Materials Science and Engineering C 49 (2015) 101–105
Contents lists available at ScienceDirect
Materials Science and Engineering C journal homepage: www.elsevier.com/locate/msec
Correction of some mechanical characteristics of human dentin under compression considering the shape effect Dmitry Zaytsev ⁎ Ural Federal University, Ekaterinburg, Russia
a r t i c l e
i n f o
Article history: Received 22 October 2014 Received in revised form 29 November 2014 Accepted 7 December 2014 Available online 26 December 2014 Keywords: Dentin Compression Shape effect Poisson's ratio
a b s t r a c t The paper is aimed to determine the true compression strength and Poisson's ratio of human dentin. The origin of the shape effect in dentin under compression is discussed, too. It was shown that the shape effect is mainly caused by the friction between the surface of the sample and the compression plates. Ratio d/h = 4 is the optimal proportion between the diagonal of compression surface and the height of dentin sample for compression testing. Inhomogeneous deformation takes place in the sample with a low aspect ratio whereas lateral deformation is suppressed in the sample with a high aspect ratio. There is significant difference between the conventional compression strength and the true compression strength. True compression strength of human dentin is 432 ± 16 MPa, the Young's modulus is 4.04 ± 0.12 GPa and Poisson's ratio of human dentin is 0.14 ± 0.04. © 2014 Elsevier B.V. All rights reserved.
1. Introduction Human dentin is the hard base of a tooth. The main mechanical function of dentin is to support the overlaying enamel, which protects the tooth from external action. Enamel re-distributes the dangerous stresses in homogeneous one into its thickness under mastication. Therefore, the uniaxial compression is the sole scheme of deformation that dentin underwent. Information on the mechanical properties of human dentin is important for the manufacturers of dental restorative materials because their properties should be close to the properties of tooth hard tissues. Study of the mechanical properties of dentin under compression attracted the interest of many researchers since 1895 [1]. However, the mechanical characteristics of dentin are different and varied in the wide diapason: the Young's modulus 8–19 GPa and the compression strength 250–350 MPa [2–4]. It has been shown that mechanical properties of the cuboid dentin samples under compression strongly depend on the ratio between the diagonal of compression surface of the sample and its height (d/h ratio, Fig. 1a). The samples with low aspect ratio behave like a brittle material while the deformation behavior of samples with high aspect ratio is close to a ductile solid. The Young's modulus of human dentin varied from 2 GPa to 11 GPa and the compression strength is 330–800 MPa [5,6]. This dependence of the mechanical properties from the sample shape explains in most cases the variation of the results given in the literature. In the other cases, the distinction
⁎ Department of Physics, Institute of Natural Sciences, Ural Federal University, Lenin Avenue, 51, 620083 Ekaterinburg, Russia. E-mail address: [email protected].
http://dx.doi.org/10.1016/j.msec.2014.12.080 0928-4931/© 2014 Elsevier B.V. All rights reserved.
in the results is caused by such experimental factors as environment and temperature of testing. The origin of the shape effect under compression connects with both intrinsic properties of the material and several experimental factors. One of them is the friction between the surface of the sample and the compression plate when the lateral deformation is hampered (Fig. 1b). The next cause of the shape effect is the difference between conventional and true stresses in the sample. Calculation of the conventional stresses is based on the initial size of sample, however, the sizes of the samples and its area of contact with the compression plates can be changed during loading and hence the level of true stresses in the samples should be distinct from the conventional stress, which was calculated by the testing machine (Fig. 2). It is not significant for a brittle material where the deformation is less than 1%, but it should be taken into account for a deformable material when the deformation can reach several tens of percent. Therefore, this difference in the stresses should be considered for human dentin under compression because its deformation reaches up to 56% for the samples with the highest aspect ratio [5–7]. One of the important mechanical parameters is the Poisson's ratio, which characterizes the competition between the lateral and the axial deformations. It is used for calculating of the Young's modulus at indentation testing. The Poisson's ratio of dentin lies in diapason 0.025–0.45 [4]. Variation of the value appears due to the measurement of Poisson's ratio that is carried out by the different methods such as Resonant Ultrasound Spectroscopy and calculation from the numerical models. No direct measurements of Poisson's ratio of dentin have been done. Besides, it should be noted that measurements of porous and inhomogeneous materials such as human dentin by means of wave transmission look like not quite correct.
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D. Zaytsev / Materials Science and Engineering C 49 (2015) 101–105
Fig. 1. Schemes: a — dimensions parameters of the sample; b — direction of friction forces.
values that reflect its intrinsic properties. The aim of this work is the correction of the compression strength, the Young's modulus and the Poisson's ratio of human dentin under compression testing.
2. Experimental procedure
Fig. 2. Сalculation of stress in the sample at compression test.
Recording of the lateral and the axial deformations of the sample during compression test allows estimating the influence of friction between the surface of the sample and the compression plate on the deformation behavior of dentin, and calculating the true mechanical
Fifty intact human molars and premolars were used in this work. They did not contain damages and were extracted from mature subjects according to the medical diagnosis and the Ethic Protocol of the Urals State Medical University at Yekaterinburg, Russia. Detailed description of the methods of sample preparation was given in Ref. [5]. The samples were cut from the teeth by means of a diamond saw with water irrigation and, further, their surfaces were abraded using the abrasive papers. Seven groups with different d/h ratios per 10 cuboid samples each were prepared. Sizes of the samples are given in Table 1. Uniaxial compression was carried out by means of a Shimadzu AGX-50 kN (Japan) testing machine at room condition. Rate of loading was 0.1 mm/min for all tests. Processing of the results including statistical analysis was carried out by Trapezium-X software. The lateral deformation of the sample was calculated by means of a Canon photo microsystem (Japan), where the width of the sample compared in situ with the etalon
Table 1 Mechanical properties of the sample of human dentin in dependence on the d/h ratio under compression. Sizes, mm
1.7 × 1.8 × 1.8
1.2 × 2.1 × 2.1
0.7 × 2.0 × 2.0
0.6 × 2.3 × 2.3
0.6 × 3.0 × 3.0
0.55 × 3.3 × 3.3
0.5 × 3.5 × 3.5
d/h ratio
1.5
2.5
4
5.5
7
8.5
10
Conventional compression strength, MPa True compression strength, MPa Young's modulus, GPa Lateral deformation / axial deformation
405 ± 23 372 ± 23 8.80 ± 0.71 0.37 ± 0.06
404 ± 13 359 ± 16 5.32 ± 0.13 0.34 ± 0.05
525 ± 34 432 ± 16 4.36 ± 0.23 0.49 ± 0.07
615 ± 30 454 ± 5 3.76 ± 0.13 0.38 ± 0.07
625 ± 12 476 ± 12 2.82 ± 0.20 0.29 ± 0.04
667 ± 61 513 ± 22 2.29 ± 0.13 0.26 ± 0.01
749 ± 46 518 ± 35 2.13 ± 0.08 0.31 ± 0.04
D. Zaytsev / Materials Science and Engineering C 49 (2015) 101–105
Fig. 3. Сalculation of the Young's modulus with correction on the lateral deformation.
(Cu wire L = 3.36 mm). The axial deformation of the sample was measured by a testing machine. Conventional compression strength was taken as maximal stress on the deformation curve, whereas true compression strength was calculated as the maximal force divided by the square of contact area of the sample, which was calculated from the width of sample at this force. The Young's modulus was calculated from the slope of linear part of the deformation curve. Poisson's ratio was determined for the samples with ratio d/h = 4. Measuring of the lateral and the axial deformations occurred at 300 MPa. This stress was less than the proportional limit and the elastic limit (~ 350 MPa) and, therefore, the deformation of the dentin sample was elastic only [5]. Poisson's ratio was taken as the ratio between the lateral and the axial deformations. Additionally, the corrected Young's modulus was obtained for the sample with ratio d/h = 4. True stress (σc) was calculated by measuring the width of sample at conventional stress of 300 MPa and the corrected curve was plotted due to the axial deformation (εp) that is similar for both types of stresses (Fig. 3). The corrected Young's modulus was determined from the slope of corrected curve. 3. Result and discussion Compression tests of the dentin samples have shown that the true compression strength is less in comparison to the conventional compression strength (Table 1). The true compression strength increases with the growth of the d/h ratio like the conventional compression strength but it occurs not so much intensive (Fig. 4a). Ratio between lateral deformation–tensile and axial deformation–compression
103
is maximal for the samples with aspect ratio d/h = 4 and minimal for the samples with the highest aspect ratio whereas it is intermediate for the samples with low aspect ratio (Fig. 4b). Corrected Young's modulus of the sample with ratio d/h = 4 has less conventional Young's modulus on ~ 7% and it is 4.04 ± 0.12 GPa. Poisson's ratio of human dentin is 0.14 ± 0.04. Observation of the back surfaces of the samples with low aspect ratio under load shows that the samples have barrel like shape and there are two types of cracks in the sample (Fig. 5a). In the first type, the crack lies along the compression axis whereas the crack of the second type is inclined at ~ 45° to the compression surface. The mechanisms of crack growth are the rapture for the first type of crack and the shift for the second one [8]. These mechanisms of crack growth can be activated simultaneously due to the value of the shear strength (50–80 MPa) that is comparable with the tensile strength (35–100 MPa) [4,9–12]. There are many cracks of the second type on the compression surface of the sample with low aspect ratio (Fig. 5b). The level of splitting forces in a sample may be accepted as the critical parameter that affects the mechanical properties of brittle materials because crack growth is suppressed when the splitting forces become small (Fig. 6) [12,13]. The basic boundary condition of the mechanics of deformable solid is the constancy of sample volume under deformation. Therefore, under compression, the height of the sample decreases while its size in the direction perpendicular to the compression axis increases. Hence, the compression loading applied to the sample must induce the tensile stress in the direction perpendicular to the compression axis. Under a small compression deformation, the stress distribution in a sample is homogeneous, while the condition does not meet under considerable deformation due to friction or interaction between the sample and the loading plates of the testing machine. As a result, the shape of the sample has a trend to be barrellike, when the highest tensile stress level takes place in the middle part of the sample (Fig. 7a). For samples having high d/h ratio, the region of inhomogeneous deformation is small in comparison to the volume of the sample (Fig. 7b). Therefore, the stress distribution in the sample may be considered as almost homogeneous. Under compression, the splitting force depends on the tensile stress in the plane normal to the compression axis. The relative proportions of tensile and compressive stresses in samples under compression testing depend on the d/h ratio of the samples. The level of tensile stress is maximal at low aspect ratio. As a result, the dentin samples with high aspect ratio, where the splitting force is lowest, demonstrate the greatest deformation and the highest compressive strength. On the other hand, the level of splitting forces is greatest in dentin samples with low aspect ratio, and hence, crack growth may be arrested only by intrinsic properties of the material. However, the ratio between
Fig. 4. Curves of dependences of d/h ratio from: a — true compression strength and conventional compression strength; b — ratio between tensile and compression deformations. Sample with d/h = 4 possesses higher ratio between the lateral and the axial deformations.
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D. Zaytsev / Materials Science and Engineering C 49 (2015) 101–105
Fig. 5. Back surface of the samples: a — the sample with low aspect ratio; b — the sample with high aspect ratio. There are two types of the cracks (black arrows) in the sample with low aspect ratio.
tensile deformation and compression deformation in samples with low aspect ratio is lower due to the inhomogeneous stress distribution than in the samples with ratio d/h = 4 where it is maximal (Fig. 4b). Ratio d/ h = 4 is optimal for compression testing insomuch as the stress distribution in this case is homogeneous and the friction is not significant as for the sample with high aspect ratio. This conclusion is supported by the finding of compression testing of dentin samples having d/h = 4.3 in liquid environments, when deformation behavior of the dentin samples in glycerin and in air was similar [14]. It is well known that glycerin may be used as lubricant. Similarity of the deformation behavior is pointed to the fact that the friction is not significant for the samples with d/h = 4. Therefore, mechanical properties of the samples with such d/h ratio are more close to the intrinsic properties of human dentin.
Acknowledgments The reported study was supported by RFBR, research project No. 14-08-31691.
4. Conclusion The observation of the samples under compression testing shows that the shape effect in human dentin is caused mainly by friction between the surface of the sample and the compression plate. Inhomogeneous deformation occurs in the sample with low aspect ratio whereas the lateral deformation is hampered in the sample with high aspect ratio. Ratio d/h = 4 is the optimal proportion between the diagonal and the height of dentin samples at compression test. There is significant difference between conventional compression strength and true compression strength in human dentin. True compression strength of human dentin is 432 ± 16 MPa, the Young's modulus is 4.04 ± 0.12 GPa and the Poisson's ratio of human dentin is 0.14 ± 0.04.
Fig. 6. Distribution of the splitting forces near the pore-like crack.
Fig. 7. The shape of a sample under compression testing: a — low d/h-ratios; b — high d/h-ratios. Inhomogeneous deformations are much more in the sample with low aspect ratio in comparison to the sample having high d/h ratio.
D. Zaytsev / Materials Science and Engineering C 49 (2015) 101–105
References [1] G.V. Black, An investigation into the physical characters of the human teeth in relation to their diseases and to practical dental operations, Dent. Cosmos 37 (1895) 353–421 (469–484, 553–571, 637–661, and 737–757). [2] N.E. Waters, Some mechanical and physical properties of teeth, Symp. Soc. Exp. Biol. 34 (1980) 99–135. [3] J.H. Kinney, S.J. Marshall, G.W. Marshall, The mechanical properties of human dentin: a critical review and re-evaluation of the dental literature, Crit. Rev. Oral Biol. Med. 14 (1) (2003) 13–29. [4] I. Cappelloni, R. Montanari, Mechanical characterization of human dentin: a critical review, Key Eng. Mater. 541 (2013) 75–96. [5] D. Zaytsev, S. Grigoriev, P. Panfilov, Deformation behavior of human dentin under uniaxial compression, Int. J. Biomater. (2012). http://dx.doi.org/10.1155/2012/ 854539. [6] D. Zaytsev, A.S. Ivashov, J.V. Mandra, P. Panfilov, On the deformation behavior of human dentin under compression and bending, Mater. Sci. Eng. C 41 (2014) 83–90.
105
[7] D. Zaytsev, P. Panfilov, On some features of the shape effect in human dentin under compression, Mater. Sci. Eng. C 45 (2014) 205–209. [8] Knott J.F., Fundamentals of Fracture Mechanics London. Butterworths 1973. [9] D. Zaytsev, A.S. Ivashov, P. Panfilov, Anisotropy of the mechanical properties of human dentin under shear testing, Mater. Lett. 138 (2015) 219–221. [10] V. Lertchirakarn, J.E.A. Palamara, H.H. Messer, Anisotropy of tensile strength of root dentin, J. Dent. Res. 80 (2) (2001) 453–456. [11] D. Zaytsev, P. Panfilov, Deformation behavior of human dentin in liquid nitrogen: a diametral compression test, Mater. Sci. Eng. C 42 (2014) 48–51. [12] L.D. Landau, E.M. Lifshitz, Theory of Elasticity, Volume 7 of Course of Theoretical Physics, 2nd ed. Pergamon Press, 1970. [13] P.W. Bridgman, Studies in Large Plastic Flow and Fracture: With Special Emphasis on the Effects of Hydrostatic Pressure, McGraw-Hill, New York, 1952. [14] D. Zaytsev, N.V. Selezneva, S.S. Grigoriev, P. Panfilov, The influence of liquid on the deformation behavior of human dentin, Prot. Met.Phys. Chem. Surf. 49 (5) (2013) 517–520.
Contents lists available at ScienceDirect
Materials Science and Engineering C journal homepage: www.elsevier.com/locate/msec
Correction of some mechanical characteristics of human dentin under compression considering the shape effect Dmitry Zaytsev ⁎ Ural Federal University, Ekaterinburg, Russia
a r t i c l e
i n f o
Article history: Received 22 October 2014 Received in revised form 29 November 2014 Accepted 7 December 2014 Available online 26 December 2014 Keywords: Dentin Compression Shape effect Poisson's ratio
a b s t r a c t The paper is aimed to determine the true compression strength and Poisson's ratio of human dentin. The origin of the shape effect in dentin under compression is discussed, too. It was shown that the shape effect is mainly caused by the friction between the surface of the sample and the compression plates. Ratio d/h = 4 is the optimal proportion between the diagonal of compression surface and the height of dentin sample for compression testing. Inhomogeneous deformation takes place in the sample with a low aspect ratio whereas lateral deformation is suppressed in the sample with a high aspect ratio. There is significant difference between the conventional compression strength and the true compression strength. True compression strength of human dentin is 432 ± 16 MPa, the Young's modulus is 4.04 ± 0.12 GPa and Poisson's ratio of human dentin is 0.14 ± 0.04. © 2014 Elsevier B.V. All rights reserved.
1. Introduction Human dentin is the hard base of a tooth. The main mechanical function of dentin is to support the overlaying enamel, which protects the tooth from external action. Enamel re-distributes the dangerous stresses in homogeneous one into its thickness under mastication. Therefore, the uniaxial compression is the sole scheme of deformation that dentin underwent. Information on the mechanical properties of human dentin is important for the manufacturers of dental restorative materials because their properties should be close to the properties of tooth hard tissues. Study of the mechanical properties of dentin under compression attracted the interest of many researchers since 1895 [1]. However, the mechanical characteristics of dentin are different and varied in the wide diapason: the Young's modulus 8–19 GPa and the compression strength 250–350 MPa [2–4]. It has been shown that mechanical properties of the cuboid dentin samples under compression strongly depend on the ratio between the diagonal of compression surface of the sample and its height (d/h ratio, Fig. 1a). The samples with low aspect ratio behave like a brittle material while the deformation behavior of samples with high aspect ratio is close to a ductile solid. The Young's modulus of human dentin varied from 2 GPa to 11 GPa and the compression strength is 330–800 MPa [5,6]. This dependence of the mechanical properties from the sample shape explains in most cases the variation of the results given in the literature. In the other cases, the distinction
⁎ Department of Physics, Institute of Natural Sciences, Ural Federal University, Lenin Avenue, 51, 620083 Ekaterinburg, Russia. E-mail address: [email protected].
http://dx.doi.org/10.1016/j.msec.2014.12.080 0928-4931/© 2014 Elsevier B.V. All rights reserved.
in the results is caused by such experimental factors as environment and temperature of testing. The origin of the shape effect under compression connects with both intrinsic properties of the material and several experimental factors. One of them is the friction between the surface of the sample and the compression plate when the lateral deformation is hampered (Fig. 1b). The next cause of the shape effect is the difference between conventional and true stresses in the sample. Calculation of the conventional stresses is based on the initial size of sample, however, the sizes of the samples and its area of contact with the compression plates can be changed during loading and hence the level of true stresses in the samples should be distinct from the conventional stress, which was calculated by the testing machine (Fig. 2). It is not significant for a brittle material where the deformation is less than 1%, but it should be taken into account for a deformable material when the deformation can reach several tens of percent. Therefore, this difference in the stresses should be considered for human dentin under compression because its deformation reaches up to 56% for the samples with the highest aspect ratio [5–7]. One of the important mechanical parameters is the Poisson's ratio, which characterizes the competition between the lateral and the axial deformations. It is used for calculating of the Young's modulus at indentation testing. The Poisson's ratio of dentin lies in diapason 0.025–0.45 [4]. Variation of the value appears due to the measurement of Poisson's ratio that is carried out by the different methods such as Resonant Ultrasound Spectroscopy and calculation from the numerical models. No direct measurements of Poisson's ratio of dentin have been done. Besides, it should be noted that measurements of porous and inhomogeneous materials such as human dentin by means of wave transmission look like not quite correct.
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D. Zaytsev / Materials Science and Engineering C 49 (2015) 101–105
Fig. 1. Schemes: a — dimensions parameters of the sample; b — direction of friction forces.
values that reflect its intrinsic properties. The aim of this work is the correction of the compression strength, the Young's modulus and the Poisson's ratio of human dentin under compression testing.
2. Experimental procedure
Fig. 2. Сalculation of stress in the sample at compression test.
Recording of the lateral and the axial deformations of the sample during compression test allows estimating the influence of friction between the surface of the sample and the compression plate on the deformation behavior of dentin, and calculating the true mechanical
Fifty intact human molars and premolars were used in this work. They did not contain damages and were extracted from mature subjects according to the medical diagnosis and the Ethic Protocol of the Urals State Medical University at Yekaterinburg, Russia. Detailed description of the methods of sample preparation was given in Ref. [5]. The samples were cut from the teeth by means of a diamond saw with water irrigation and, further, their surfaces were abraded using the abrasive papers. Seven groups with different d/h ratios per 10 cuboid samples each were prepared. Sizes of the samples are given in Table 1. Uniaxial compression was carried out by means of a Shimadzu AGX-50 kN (Japan) testing machine at room condition. Rate of loading was 0.1 mm/min for all tests. Processing of the results including statistical analysis was carried out by Trapezium-X software. The lateral deformation of the sample was calculated by means of a Canon photo microsystem (Japan), where the width of the sample compared in situ with the etalon
Table 1 Mechanical properties of the sample of human dentin in dependence on the d/h ratio under compression. Sizes, mm
1.7 × 1.8 × 1.8
1.2 × 2.1 × 2.1
0.7 × 2.0 × 2.0
0.6 × 2.3 × 2.3
0.6 × 3.0 × 3.0
0.55 × 3.3 × 3.3
0.5 × 3.5 × 3.5
d/h ratio
1.5
2.5
4
5.5
7
8.5
10
Conventional compression strength, MPa True compression strength, MPa Young's modulus, GPa Lateral deformation / axial deformation
405 ± 23 372 ± 23 8.80 ± 0.71 0.37 ± 0.06
404 ± 13 359 ± 16 5.32 ± 0.13 0.34 ± 0.05
525 ± 34 432 ± 16 4.36 ± 0.23 0.49 ± 0.07
615 ± 30 454 ± 5 3.76 ± 0.13 0.38 ± 0.07
625 ± 12 476 ± 12 2.82 ± 0.20 0.29 ± 0.04
667 ± 61 513 ± 22 2.29 ± 0.13 0.26 ± 0.01
749 ± 46 518 ± 35 2.13 ± 0.08 0.31 ± 0.04
D. Zaytsev / Materials Science and Engineering C 49 (2015) 101–105
Fig. 3. Сalculation of the Young's modulus with correction on the lateral deformation.
(Cu wire L = 3.36 mm). The axial deformation of the sample was measured by a testing machine. Conventional compression strength was taken as maximal stress on the deformation curve, whereas true compression strength was calculated as the maximal force divided by the square of contact area of the sample, which was calculated from the width of sample at this force. The Young's modulus was calculated from the slope of linear part of the deformation curve. Poisson's ratio was determined for the samples with ratio d/h = 4. Measuring of the lateral and the axial deformations occurred at 300 MPa. This stress was less than the proportional limit and the elastic limit (~ 350 MPa) and, therefore, the deformation of the dentin sample was elastic only [5]. Poisson's ratio was taken as the ratio between the lateral and the axial deformations. Additionally, the corrected Young's modulus was obtained for the sample with ratio d/h = 4. True stress (σc) was calculated by measuring the width of sample at conventional stress of 300 MPa and the corrected curve was plotted due to the axial deformation (εp) that is similar for both types of stresses (Fig. 3). The corrected Young's modulus was determined from the slope of corrected curve. 3. Result and discussion Compression tests of the dentin samples have shown that the true compression strength is less in comparison to the conventional compression strength (Table 1). The true compression strength increases with the growth of the d/h ratio like the conventional compression strength but it occurs not so much intensive (Fig. 4a). Ratio between lateral deformation–tensile and axial deformation–compression
103
is maximal for the samples with aspect ratio d/h = 4 and minimal for the samples with the highest aspect ratio whereas it is intermediate for the samples with low aspect ratio (Fig. 4b). Corrected Young's modulus of the sample with ratio d/h = 4 has less conventional Young's modulus on ~ 7% and it is 4.04 ± 0.12 GPa. Poisson's ratio of human dentin is 0.14 ± 0.04. Observation of the back surfaces of the samples with low aspect ratio under load shows that the samples have barrel like shape and there are two types of cracks in the sample (Fig. 5a). In the first type, the crack lies along the compression axis whereas the crack of the second type is inclined at ~ 45° to the compression surface. The mechanisms of crack growth are the rapture for the first type of crack and the shift for the second one [8]. These mechanisms of crack growth can be activated simultaneously due to the value of the shear strength (50–80 MPa) that is comparable with the tensile strength (35–100 MPa) [4,9–12]. There are many cracks of the second type on the compression surface of the sample with low aspect ratio (Fig. 5b). The level of splitting forces in a sample may be accepted as the critical parameter that affects the mechanical properties of brittle materials because crack growth is suppressed when the splitting forces become small (Fig. 6) [12,13]. The basic boundary condition of the mechanics of deformable solid is the constancy of sample volume under deformation. Therefore, under compression, the height of the sample decreases while its size in the direction perpendicular to the compression axis increases. Hence, the compression loading applied to the sample must induce the tensile stress in the direction perpendicular to the compression axis. Under a small compression deformation, the stress distribution in a sample is homogeneous, while the condition does not meet under considerable deformation due to friction or interaction between the sample and the loading plates of the testing machine. As a result, the shape of the sample has a trend to be barrellike, when the highest tensile stress level takes place in the middle part of the sample (Fig. 7a). For samples having high d/h ratio, the region of inhomogeneous deformation is small in comparison to the volume of the sample (Fig. 7b). Therefore, the stress distribution in the sample may be considered as almost homogeneous. Under compression, the splitting force depends on the tensile stress in the plane normal to the compression axis. The relative proportions of tensile and compressive stresses in samples under compression testing depend on the d/h ratio of the samples. The level of tensile stress is maximal at low aspect ratio. As a result, the dentin samples with high aspect ratio, where the splitting force is lowest, demonstrate the greatest deformation and the highest compressive strength. On the other hand, the level of splitting forces is greatest in dentin samples with low aspect ratio, and hence, crack growth may be arrested only by intrinsic properties of the material. However, the ratio between
Fig. 4. Curves of dependences of d/h ratio from: a — true compression strength and conventional compression strength; b — ratio between tensile and compression deformations. Sample with d/h = 4 possesses higher ratio between the lateral and the axial deformations.
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D. Zaytsev / Materials Science and Engineering C 49 (2015) 101–105
Fig. 5. Back surface of the samples: a — the sample with low aspect ratio; b — the sample with high aspect ratio. There are two types of the cracks (black arrows) in the sample with low aspect ratio.
tensile deformation and compression deformation in samples with low aspect ratio is lower due to the inhomogeneous stress distribution than in the samples with ratio d/h = 4 where it is maximal (Fig. 4b). Ratio d/ h = 4 is optimal for compression testing insomuch as the stress distribution in this case is homogeneous and the friction is not significant as for the sample with high aspect ratio. This conclusion is supported by the finding of compression testing of dentin samples having d/h = 4.3 in liquid environments, when deformation behavior of the dentin samples in glycerin and in air was similar [14]. It is well known that glycerin may be used as lubricant. Similarity of the deformation behavior is pointed to the fact that the friction is not significant for the samples with d/h = 4. Therefore, mechanical properties of the samples with such d/h ratio are more close to the intrinsic properties of human dentin.
Acknowledgments The reported study was supported by RFBR, research project No. 14-08-31691.
4. Conclusion The observation of the samples under compression testing shows that the shape effect in human dentin is caused mainly by friction between the surface of the sample and the compression plate. Inhomogeneous deformation occurs in the sample with low aspect ratio whereas the lateral deformation is hampered in the sample with high aspect ratio. Ratio d/h = 4 is the optimal proportion between the diagonal and the height of dentin samples at compression test. There is significant difference between conventional compression strength and true compression strength in human dentin. True compression strength of human dentin is 432 ± 16 MPa, the Young's modulus is 4.04 ± 0.12 GPa and the Poisson's ratio of human dentin is 0.14 ± 0.04.
Fig. 6. Distribution of the splitting forces near the pore-like crack.
Fig. 7. The shape of a sample under compression testing: a — low d/h-ratios; b — high d/h-ratios. Inhomogeneous deformations are much more in the sample with low aspect ratio in comparison to the sample having high d/h ratio.
D. Zaytsev / Materials Science and Engineering C 49 (2015) 101–105
References [1] G.V. Black, An investigation into the physical characters of the human teeth in relation to their diseases and to practical dental operations, Dent. Cosmos 37 (1895) 353–421 (469–484, 553–571, 637–661, and 737–757). [2] N.E. Waters, Some mechanical and physical properties of teeth, Symp. Soc. Exp. Biol. 34 (1980) 99–135. [3] J.H. Kinney, S.J. Marshall, G.W. Marshall, The mechanical properties of human dentin: a critical review and re-evaluation of the dental literature, Crit. Rev. Oral Biol. Med. 14 (1) (2003) 13–29. [4] I. Cappelloni, R. Montanari, Mechanical characterization of human dentin: a critical review, Key Eng. Mater. 541 (2013) 75–96. [5] D. Zaytsev, S. Grigoriev, P. Panfilov, Deformation behavior of human dentin under uniaxial compression, Int. J. Biomater. (2012). http://dx.doi.org/10.1155/2012/ 854539. [6] D. Zaytsev, A.S. Ivashov, J.V. Mandra, P. Panfilov, On the deformation behavior of human dentin under compression and bending, Mater. Sci. Eng. C 41 (2014) 83–90.
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