, Endo H. Determination of fracture toughness of human permanent and primary enamel using an indentation microfracture method. J Mater Sci Mater Med 2012;23:2047–54. [35] Budiansky B, O’Connell R. Elastic moduli of a cracked solid. Int J Solids Struct 1976;12:81–97.
d e n t a l m a t e r i a l s 3 0 ( 2 0 1 4 ) 1224–1233
[36] Case ED, Kim Y. The effect of surface-limited microcracks on the effective Young’s modulus of ceramics. Part I. Analysis. J Mater Sci 1993;28:1885–900. [37] Kim Y, Case ED. The effect of surface-limited microcracks on the effective Young’s modulus of ceramics. Part II. Application of analysis to particular microcrack geometries. J Mater Sci 1993;28:1901–9. [38] Kim Y, Case ED. The effect of surface-limited microcracks on the effective Young’s modulus of ceramics. Part III. Experiments. J Mater Sci 1993;28:1910–8. [39] Fan X, Case E.D, Baumann MJ. The effect of indentation-induced microcracks on the elastic modulus of hydroxyapatite. J Mater Sci 2012;47: 6333–45. [40] Salganik RL. Mechanics of bodies with many cracks. Mech Solids 1973;8(4):135–43. [41] Hoenig A. Elastic moduli of non-randomly crack body. Int J Solids Struct 1979;15:137–54. [42] Laws N, Brockenbrough JR. The effect of microcrack systems on the loss of stiffness of brittle solids. Int J Solids Struct 1987;23:1247–68.
1233
[43] Feng XQ, Yu SW. Estimate of effective elastic moduli with microcrack interaction effects. Theor Appl Fract Mech 2000;34:225–33. [44] Chen X, Hutchinson J, Evans A. The mechanics of indentation induced lateral cracking. J Am Ceram Soc 2005;88:1233–8. [45] Chen J, Bull SJ. On the relationship between plastic zone radius and maximum depth during nanoindentation. Surf Coat Technol 2006;201:4289–93. [46] Garrido MA, Giráldez I, Ceballos L, Gómez del Río MT, Rodríguez J. Nanotribological behaviour of tooth enamel rod affected by bleaching treatment. Wear 2011;271:2334–9. [47] Kinney JH, Balooch M, Marshall GW, Marshall SJ. A micromechanics model of the elastic properties of human dentine. Arch Oral Biol 1999;44:813–22. [48] Park S, Wang DH, Zhang D, Romberg E. Mechanical properties of human enamel as a function of age and location in the tooth. J Mater Sci Mater Med 2008;19:2317–24. [49] He LH, Swain MV. Understanding the mechanical behavior of human enamel from its structural and compositional characteristics. J Mech Behav Biomed Mater 2008;1:18–29.
Available online at www.sciencedirect.com
ScienceDirect journal homepage: www.intl.elsevierhealth.com/journals/dema
On the possibility of estimating the fracture toughness of enamel Miguel Ángel Garrido a,∗ , Isabel Giráldez b , Laura Ceballos b , Jesús Rodríguez a a
Departamento de Tecnología Mecánica, Escuela Superior de Ciencias Experimentales y Tecnología, Universidad Rey Juan Carlos, C/ Tulipán s/n, Móstoles, Madrid, Spain b Departamento de Estomatología, Facultad de Ciencias de la Salud, Universidad Rey Juan Carlos, Avda. Atenas s/n, Alcorcón, Madrid, Spain
a r t i c l e
i n f o
a b s t r a c t
Article history:
Objectives. There are many works that have attempted to estimate the fracture toughness
Received 9 July 2013
of enamel by indentation techniques using equations whose success in determining the
Received in revised form
actual value of fracture toughness, rely on a particular three-dimensional pattern consisting
15 July 2014
of cracks growing from the edges of the indentation. Recently, an alternative methodology
Accepted 8 August 2014
based on an energetic approach has been developed to estimate the fracture toughness of coatings by depth sensing indentation that is not less affected by the cracks pattern generated. In this work, the energetic approach to indentation fracture toughness of bovine
Keywords:
enamel is presented and compared with those toughness values obtained using the tradi-
Enamel
tional expressions reported in the literature.
Indentation fracture toughness
Methods. Indentation tests were carried out using a diamond Berkovich indenter onto the
Energetic approach
enamel surface of eight incisors from bovines of two years old. A continuous stiffness measurement methodology was used with a frequency of 45 Hz and displacement amplitude of 2 nm up to a maximum penetration depth of 2000 nm. Results. The results showed that some modifications in the energetic methodology should be performed in order to apply it successfully. Significance. The fracture toughness values obtained using the traditional equation and applying the energetic methodology, were significantly different, although the values were within the range obtained by other authors. © 2014 Academy of Dental Materials. Published by Elsevier Ltd. All rights reserved.
1.
Introduction
Enamel is the hardest and stiffest tissue of mammals. Its microstructure consists of rods encapsulated by thin protein rich sheaths that are arranged parallel in a direction
perpendicular to the dentino-enamel junction (DEJ) from dentin to the outer enamel surface. The enamel microstructure of all mammals appears to be very similar on a histochemical and anatomic basis [1–5]. Numerous methods have been employed to experimentally measure the fracture toughness (KC ) of the enamel. The
∗ Corresponding author at: Escuela Superior de Ciencias Experimentales y Tecnología, Departamento de Tecnología Mecánica, Universidad Rey Juan Carlos, C/ Tulipán s/n, E 28933 Móstoles, Madrid, Spain. Tel.: +34 914887177; fax: +34 914888150. E-mail address: [email protected] (M.Á. Garrido). http://dx.doi.org/10.1016/j.dental.2014.08.364 0109-5641/© 2014 Academy of Dental Materials. Published by Elsevier Ltd. All rights reserved.
1225
d e n t a l m a t e r i a l s 3 0 ( 2 0 1 4 ) 1224–1233
determination of KC by indentation techniques is based on measuring the size of cracks induced in a material during indentation [6]. Several expressions are available to determine KC by this technique, depending on the indenter geometry and crack morphology [7–10]. In 1987, Laugier [10] determined that:
KC = xV
a 1/2 E 2/3 P l
H
(1)
c3/2
where a is the half-diagonal of the indentation impression, l is the crack length measured from the indentation imprint edge, and c is the crack length measured from the center of the indentation imprint. Additionally, Laugier [10] showed that the radial and half-penny models make similar predictions when xv = 0.015. The Eq. (1) was developed for ceramic materials and for the symmetrical Vickers indentations. Some efforts have been made to obtain similar equation but properly modified for Berkovich indentations. Ouchterlony [11] investigated the nature of the radial cracking and determined a modification factor for stress intensity factor to account the number of radial cracks, n, formed.
k1 =
n/2 1 + (n/2) sin(2/n)
(2)
The ratio of k1 values for n = 4 (Vickers indenter) and n = 3 (Berkovich indenter) is 1.073. Introducing this ratio in Eq. (1), a modified Laugier model adapted to Berkovich indentations can be obtained, according to Eq. (3).
KC = 1.073 · xV
a 1/2 E 2/3 P l
H
c3/2
(3)
However, these equations provide an actual value of the fracture toughness by indentation only if a particular pattern of cracks running from the indentation imprint vertices is obtained. This work has been developed to answer the question about what happens if cracks generated by indentation do not show that characteristic pattern. Concerning this point, there could be an alternative methodology based on the energy released during cracking process, as a measurement of the fracture toughness [12]. This energetic method calculates the fracture toughness comparing the area limiting by the load-displacement curve with crack generation and the hypothetical one if cracking does not occur [12]. For a thin film, Li et al. [12] determined the critical stressintensity factor assuming Mode I:
KIC =
E (1 − v2 )2CR
1/2 U ·
t
(4)
where E is the elastic modulus and is the Poisson’s ratio, KIC is the indentation fracture toughness of the material, U is a measure of the fracture energy, t is the film thickness and
2CR is the crack length. The product of 2CR and t gives the cracked area, Acrack , therefore, Eq. (4) can be written as:
KIC =
E (1 − 2 )
1/2 U ·
(5)
Acrack
Eq. (5) allows obtaining fracture toughness values from indentation tests. Under this testing method, the strain energy, U, can be obtained from the area between the hypothetical loading curve if no cracking exists and the experimental one. This equation can be used even when a particular cracks pattern is not obtained, although the critical point of Eq. (5) is to determine again the cracked area. Hereafter, Eq. (3) is referred in this paper as the traditional equation to distinguish the equation using the energetic methodology, Eq. (5). There is some scatter in the literature about the actual fracture toughness of enamel [13]. Hassan et al. [14] have reported values of human tooth enamel, using a Vickers indenter, in the range of 0.7–1.37 MPa m1/2 . Xu et al. [15] reported fracture toughness values of 0.84 MPa m1/2 for labial human enamel also using a Vickers indenter in the same range of those determined by Padmanabhan et al. [16]. Bajaj and Arola [17] reported fracture toughness values for human enamel that ranged from 1.79 MPa m1/2 to 2.37 MPa m1/2 obtained from Rcurve analysis, and Baldassarri et al. [18] obtained values of 0.5 MPa m1/2 and 1.3 MPa m1/2 for transversal and midsagital enamel orientation, respectively, using a Vickers indenter on rat tooth. There are various reasons that may cause the large variation in reported fracture toughness values. The biological nature of the enamel suggests that the compositional variations among specimens could impact to the toughness values [19,20]. Additionally, the microstructure orientation of enamel could also contribute to this variability [21]. Therefore, the aim of this study was to compare the fracture toughness values obtained from the energetic methodology using depth sensing indentation technique with those obtained applying the traditional equation based in a specific pattern of cracks.
2.
Materials and methods
2.1.
Preparation of specimens
Eight incisors were extracted from bovines of two years old. Teeth were cleaned and stored in artificial saliva (Table 1) prior to the tests. The labial surfaces of the specimens were polished
Table 1 – The composition of artificial saliva. Composition Sodium carboxymethyl cellulose Sorbitol Sodium chloride Potassium phosphate Calcium chloride, dihydrate Magnesium chloride, hexahydrate Dibasic sodium diphosphate Purified water
g/100 g of solution 1.000 3.000 0.084 0.120 0.015 0.005 0.034 bal
1226
d e n t a l m a t e r i a l s 3 0 ( 2 0 1 4 ) 1224–1233
in two steps, using a mechanical grinder (Labopol-5, Struers, Copenhagen, Denmark) with polishing cloths of alumina suspension slurry of 3 m and 0.020 m, respectively. This polish sequence provided an average roughness lower than 20 nm. This value was checked by atomic force microscopy measurements. Afterwards, the specimens were kept fully hydrated in artificial saliva at room temperature before the indentation tests. The composition of saliva is detailed in Table 1.
2.2.
Indentation tests
A preliminary study of Vickers indentations on the bovine enamel was carried out, following the ASTM standard requirements. Two different maximum loads were applied: 5 N, 20 N. The time to load and the dwell time was 15 s. Additionally, hardness and elastic modulus of bovine enamel surfaces were studied by depth sensing indentation (Nanoindenter XP-MTS System Corporation) using a Berkovich diamond indenter with a tip radius of 50 nm. A Continuous Stiffness Measurement module (CSM) [22,23] was chosen. A maximum penetration depth of 300 nm was fixed for all indentation tests. Three rows of 20 indentations were performed on each sample. Each indentation was separated by 100 m from each other. During the loading branch, continuous loading–unloading cycles with amplitude of 2 nm and a frequency of 45 Hz were superimposed. The Oliver–Pharr methodology [22] was applied on each of these partial unloading cycles, providing values of elastic modulus (E), and hardness (H), as a continuous function of load or penetration depth. Before each batch tests, the Berkovich indenter was calibrated on a standard fused silica.
2.3.
Fracture toughness analysis
Additionally, 5 indentations were done on each sample using the CSM module up to a maximum penetration depth of 2000 nm. This penetration depth was sufficient to generate a characteristic pattern of cracks, allowing us to apply Eqs. (3) and (5). After the indentation tests, the residual imprints were observed by scanning electron microscopy (SEM) (Hitachi S-3400N) under low vacuum condition. The fracture toughness of the enamel was determined using the traditional methodology and the energetic one. The application of each methodology has difficulties. To estimate the fracture toughness by the traditional methodology implies to obtain a characteristic pattern of cracks. This problem was overcome choosing adequate indentation test conditions. To determine the fracture toughness by the energetic methodology, it is necessary to calculate the energy released during the cracking process, U. The cracking area, Acrack , has to be estimated for both methodologies.
2.3.1.
Determination of U
The proposed procedure to calculate the energy released during the cracking process, is based on the contact stiffness variation due to cracking. Besides the cracking process, other phenomena may affect the contact stiffness during indentation, as quasi-plastic deformation phenomena associated with the movement of the water and protein phase. However,
given the conditions under which the indentation tests were made, with very high strain rates, the contribution of these phenomena to the variation of the contact stiffness, measured from the elastic unloading branch of each cycle, can be considered negligible. Additionally, it is well known that using a pyramidal indenter, the ratio of the contact area to the depth of the indentation, remains constant for an increasing indenter load, i.e. the Berkovich indentation exhibits geometrical similarity. Consequently, the strain within the material is constant, independent of the penetration depth, except for very small values, where the apex radius effect is dominant. Therefore, the plasticity effects should be equally important at low or high penetrations. If at low penetration depths, a linear tendency of the contact stiffness versus square root of the contact area is obtained, any different trend observed at high depths should not be a consequence of the plastic deformation of the enamel during the indentation process. Therefore, when the contact stiffness versus square root of the projected contact area does not show a linear tendency, it is, most probably, a consequence of a cracking process developed during the indentation process. It is well known [24–28] that the area under the loaddisplacement curve during the unloading process is the corresponding elastic work. However, if cracking occurs, part of the elastic energy stored during the indentation test will be released to create new crack surfaces. This will be reflected as a change in the contact stiffness and, consequently, the relationship between contact stiffness, S, and the squared root of the contact area, A, will not be linear, as the contact theory predicts [22,29]. Therefore, the linear tendency of contact stiffness versus squared root of the contact area will only remain before cracking occurs. Then, it is possible to identify in the S–A1/2 curve, a critical point associated with the onset of cracking (S* , A1/2* ). According to the Oliver–Pharr methodology [22], it is possible to calculate the load and penetration depth values corresponding to the critical point associated with the cracking initiation, P* and h* . For values (P, h) lower than the critical point, the indentation process will take place without cracking. For higher values, the indentation process will be characterized by the nucleation and propagation of cracks. The load-penetration depth curve for values lower than the critical one can be extrapolated according to the Kick’s law [27], P = E · h2 , to the maximum penetration depth or indentation load, obtaining a hypothetical curve characteristic of a non cracking process.
2.3.2.
Determination of Acrack
Following the assumption of Laugier to develop the indentation fracture toughness model for Berkovich indentation [10], the radial crack geometry inside the enamel was postulated in this work (see Fig. 1a). Under this assumption, the authors propose two possible ways to estimate the cracked area: using the crack lengths directly; and through the contact stiffness variation. The cracked area estimation through the crack length was determined assuming that the crack from the edge of the indentation imprint, run through the protein interlayer (Fig. 1b). This phenomenon was previously observed by other authors [14,30–33]. Under this assumption, the crack path around each prism was approached as a half of a
d e n t a l m a t e r i a l s 3 0 ( 2 0 1 4 ) 1224–1233
1227
where N corresponding to the number of cracks, a and c are cracks lengths defined in Fig. 1. Considering that the experimental conditions of the indentations were carefully selected to obtain a characteristic pattern composed of three cracks, each one running from each indenter Berkovich corner, N = 3. The cracking area estimation from the contact stiffness can be expressed as a function of the elastic modulus of the enamel according to the Oliver and Pharr methodology [22]. Therefore, the variation of the contact stiffness observed from the critical point (P* , h* ), is associated with a progressive decrease in the corresponding elastic modulus. The decrease of this property can be used to estimate the cracked area, mainly following the results of Budiansky and O’Connell [35], Case and Kim [36–38], Fan et al. [39], and others authors [40–42]. According to these studies, the elastic modulus, E, of the microcracked body can be expressed by: E = ES (1 − fε)
(10)
where Es refers to the non-microcraked elastic modulus. The function f depends on the spatial orientation of the microcracks [35,37]. The crack damage parameter ε can be expressed [35,40–42] as:
ε= Fig. 1 – (a) Crack geometry idealization along the load direction; (b) schematic of the sinuous path followed by the crack through the rods interfaces.
circumference (Fig. 1b). Additionally, this sinuous crack propagation, was considered effective from the end of the indentation imprint, a, to the end of the crack, c (Fig. 1b). Therefore, the length of each crack, Lc , could be written as: Lc =
(c − a) + a 2
(6)
In order to obtain the cracked area, an elliptical geometry of the crack shape in the depth direction was expected (Fig. 1a). The major axis was equal to the crack length, Lc , and the minor one was assumed equal to c/2 (Fig. 1a). Similar postulation was previously pointed out by Hayashi-Sakai et al. [34]. Therefore, the area of each crack, Ac , was written as: Ac =
Lc c 2 2
(7)
If N cracks were generated during the indentation process, the total cracked area, Acrack could be written as: Acrack = N · Ac
(8)
Substituting Eqs. (6) and (7) into Eq. (8), the total cracked area result:
Acrack = N ·
2 2 (2 − ) c + a·c 8 8
2
A2 P
· NV
(11)
where Nv is the volume number density of microcracks, A2 is the mean of the square of the cracked area and P is the mean crack perimeter. The volume number of density is defined by the relation between the number of cracks, N, and a representative volume, V [43]: NV =
N V
(12)
In this study and based on the cylindrical cavity model for cracked brittle solids subjected to indentation tests [44], a cylindrical representative volume of radius equal to the mean crack length, c, and total depth of c/2 was considered (Fig. 1). Therefore, the volume of material affected by the indentation process could be estimated by Eq. (13): V = · c2 ·
c 2
(13)
According to Fig. 1:
A=
2
c 1 2 2
P=
c 2
(14)
(15)
Substituting Eqs. (14) and (15) into Eq. (11):
(9)
ε=
c3 · NV 16
(16)
1228
d e n t a l m a t e r i a l s 3 0 ( 2 0 1 4 ) 1224–1233
Assuming radial crack geometry, as defined Laugier [10], with aligned crack orientation [25]: f =
16 (1 − 2 ) 3
(17)
where Q is a constant bounded between two limits. Substituting Eq. (22) into Eq. (24), this constant was equal to: Q1 =
2 34 64
(26)
where is the Poisson’s ratio of the non-microcracked body. Substituting Eqs. (16) and (17) into Eq. (10):
Similarly, if Eq. (23) is substituted into Eq. (24), Q constant was equal to:
1 − 2 3 N ES − E = c ES 3 V
Q2 =
(18)
The first term of the Eq. (18) represents the elastic modulus variation regarding to the value of the non-microcraked solid. If E represents the elastic modulus evolution, Eq. (18) could be written as: E 1 − 2 3 N = c ES 3 V
(19)
Using the Eq. (14), the total cracked area, Acrack could be estimated by Eq. (20):
c 2 1
Acrack = N 2 2
(20)
where N represents the number of cracks generated by the indentation process. Substituting Eq. (19) into Eq. (20), the total cracked area could estimate as: Acrack =
3 V E 8 (1 − 2 )c ES
(21)
Considering the sinuous path of the crack, through the protein interlayer (Figs. 1 and 2), the c parameter in Eqs. (13) and (21), changed to (/2)·c. However, concerning the volume affected by the indentation process, V (Eq. (13)), two limiting situations could be identified. A first circumstance in which the longitudinal axes of the prisms were oriented perpendicularly to the surface; and another situation where the longitudinal axes were oriented parallel to the surface. Considering the first extreme situation, the volume affected, V1 , can be written as: V1 = ·
2 c 2
c
2
=
3 2
c
(22)
For the second extreme situation, the volume, V2 , was: V2 = ·
2 c 2
c
22
=
2
3 2
c
(23)
Similarly, the cracked area (Eq. (21)) could be written as: Acrack =
3 V E 8 (1 − 2 )((/2)c) ES
(24)
Substituting Eq. (22) or (23) into Eq. (24), the total cracked area could be written as: Acrack = Q
E c2 (1 − 2 ) ES
(25)
34 64
(27)
Eq. (25) represented the total cracked area depending on the elastic modulus decrease with the penetration depth, recorded during the indentation process. Eqs. (9) and (25) represented the total cracked area obtained from different approximations. Now, it is possible to apply the energetic methodology through Eq. (2).
3.
Results
Fig. 2a and b shows two optical images of representative residual imprints obtained from Vickers indentations. For both loads, it is observed that the cracks progressed randomly from the vertices and faces of the indentation. Besides, the prisms geometry is clearly revealed, which suggests that the cracks predominantly run surrounding the hydroxyapatite rods, i.e. through protein interfaces. Fig. 2c shows a detail of a crack path, observing a preferential propagation along the protein interlayer. Similar phenomenon has been observed by other researchers [14,30–33]. Fig. 3 shows two SEM images of typical residual imprints from depth sensing indentations with Berkovich tip at different maximum penetration depths. Their corresponding curves of the contact stiffness versus squared root of the contact area for 300 nm (Fig. 3a) and 2000 nm (Fig. 3b) penetrations depths were also included. Indentations at maximum penetration depth of 300 nm were characterized by the absence of cracks at the edges of the residual imprint (Fig. 3a). The contact stiffness, S, showed a linear variation with the square root of the contact area, A, between the indenter and the material surface (Fig. 3a); then, the indentation process was carried out without cracking. By contrast, the indentations obtained at higher penetration depths of 2000 nm were characterized by the presence of cracks extended from the edges of the residual imprints (Fig. 3b). In addition, it was also observed how the cracks deflected during their propagation, and the contact stiffness lost its linear variation with the square root of contact area for values above a threshold one (Fig. 3b). Fig. 3a shows a SEM image of a representative Berkovich indentation up to maximum penetration depth of 300 nm. The size of the residual imprint ranged from 1.5 to 2 m, however, the zone affected by the indentation process is higher than the indentation imprint [45]. Consequently several rods participate in the indentation process. Therefore, it can be assumed that the mechanical properties obtained through the indentation test, were average values. Similar reasoning could be made for indentations up to 2000 nm of penetration depth.
d e n t a l m a t e r i a l s 3 0 ( 2 0 1 4 ) 1224–1233
1229
Fig. 2 – Typical cracks pattern obtained from Vickers indentations on enamel: (a) 5 N; (b) 20 N; (c) detail of a crack propagating along the protein interlayer.
Table 2 summarizes the hardness and elastic modulus of bovine enamel obtained from indentation tests at maximum penetration depth of 300 nm, where the loading process did not cause any cracking. The methodology followed to obtain
both properties was described by Oliver and Pharr [22]. The elastic modulus and hardness values remained roughly constant regardless of the penetration depth (Fig. 4a). Therefore, indentation size effect could be considered negligible under
Fig. 3 – Typical pattern of cracks obtained from nanoindentations carried out at different maximum penetration depths: (a) 300 nm; (b) 2000 nm. Their corresponding indentation contact stiffness versus squared root of the contact area plots were put over each image.
1230
d e n t a l m a t e r i a l s 3 0 ( 2 0 1 4 ) 1224–1233
Table 2 – Experimental results of elastic modulus, E, hardness, H and indentation fracture toughness, KC . Mean values and standard deviations (in brackets). The methodologies and equations followed to obtain the each property are included in the last column. Property
Value
Equation/Reference
Elastic modulus, E (GPa) Hardness, H (GPa) Crack length, c (m) Imprint size, a (m) Indentation fracture √ toughness, KC (MPa m)
92 (6) 4.7 (0.3) 12 (3) 7 (2) 0.7 (0.2)
Oliver–Pharr [22] Oliver–Pharr [22] – – Eq. (3) [10,11]
these test conditions. Additionally, the force-penetration curves fit a quadratic law, as predicted by the Kick’s law [27] (Fig. 4b). Table 2 includes fracture toughness data of enamel provided by the traditional equation, Eq. (3), obtained from the results reported by Laugier [10] and Ouchterlony [11].
Fig. 4 – (a) Elastic modulus and hardness values versus penetration depth of a representative indentation test on bovine enamel following the continuous stiffness measurement methodology up to maximum penetration depth of 300 nm; (b) load, P, versus squared penetration depth, h2 , curve for a representative indentation test on enamel up to 300 nm of penetration depth.
Fig. 5 – Elastic modulus evolution with the penetration depth during a representative Berkovich indentation test up to maximum penetration depth of 2500 nm.
Fig. 5 shows a representative variation of the elastic modulus versus penetration depth, corresponding to the Berkovich indentation on the enamel up to maximum penetration depth of 2500 nm. This figure revealed that the elastic modulus remained constant with the indentation depth up to a critical point was reached. The elastic modulus gradually decreased from this penetration depth. Consequently, two regions could be identified concerning to the elastic modulus evolution with the penetration depth. Both sections were fitted by linear tendencies. The critical point could be experimentally determined as the intersection between the linear fits of the E constant and E decreasing regions. This point corresponded to a penetration depth of 1000 nm (Fig. 5). The average value of the elastic modulus variation, E, is included in Table 3. Fig. 6a shows a typical curve of contact stiffness versus squared root of the contact area. The linearity only remained up to a squared root of the contact area equivalent to a maximum penetration depth of 1000 nm, roughly. The critical point of cracking onset (S* , A1/2* ) was calculated for each test. Following the methodology described in the Fracture toughness analysis section, the critical point associated with the cracking initiation, P* and h* , was determined on each test (Fig. 6b). The load-penetration depth curve for values lower than the critical one was extrapolated to the maximum penetration depth of 2000 nm, obtaining the hypothetical curve without associated cracking process. Fig. 6b shows a comparative example between the experimental indentation curve, with cracking, and the hypothetical indentation curve, without cracking. The difference in area between the two force–displacement curves gave the energy released during the cracking process, U. The average values and standard deviations are included in Table 3. After that, the cracked area was determined using both possibilities: through the equation based measurement of crack length, c, Eq. (9); and the equation based on the crack length, c, and the elastic modulus variation due to the cracking process, Eq. (25). Table 3 summarizes the average cracked area value calculated by both equations and the fracture toughness
1231
d e n t a l m a t e r i a l s 3 0 ( 2 0 1 4 ) 1224–1233
Table 3 – Experimental results of fracture toughness of the enamel using the alternative energetic methodology. Average values of the energy released to create new crack surfaces, U; the elastic modulus variation, E; and the cracked area Acrack were included. The corresponding standard deviations have been also included (in brackets). Property U (mN nm) E (GPa) Acrack (m2 ) Acrack (m2 ) Crack length, c (m) √ Indentation fracture toughness, KC (MPa m) by crack measurement √ Indentation fracture toughness, KC (MPa m) by E measurement
via the alternative energetic methodology using both expressions for the cracked area.
4.
Discussion
According to the results of Vickers indentations, in which a random patter of cracks was obtained, the applicability of traditional equations (Eq. (3)) is compromised. Consequently, the possibility of using an alternative methodology, that would not have to depend on the generation of a particular pattern of cracks, has been explored. In that sense, it seems that energetic methodology fulfills this condition (Eq. (5)). To check its reliability, an experimental campaign of Berkovich indentations at high penetration depths was designed, so that a characteristic pattern of cracks was obtained allowing the application of both methodologies. Both methodologies need to know the elastic modulus of the indented material; therefore, additional indentation tests at small penetration depths, under free induced cracks conditions, were made on the enamel. Concerning elastic modulus and hardness obtained from indentations up to maximum penetration depth of 300 nm, the average values included in Table 2 were very similar to
Value 2710 (126) 16 (3) 152 (10) 86 (3) 135 (3) 12 (3) 1.34 (0.14) 1.77 (0.23) 1.41 (0.23)
Equation/Reference – – According Eq. (9) Eq. (25) using Q1 Eq. (25) using Q2 – Using Eq. (5) and Acrack from Eq. (9) Using Eq. (5) and Acrack from Eq. (25) with Q1 Using Eq. (5) and Acrack from Eq. (25) with Q2
those reported in previous works [15,46–49]. Furthermore, the contact stiffness evolution versus squared root of the contact area was linear (Fig. 3a). Both circumstances let to assume that the indentation process designed to obtain the elastic modulus and hardness, was developed under free cracks conditions, or at least, was developed so that the cracks inherent to the enamel microstructure did not have a significant effect. Figs. 2 and 3b show how the cracks appeared zigzagged. Similar mechanism was also observed by other researchers [14,32,33], suggesting that it is more difficult for cracks to cut through the enamel rods that to go around them. The fracture toughness value obtained through the traditional methodology (Table 2) was in the order of 0.7 MPa*m1/2 . This value is in the range of the results reported by other authors [14]. When the fracture toughness values obtained by both methodologies, traditional (Table 2) and energetic (Table 3) were compared, a considerable difference could be observed. The toughness obtained by following the energetic methodology was, at least, two times the value obtained by the traditional one. These values were similar to those reported by other authors who have used fracture mechanics tests to evaluate the fracture toughness of enamel. Bechtle et al. [30] used a three-point bending test to estimate the fracture
Fig. 6 – (a) Contact stiffness versus squared root of the contact area for a representative indentation test; (b) indentation load versus penetration depth of an experimental and hypothetical indentation curve.
1232
d e n t a l m a t e r i a l s 3 0 ( 2 0 1 4 ) 1224–1233
toughness of enamel. They reported values that range from 0.8 to 1.5 MPa*m1/2 at the beginning of crack propagation up to 4.4 MPa*m1/2 for 500 m crack extension. In a recently published work, the fracture toughness values derived from CT samples consisting of human enamel cubes mounted in epoxy resin ranged from 0.5 to 0.8 MPa*m1/2 at the beginning of crack propagation up to 2.4 MPa*m1/2 at 1.5 mm crack extension [17].
5.
Conclusions
In this work the suitability of the equation traditionally used for Berkovich indentation to determine the indentation toughness of the enamel has been analysed. The difficulty of obtaining an appropriate crack pattern requires the investigation of alternative ways to estimate the fracture toughness. In this work, an alternative energetic methodology, adapted to the depth sensing indentation results, has been developed. The method is based on the contact stiffness variation due to the cracking process. The values obtained using this methodology, were clearly different than those reported by other authors that use traditional indentation equations.
references
[1] Habelitz S, Marshall SJ, Marshall GW, Balooch M. Mechanical properties of human dental enamel on the nanometre scale. Arch Oral Biol 2001;46:173–83. [2] Poole DFG, Brooks AW. The arrangement of crystallites in enamel prisms. Arch Oral Biol 1961;5:14–27. [3] Ge J, Cui FZ, Wang XM, Feng HL. Property variations in the prism and the organic sheath within enamel by nanoindentation. Biomaterials 2005;26:3333–9. [4] Oesterle LJ, Shellhart WC, Belanger GK. The use of bovine enamel in bonding studies. Am J Orthod Dentofac Orthop 1998;114:514–20. [5] Sanches RP, Otani C, Damiao AJ, Miyakawa W. AFM characterization of bovine enamel and dentine after acid etching. Micron 2009;40:502–6. [6] Kruzic JJ, Kim DK, Koester KJ, Ritchie RO. Indentation techniques for evaluating the fracture toughness of biomaterials and hard tissues. J Mech Behav Biomed 2009;2:384–95. [7] Lawn BR, Evans AG, Marshall DB. Elastic/plastic indentation damage in ceramics: the median/radial crack system. J Am Ceram Soc 1980;63:574–81. [8] Anstis GR, Chantikul P, Lawn BR, Marshall DB. A critical evaluation of indentation techniques for measuring fracture toughness: I. Direct crack measurements. J Am Ceram Soc 1981;64:533–8. [9] Niihara K, Morena R, Hasselmann DPH. A fracture mechanics analysis of indentation-induced Palmqvist crack in ceramics. J Mater Sci Lett 1983;2:221–3. [10] Laugier MT. Palmqvist indentation toughness in WC–Co composites. J Mater Sci Lett 1987;6:897–900. [11] Ouchterlony F. Stress intensity factors for the expansion loaded star crack. Eng Fract Mech 1976;8:447–8. [12] Li X, Diao D, Bhushan B. Fracture mechanisms of thin amorphous carbon films in nanoindentation. Acta Mater 1997;45:4453–61. [13] Soderholm KJ. Review of the fracture toughness approach. Dent Mater 2010;26:63–77. [14] Hassan R, Caputo AA, Bunshash RF. Fracture toughness of human enamel. J Dent Res 1981;60:820–7.
[15] Xu HHK, Smith DT, Jahanmir S, Romberg E, Kelly JR, Thompson VP, et al. Indentation damage and mechanical properties of human enamel and dentin. J Dent Res 1998;77:472–80. [16] Padmanabhan SK, Balakrishnan A, Chu MC, Kim TN, Cho SJ. Micro-indentation fracture behavior of human enamel. Dent Mater 2010;26:100–4. [17] Bajaj D, Arola DD. On the R-curve behavior of human tooth enamel. Biomaterials 2009;30:4037–46. [18] Baldassarri M, Margolis HC, Beniash E. Compositional determinants of mechanical properties of enamel. J Dent Res 2008;87: 645–9. [19] Van der Linden AHIM, Booij M, ten Bosch JJ, Arends T. Protein loss of bovine dental enamel during in-vitro subsurface demineralization. Arch Oral Biol 1985;30:645–50. [20] Robinson C, Lowe NR, Weatherell JA. Amino acid composition, distribution and origin of “tuft” protein in human and bovine dental enamel. Arch Oral Biol 1975;20:29–42. [21] Rasmussen ST, Patchin RE. Fracture properties of human enamel and dentin in an aqueous environment. J Dent Res 1984;63:1362–8. [22] Oliver WC, Pharr GM. An improved technique for determining hardness and elastic modulus using load and displacement sensing indentation experiments. J Mater Res 1992;7:1564–83. [23] Pethica JB, Oliver WC. Mechanical properties of nanometer volumes of material: use of the elastic response of small area indentations. Materials Research Society Symposium Proceedings. Materials Research Society, Pittsburgh, PA 1989;130:13–23. [24] Shorshorov MKh, Bulychev SI, Alekhin VO. Work of plastic and elastic deformation during indenter indentation. Sov Phys Dokl 1981;26:769–71. [25] Cheng Y-T, Cheng C-M. Relationships between hardness, elastic modulus, and the work of indentation. Appl Phys Lett 1998;73:614–6. [26] Oliver WC, Pharr GM. Review. Measurement of hardness and elastic modulus by instrumented indentation: advances in understanding and refinements to methodology. J Mater Res 2004;19:3–20. [27] Cheng Y-T, Cheng C-M. Scaling, dimensional analysis, and indentation measurements. Mater Sci Eng R 2004;44: 91–149. [28] Jha KK, Zhang S, Suksawang N, Wang T-L, Agarwal A. Work-of-indentation as a means to characterize indenter geometry and load–displacement response of a material. J Phys D: Appl Phys 2013;46:1–11. [29] Pharr GM. Measurement of mechanical properties by ultra-low load indentation. Mater Sci Eng A 1998;253:151–9. [30] Bechtle S, Habelitz S, Klocke A, Fett T, Schneider GA. The fracture behaviour of dental enamel. Biomaterials 2010;31:375–84. [31] Kunjalukkal S, Balakrishnan A, Chu M-C, Nam T, Jai S. Micro-indentation fracture behavior of human enamel. Dent Mater 2010;26:100–4. [32] Rasmussen ST, Patchin RE, Scott DB, Heuer AH. Fracture properties of human enamel and dentin. J Dent Res 1976;55:154–64. [33] Boyde A. Microstructure of enamel. In: Chadwick DJ, Cardew G, editors. Ciba Foundation Symposium, 205: dental enamel. New York: Wiley; 1997. p. 18–31. [34] Hayashi-Sakai S, Sakai J, Sakamoto M, Endo H. Determination of fracture toughness of human permanent and primary enamel using an indentation microfracture method. J Mater Sci Mater Med 2012;23:2047–54. [35] Budiansky B, O’Connell R. Elastic moduli of a cracked solid. Int J Solids Struct 1976;12:81–97.
d e n t a l m a t e r i a l s 3 0 ( 2 0 1 4 ) 1224–1233
[36] Case ED, Kim Y. The effect of surface-limited microcracks on the effective Young’s modulus of ceramics. Part I. Analysis. J Mater Sci 1993;28:1885–900. [37] Kim Y, Case ED. The effect of surface-limited microcracks on the effective Young’s modulus of ceramics. Part II. Application of analysis to particular microcrack geometries. J Mater Sci 1993;28:1901–9. [38] Kim Y, Case ED. The effect of surface-limited microcracks on the effective Young’s modulus of ceramics. Part III. Experiments. J Mater Sci 1993;28:1910–8. [39] Fan X, Case E.D, Baumann MJ. The effect of indentation-induced microcracks on the elastic modulus of hydroxyapatite. J Mater Sci 2012;47: 6333–45. [40] Salganik RL. Mechanics of bodies with many cracks. Mech Solids 1973;8(4):135–43. [41] Hoenig A. Elastic moduli of non-randomly crack body. Int J Solids Struct 1979;15:137–54. [42] Laws N, Brockenbrough JR. The effect of microcrack systems on the loss of stiffness of brittle solids. Int J Solids Struct 1987;23:1247–68.
1233
[43] Feng XQ, Yu SW. Estimate of effective elastic moduli with microcrack interaction effects. Theor Appl Fract Mech 2000;34:225–33. [44] Chen X, Hutchinson J, Evans A. The mechanics of indentation induced lateral cracking. J Am Ceram Soc 2005;88:1233–8. [45] Chen J, Bull SJ. On the relationship between plastic zone radius and maximum depth during nanoindentation. Surf Coat Technol 2006;201:4289–93. [46] Garrido MA, Giráldez I, Ceballos L, Gómez del Río MT, Rodríguez J. Nanotribological behaviour of tooth enamel rod affected by bleaching treatment. Wear 2011;271:2334–9. [47] Kinney JH, Balooch M, Marshall GW, Marshall SJ. A micromechanics model of the elastic properties of human dentine. Arch Oral Biol 1999;44:813–22. [48] Park S, Wang DH, Zhang D, Romberg E. Mechanical properties of human enamel as a function of age and location in the tooth. J Mater Sci Mater Med 2008;19:2317–24. [49] He LH, Swain MV. Understanding the mechanical behavior of human enamel from its structural and compositional characteristics. J Mech Behav Biomed Mater 2008;1:18–29.
