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Early stage-stress relaxation in compact bone
Journal of Biomechanics 32 (1999) 93 — 97
Technical Note
Early stage-stress relaxation in compact bone Toshiharu Goto, Naoki Sasaki*, Kunio Hikichi Division of Biological Sciences, Graduate School of Science, Hokkaido University, Kitaku, Sapporo 060, Japan Received in final form 10 September 1998
Abstract The early stage of stress relaxation, up to 10 s after strain application, in compact bone was investigated in order to find the limit of the applicability of the empirical equation (Sasaki et al., 1993. Journal of Biomechanics 26, 1369—1376), E (t)"E +A exp [!(t/q )@]#A exp (!t/q ),, A #A "1, 0(b)1, where E is the relaxation modulus value at t"0, A and A are portions of the first term (the KWW relaxation) and the second term (the Debye relaxation), respectively, and q and q are relaxation times of respective relaxation processes. The relaxation modulus was generally well fitted to the empirical equation, although magnification revealed that in the short-time response region there was an inconsistency between the empirical equation and the measured data. The residual deviation increases with the decrease in time of the initial datum point which was used for the regression analysis. This tendency was concluded to be due to material properties. These results indicate the existence of a new relaxation in the short time response region (t(2—3 s) of the stress relaxation process that cannot be described by the empirical equation. 1999 Elsevier Science Ltd. All rights reserved. Keywords: Compact bone; Stress relaxation; Empirical formula for relaxation modulus; KWW relaxation; Response in short-time region
1. Introduction
Watts (KWW) function and an exponential decay function (Sasaki et al., 1993):
The major constituents of bone are a stiff hydroxyapatite (HAP)-like mineral and a pliant collagen matrix. Interaction between both phases is considered to determine the mechanical properties of a bone element, though as yet there have been no complete models that describe the mechanical structure responsible for the interaction in bone. Bone is not a perfectly elastic material, and the importance of molecular motion in the collagen phase of bone, in conjunction with water content of the system, for determining the bone’s non-elastic properties has been pointed out (Lakes and Katz, 1979). To understand the viscoelastic behavior of bone, a relaxation function for the relaxation phenomena in bone must be established. In our previous work, we found that stress relaxation of cortical bone could be generally expressed by a combination of the Kohlrausch—Williams—
E (t)"E +A exp [!(t/q )@]#A exp (!t/q ), , (1) [A #A "1, 0(A , A )1, 0(b)1], where E is the initial value, E (0), q and q are character istic times, A and A are portions of the relaxation processes described by the KWW function and the exponential (Debye) function, respectively, and b is a parameter describing the shape of the relaxation modulus. The values of q and q have been reported to be about 10 and 105—6 s, respectively (Sasaki and Enyo, 1995; Sasaki et al., 1993). Thus, the early stage of relaxation can be characterized by the KWW function. This empirical equation has been shown to be applicable to the data in the literature (Lugassy and Korostoff, 1969). The E value determined by fitting to Eq. (1) was generally slightly smaller than the Young’s modulus value obtained by dynamic mechanical or ultrasonic measurements (Currey, 1984; Goto et al., 1996). This discrepancy suggests a limit of the empirical equation
* Corresponding author. Tel.: 81 11 706 2659; fax: 81 11 706 4992.
0021-9290/99/$ — see front matter 1999 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 1 - 9 2 9 0 ( 9 8 ) 0 0 1 3 8 - 9
94
T. Goto et al. / Journal of Biomechanics 32 (1999) 93 — 97
and the existence of a new relaxation process that is not described by the KWW function. The aim of this study was to find the limit in time beyond which the KWW function is not applicable. For this, stress relaxation measurements over time were made up to 10 s after the strain application using a digitizer with sampling times of less than 5 ms.
2. Experimental Bone samples used in this study were obtained from the mid-diaphysis of an 18-month-old bovine femur. Op-
tical microscopic examination showed that all the samples were generally plexiform but partly transformed into Haversian bone. The samples were cut using a band saw. The cut sections were shaped into rectangular plates by emery paper under tap water, with the longer edges parallel to the longitudinal axis of the bone. The details of sample preparation are presented elsewhere (Sasaki et al., 1993). The relaxation Young’s modulus was measured by three-point bending of the rectangular sample plate, as shown in Fig. 1a. A Kyowa Electric Works (KEW) LTS1K strain gage transducer was used as both the deformation generator and the force sensor. The LTS1K
Fig. 1. (a) Schematic diagram of the relaxation Young’s modulus measuring system. (b) A typical load-time curve showing relaxation of bone (䉫) in comparison with stainless steel (䊉), which does not visibly relax.
T. Goto et al. / Journal of Biomechanics 32 (1999) 93 — 97
gage was set on an auto-microstage, and an operation of the auto-microstage was made by a rectangular pulse generated by a function synthesizer (Toa FS-111A) to guarantee identical deformation for a series of measurements. Bending deformation was detected by a Yokogawa Electric Works (YEW) 3612 strain transducer with a YEW 3624-05 sensor of the non-contacting type. Over a 50 s time interval, an electric signal from the LTS1K was passed on to a KEW DPM-613A strain amplifier, then to an Autnics S-210 auto-digitizer, and finally to a personal computer using a GPIB. The electric signal was recorded every 0.05 s by the S210. From 50 s up to 5;10 s, an electric signal generated from LTS1K was directly recorded by a KEW data logger, UCAM 10A, and finally passed on to a microcomputer. A maximum strain of less than 0.05% was applied within 0.025 s. This value is well within the frequency range examined by other authors in dynamic studies (Lakes et al., 1979). Fig. 1b shows the stress response from a high carbon stainless steel plate, which can be regarded as an ideally elastic material, in comparison with the stress relaxation of a bone specimen. From the stress response by the stainless steel plate, it was found that there were no overshoot in the applied strain. All the measurements were executed at 30.0°C in saline solution. Stress relaxation measurements were performed on four specimens, and parameter fitting to the empirical equation (1) was made for the average relaxation modulus curve. The fitting was carried out on the basis of the Symplex method of least squares for five independent parameters, E , b, q , q , and A . In order to increase the degree of compactness of the sum of squares (SS) of deviations of
95
empirically determined modulus values from those estimated by proposed equation (1), the least square procedures were repeated 10 000 times.
3. Results and discussion Fig. 2 shows the relaxation modulus, E (t), plotted against time for compact bone. The data points are averaged values over four stress relaxation measurements. The solid line represents the empirical equation presented above (Eq. (1)). The fitting seems to be satisfactory. The parameters were determined to be: E " 14.9 GPa, b"0.40, q "1.5;10 s, q "4.0;10 s, and A "0.15. There is a notable inconsistency between the data and fitting curve up to about 3 s. According to Fig. 1b, as mentioned above, there is no overshoot in the applied strain in this strain rate using this system. The overshoot against Eq. (1) observed for bone specimens is concluded to be a characteristic of the material. This strongly suggests that there is a relaxation process different from the KWW- and the Debye-type relaxation processes in the short time response region. Fig. 3a shows the residual deviation (defined as SS/N where N is the number of datum points) plotted against the time, ¹, of the initial datum point which was used for the regression analysis. There are two peaks: one at the short time response region and the other at about ¹"50 s. Using data points containing the data from these two regions, deviation in modulus value from that estimated by using Eq. (1) increases to increase the residual deviation. The data at about 50 s are considered to originate from the incomplete matching of the strain
Fig. 2. The average values of relaxation Young’s modulus plotted against logarithm of time. The solid line represents the empirical relation described by Eq. (1).
96
T. Goto et al. / Journal of Biomechanics 32 (1999) 93 — 97
Fig. 4. Logarithm of [ -dE (t)/d log t, plotted against log t, where E (t) was estimated from Eq. (1) using parameters determined for Fig. 2; E "14.9 GPa, q "1.5;10 s, q "4.0;10 s, b"0.40, and A "0.15.
Fig. 3. (a) Residual deviation plotted against the time, ¹, of the initial relaxation datum point used for the regression analysis. (b) Residual deviation plotted against ¹ for the data obtained using only the UCAM 10A system.
Fig. 4 shows the relaxation spectrum depicted by using parameters obtained for Fig. 2. At the longest t, a peak corresponding to the Debye-type relaxation was observed, and in the t&10 s region, a slight peak of the KWW-type relaxation was observed. A peak of the KWW-type relaxation was also observed by Lakes and Katz (1979). They attributed a few peaks observed at t&10—10\ s to bone structures such as lamellae or osteons. There is a possibility that the relaxation observed in the short time response region in our experiments is related to that observed by Lakes and Katz (1979). A quantitative discussion must be made after determining the function that describes the relaxation Young’s modulus in a short time after the excitation. Acknowledgements
amplifier-autodigitizer system to the UCAM 10A data logging system, because the peak of the residual deviation is located at the transforming region of these two systems. Using the UCAM 10A system and measuring from 10 s to 5;10 s, there is no peak of the residual deviation observed at around 50 s (Fig. 3b). This indicates that a peak observed around 50 s in these measurements must be artefactual. The peak of the residual deviation at the short time response region corresponds to the inconsistency of Eq. (1) and the data in Fig. 2. In this region, the inconsistency is remarkable and the residual deviation value increases when ¹ approaches 0. This supports the proposal of the existence of a new relaxation process in the short time response region.
The authors thank Yukimasa Matsuda and Atsuhiro Kondo for their cooperation. References Currey, J.D., 1984. The Mechanical Adaptations of Bones. Princeton University Press, Princeton, NJ. Goto, T., Sasaki, N., Hikichi, K., 1996. Viscoelastic properties of bone as a function of temperature. Journal of Japanese Society of Biorheology 10, 37—45. Lugassy, A.A., Korostoff, E., 1969. Viscoelastic behavior of bovine femoral cortical bone and sperm whale dentin. In: Korostoff, K. (Ed.). Research in Dental and Medical Materials. Plenum Press, New York. Lakes, R.S., Katz, J.L., Sternstein, S.S., 1979. Viscoelastic properties of wet cortical bone. 1. Torsional and biaxial studies. Journal of Biomechanics 12, 657—678.
T. Goto et al. / Journal of Biomechanics 32 (1999) 93 — 97 Lakes, R.S., Katz, J.L., 1979. Viscoelastic properties of wet cortical bone. 2. Relaxation mechanisms. Journal of Biomechanics 12, 679—687. Ngai, K.L., 1987. Evidences for universal behavior of condensed matter at low frequencies/long times. In: Ramakrishnan, T.V., Lakshimi, M.R. (Eds.), Non-Debye Relaxation in Condensed Matter. World Scientific, Singapore.
97
Sasaki, N., Nakayama, Y., Yoshikawa, M., Enyo, A., 1993. Stress relaxation function of bone and bone collagen. Journal of Biomechanics 26, 1369—1376. Sasaki, N., Enyo, A., 1995. Viscoelastic properties of bone as a function of water content. Journal of Biomechanics 28, 809—815.
Technical Note
Early stage-stress relaxation in compact bone Toshiharu Goto, Naoki Sasaki*, Kunio Hikichi Division of Biological Sciences, Graduate School of Science, Hokkaido University, Kitaku, Sapporo 060, Japan Received in final form 10 September 1998
Abstract The early stage of stress relaxation, up to 10 s after strain application, in compact bone was investigated in order to find the limit of the applicability of the empirical equation (Sasaki et al., 1993. Journal of Biomechanics 26, 1369—1376), E (t)"E +A exp [!(t/q )@]#A exp (!t/q ),, A #A "1, 0(b)1, where E is the relaxation modulus value at t"0, A and A are portions of the first term (the KWW relaxation) and the second term (the Debye relaxation), respectively, and q and q are relaxation times of respective relaxation processes. The relaxation modulus was generally well fitted to the empirical equation, although magnification revealed that in the short-time response region there was an inconsistency between the empirical equation and the measured data. The residual deviation increases with the decrease in time of the initial datum point which was used for the regression analysis. This tendency was concluded to be due to material properties. These results indicate the existence of a new relaxation in the short time response region (t(2—3 s) of the stress relaxation process that cannot be described by the empirical equation. 1999 Elsevier Science Ltd. All rights reserved. Keywords: Compact bone; Stress relaxation; Empirical formula for relaxation modulus; KWW relaxation; Response in short-time region
1. Introduction
Watts (KWW) function and an exponential decay function (Sasaki et al., 1993):
The major constituents of bone are a stiff hydroxyapatite (HAP)-like mineral and a pliant collagen matrix. Interaction between both phases is considered to determine the mechanical properties of a bone element, though as yet there have been no complete models that describe the mechanical structure responsible for the interaction in bone. Bone is not a perfectly elastic material, and the importance of molecular motion in the collagen phase of bone, in conjunction with water content of the system, for determining the bone’s non-elastic properties has been pointed out (Lakes and Katz, 1979). To understand the viscoelastic behavior of bone, a relaxation function for the relaxation phenomena in bone must be established. In our previous work, we found that stress relaxation of cortical bone could be generally expressed by a combination of the Kohlrausch—Williams—
E (t)"E +A exp [!(t/q )@]#A exp (!t/q ), , (1) [A #A "1, 0(A , A )1, 0(b)1], where E is the initial value, E (0), q and q are character istic times, A and A are portions of the relaxation processes described by the KWW function and the exponential (Debye) function, respectively, and b is a parameter describing the shape of the relaxation modulus. The values of q and q have been reported to be about 10 and 105—6 s, respectively (Sasaki and Enyo, 1995; Sasaki et al., 1993). Thus, the early stage of relaxation can be characterized by the KWW function. This empirical equation has been shown to be applicable to the data in the literature (Lugassy and Korostoff, 1969). The E value determined by fitting to Eq. (1) was generally slightly smaller than the Young’s modulus value obtained by dynamic mechanical or ultrasonic measurements (Currey, 1984; Goto et al., 1996). This discrepancy suggests a limit of the empirical equation
* Corresponding author. Tel.: 81 11 706 2659; fax: 81 11 706 4992.
0021-9290/99/$ — see front matter 1999 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 1 - 9 2 9 0 ( 9 8 ) 0 0 1 3 8 - 9
94
T. Goto et al. / Journal of Biomechanics 32 (1999) 93 — 97
and the existence of a new relaxation process that is not described by the KWW function. The aim of this study was to find the limit in time beyond which the KWW function is not applicable. For this, stress relaxation measurements over time were made up to 10 s after the strain application using a digitizer with sampling times of less than 5 ms.
2. Experimental Bone samples used in this study were obtained from the mid-diaphysis of an 18-month-old bovine femur. Op-
tical microscopic examination showed that all the samples were generally plexiform but partly transformed into Haversian bone. The samples were cut using a band saw. The cut sections were shaped into rectangular plates by emery paper under tap water, with the longer edges parallel to the longitudinal axis of the bone. The details of sample preparation are presented elsewhere (Sasaki et al., 1993). The relaxation Young’s modulus was measured by three-point bending of the rectangular sample plate, as shown in Fig. 1a. A Kyowa Electric Works (KEW) LTS1K strain gage transducer was used as both the deformation generator and the force sensor. The LTS1K
Fig. 1. (a) Schematic diagram of the relaxation Young’s modulus measuring system. (b) A typical load-time curve showing relaxation of bone (䉫) in comparison with stainless steel (䊉), which does not visibly relax.
T. Goto et al. / Journal of Biomechanics 32 (1999) 93 — 97
gage was set on an auto-microstage, and an operation of the auto-microstage was made by a rectangular pulse generated by a function synthesizer (Toa FS-111A) to guarantee identical deformation for a series of measurements. Bending deformation was detected by a Yokogawa Electric Works (YEW) 3612 strain transducer with a YEW 3624-05 sensor of the non-contacting type. Over a 50 s time interval, an electric signal from the LTS1K was passed on to a KEW DPM-613A strain amplifier, then to an Autnics S-210 auto-digitizer, and finally to a personal computer using a GPIB. The electric signal was recorded every 0.05 s by the S210. From 50 s up to 5;10 s, an electric signal generated from LTS1K was directly recorded by a KEW data logger, UCAM 10A, and finally passed on to a microcomputer. A maximum strain of less than 0.05% was applied within 0.025 s. This value is well within the frequency range examined by other authors in dynamic studies (Lakes et al., 1979). Fig. 1b shows the stress response from a high carbon stainless steel plate, which can be regarded as an ideally elastic material, in comparison with the stress relaxation of a bone specimen. From the stress response by the stainless steel plate, it was found that there were no overshoot in the applied strain. All the measurements were executed at 30.0°C in saline solution. Stress relaxation measurements were performed on four specimens, and parameter fitting to the empirical equation (1) was made for the average relaxation modulus curve. The fitting was carried out on the basis of the Symplex method of least squares for five independent parameters, E , b, q , q , and A . In order to increase the degree of compactness of the sum of squares (SS) of deviations of
95
empirically determined modulus values from those estimated by proposed equation (1), the least square procedures were repeated 10 000 times.
3. Results and discussion Fig. 2 shows the relaxation modulus, E (t), plotted against time for compact bone. The data points are averaged values over four stress relaxation measurements. The solid line represents the empirical equation presented above (Eq. (1)). The fitting seems to be satisfactory. The parameters were determined to be: E " 14.9 GPa, b"0.40, q "1.5;10 s, q "4.0;10 s, and A "0.15. There is a notable inconsistency between the data and fitting curve up to about 3 s. According to Fig. 1b, as mentioned above, there is no overshoot in the applied strain in this strain rate using this system. The overshoot against Eq. (1) observed for bone specimens is concluded to be a characteristic of the material. This strongly suggests that there is a relaxation process different from the KWW- and the Debye-type relaxation processes in the short time response region. Fig. 3a shows the residual deviation (defined as SS/N where N is the number of datum points) plotted against the time, ¹, of the initial datum point which was used for the regression analysis. There are two peaks: one at the short time response region and the other at about ¹"50 s. Using data points containing the data from these two regions, deviation in modulus value from that estimated by using Eq. (1) increases to increase the residual deviation. The data at about 50 s are considered to originate from the incomplete matching of the strain
Fig. 2. The average values of relaxation Young’s modulus plotted against logarithm of time. The solid line represents the empirical relation described by Eq. (1).
96
T. Goto et al. / Journal of Biomechanics 32 (1999) 93 — 97
Fig. 4. Logarithm of [ -dE (t)/d log t, plotted against log t, where E (t) was estimated from Eq. (1) using parameters determined for Fig. 2; E "14.9 GPa, q "1.5;10 s, q "4.0;10 s, b"0.40, and A "0.15.
Fig. 3. (a) Residual deviation plotted against the time, ¹, of the initial relaxation datum point used for the regression analysis. (b) Residual deviation plotted against ¹ for the data obtained using only the UCAM 10A system.
Fig. 4 shows the relaxation spectrum depicted by using parameters obtained for Fig. 2. At the longest t, a peak corresponding to the Debye-type relaxation was observed, and in the t&10 s region, a slight peak of the KWW-type relaxation was observed. A peak of the KWW-type relaxation was also observed by Lakes and Katz (1979). They attributed a few peaks observed at t&10—10\ s to bone structures such as lamellae or osteons. There is a possibility that the relaxation observed in the short time response region in our experiments is related to that observed by Lakes and Katz (1979). A quantitative discussion must be made after determining the function that describes the relaxation Young’s modulus in a short time after the excitation. Acknowledgements
amplifier-autodigitizer system to the UCAM 10A data logging system, because the peak of the residual deviation is located at the transforming region of these two systems. Using the UCAM 10A system and measuring from 10 s to 5;10 s, there is no peak of the residual deviation observed at around 50 s (Fig. 3b). This indicates that a peak observed around 50 s in these measurements must be artefactual. The peak of the residual deviation at the short time response region corresponds to the inconsistency of Eq. (1) and the data in Fig. 2. In this region, the inconsistency is remarkable and the residual deviation value increases when ¹ approaches 0. This supports the proposal of the existence of a new relaxation process in the short time response region.
The authors thank Yukimasa Matsuda and Atsuhiro Kondo for their cooperation. References Currey, J.D., 1984. The Mechanical Adaptations of Bones. Princeton University Press, Princeton, NJ. Goto, T., Sasaki, N., Hikichi, K., 1996. Viscoelastic properties of bone as a function of temperature. Journal of Japanese Society of Biorheology 10, 37—45. Lugassy, A.A., Korostoff, E., 1969. Viscoelastic behavior of bovine femoral cortical bone and sperm whale dentin. In: Korostoff, K. (Ed.). Research in Dental and Medical Materials. Plenum Press, New York. Lakes, R.S., Katz, J.L., Sternstein, S.S., 1979. Viscoelastic properties of wet cortical bone. 1. Torsional and biaxial studies. Journal of Biomechanics 12, 657—678.
T. Goto et al. / Journal of Biomechanics 32 (1999) 93 — 97 Lakes, R.S., Katz, J.L., 1979. Viscoelastic properties of wet cortical bone. 2. Relaxation mechanisms. Journal of Biomechanics 12, 679—687. Ngai, K.L., 1987. Evidences for universal behavior of condensed matter at low frequencies/long times. In: Ramakrishnan, T.V., Lakshimi, M.R. (Eds.), Non-Debye Relaxation in Condensed Matter. World Scientific, Singapore.
97
Sasaki, N., Nakayama, Y., Yoshikawa, M., Enyo, A., 1993. Stress relaxation function of bone and bone collagen. Journal of Biomechanics 26, 1369—1376. Sasaki, N., Enyo, A., 1995. Viscoelastic properties of bone as a function of water content. Journal of Biomechanics 28, 809—815.