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Nonlinear stress relaxation of trabecular bone
Mechanics Research Communications 36 (2009) 275–283
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Mechanics Research Communications journal homepage: www.elsevier.com/locate/mechrescom
Nonlinear stress relaxation of trabecular bone Virginio Quaglini *, Valentina La Russa, Stefano Corneo Department of Structural Engineering, Politecnico of Milano, Piazza Leonardo da Vinci 32 – 20133 Milano, Italy
a r t i c l e
i n f o
Article history: Received 9 September 2008 Available online 21 November 2008
Keywords: Viscoelasticity Stress relaxation Nonlinearity Trabecular bone
a b s t r a c t The viscoelastic behavior of bovine trabecular bone is investigated by means of relaxation tests carried out either in compression or in bending configurations. The experimental curves show that the stress relaxation process has two distinct stages: a long-term process depending only on the time variable, and a short-term process whose magnitude is affected from elastic stresses. An analytical expression describing the normalized relaxation curves is proposed, and the relevant characteristic times are identified by fitting the experimental data. The characteristic times are found to be material properties not depending on the particular loading configuration. Ó 2008 Elsevier Ltd. All rights reserved.
1. Introduction In the last forty years, the viscoelasticity of cortical bone has been the subject of a number of experimental and theoretical studies. Creep tests (Caler and Carter, 1989; Bowman et al., 1994; Fondrk et al., 1988), relaxation tests (Sasaki et al., 1993; Iyo et al., 2004) and dynamic–mechanical analyses (Lakes et al., 1979; Yamashita et al., 2001) were carried out using either human or bovine models, in tension, compression, bending and in torsion. The viscoelastic features of bone were explained as a result of several processes including thermoelastic coupling, piezoelectric coupling, motion of fluids in canals and interstitia of bone and intrinsic viscoelasticity of collagen fibers (Lakes and Katz, 1979a). The main outcomes can be summarized in the following points: (1) bone viscoelasticity can be handled within the framework of the quasi-linear viscoelasticity (a description of the theory can be found in Fung, 1993): the ‘‘elastic” stress–strain relationship is generally nonlinear, but the memory is independent of the magnitude of stress or strain (Lakes and Katz, 1979b); (2) the stress relaxation process can be split into two stages, one relevant to the short-term behavior, and the second one to the long-term dynamics; for both processes, analytical expressions based on exponential-law functions were proposed (Sasaki et al., 1993; Iyo et al., 2004, 2006); (3) the short-term relaxation process is likely to be isotropic, while the long-term dynamics is affected by the orientation of the applied strain. Although drawn for the cortical bone, these findings in part apply to trabecular bone as well. Deligianni et al. (1994) performed stress relaxation tests at different values of the initial stress in the range between 0.33 and 0.66 of the ultimate stress, and confirmed both the assumptions about the separation of stress and time variables within the relaxation function, and the anisotropy of the relaxation properties. Guedes et al. (2006) proposed power-law expressions for the relaxation function, and
* Corresponding author. Tel.: +39 02 23994248; fax: +39 02 23994286. E-mail address: [email protected] (V. Quaglini). 0093-6413/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.mechrescom.2008.10.012
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showed that these expressions well match the exponential model proposed by Iyo et al. (2004) in the short-term process, but diverge over large times. In this study, we investigated the viscoelastic behavior of trabecular bone by means of stress relaxation tests carried out either in compression or in bending load configuration at different values of the initial stress in a bovine model. The aim was to assess the reliability of the quasi-linear viscoelastic model within a wide range of stress conditions, and to develop a suitable analytical expression for the stress relaxation function consistent with the experimental data. 2. Materials and methods 2.1. Test specimens The tests were performed on specimens of trabecular bone harvested from bovine femoral heads. Eleven donor animals (age ranging from 18 to 24 months) taken from a slaughter were included into the protocol. Each animal was identified with a label as B #X, where X = 1, 2, 3, . . . After removing of fat, muscles and other soft tissues, bovine femurs were placed in a custom vice and the heads were resected with the help of a die along a plane perpendicular to the femoral neck’s axis. Two distinct types of specimens were machined: cylindrical specimens (height 20 mm, diameter 10 mm) and beamshaped specimens (cross-section 5 5 mm, length approximately 40 mm). The specimens were extracted from the core of the femoral heads, with their long axes perpendicular to the head’s resection plane, in order to minimize the influences of non homogeneity (see, for example, Schoenfeld et al. (1974) and Vahey et al. (1987)) and anisotropy (Deligianni et al., 1994) of the tissue, so only one or two specimens were obtained from each femur, for a total of 42 specimens (21 cylindrical specimens and 21 beam specimens). The specimens were defatted from the bone marrow according to a standard procedure consisting of immersion in 1:3 solution of water and chloride, followed by three repetitions of ultrasonic cleaning in demineralized water for 5 min and subsequent air-drying for 2 min. After preparation, all specimens were stored frozen at 24 °C. Prior to testing, the specimens were thawed in physiological saline and conditioned at ambient temperature for at least 1 h. 2.2. Experimental protocol The tests were carried out on an MTS Synergie 200 H (MTS, Minneapolis, MN) testing machine equipped with a 1000 kN load cell. A custom environmental chamber comprising a thermostatic bath filled with saline solution was mounted on the testing machine, allowing control of the degree of moisture and the temperature (37 ± 1 °C) of the specimen throughout the test. Relaxation tests were performed either in compression on the cylindrical specimens or in the three-point bending configuration on the beam-shaped specimens. In order to harmonize the testing protocol and the procedure for data analysis between the two distinct load configurations, a suitable notation was introduced. Let P and M indicate the internal force and moment produced within the specimen from the external load, namely, the axial force in the compression test (P) and the bending moment at the midspan cross-section in the three-point bending test (M). Preliminary relaxation tests either in compression or in bending were carried out to assess the suitability of the storage procedure to preserve the viscoelastic properties of the trabecular bone specimens over time, and to determine the minimum period required by the bone tissue for complete recovery from the previously applied stress history. The cylindrical specimens were mechanically preconditioned by execution of 10 loading cycles between 0 and 72 N, which corresponded to about 60% of the compressive load at the limit of elasticity (PY = 1412 ± 101.6 N) previously determined. The preconditioned specimens were assayed in stress relaxation tests consisting in the application of a compressive strain at the constant rate of 0.9 mm/mm min1, until a specified preset load was attained; the strain was then held constant and the decrease in load recorded over a period of 600 s. Eleven cylindrical specimens were submitted for the execution of the stress relaxation test at five different values of the internal force (P0 = 120, 240, 360, 480 and 600 N), for a total of 55 tests. The beam-shaped specimens were tested in a three-point bending configuration, relying on two supports with a span width of 15 mm, according to a similar procedure. A set of preliminary tests was carried out on dummy specimens, and the bending moment at the midspan of the specimen was determined at the limit of elasticity (MY = 361.1 ± 53.6 N mm). Mechanical preconditioning was carried out by loading the specimen so that the midspan bending moment ranged between 0 and 180 N mm, the upper boundary corresponding again to about 60% of the yield limit. In the stress relaxation tests, the specimens were loaded at their midspan at a constant rate of 2 mm/mm min1 until a preset value of bending moment was attained; the deflection was then held constant for 600 s, and the relevant decrease in the internal moment at the midspan recorded. Thirteen beam-shaped specimens were tested each at four different values of the preset moment (M0 = 37.5, 75, 112.5 and 150 N mm) for a total of 52 tests. Throughout the experiments the specimens were kept immersed in physiological saline at 37 ± 1 °C within the environmental chamber on the testing machine. After each test, the specimen was removed from the testing bench and left to recover from the previous loading history for a period of 1 h in saline solution.
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2.3. Data analysis The response of a viscoelastic solid in a relaxation test depends on both the initial load and on the time elapsed from the beginning of the test, so it is possible to describe the history of the internal force P, (or the internal moment M as relevant), by means of an analytical function of both P0 (or M0) and t as independent variables, where P0 (M0) indicates the elastic value of the internal load (moment) applied at the beginning of the test (t = 0 s), before relaxation occurs. Under the condition P(t) 6 PY (M(t) 6 MY), where PY (MY) represents the value of the internal force (internal moment) at the limit of elasticity, we can introduce either for cylindric specimens tested in compression or for beam specimens tested in bending, a generalized stress variable S independent of the shape and size of the specimen according to the following definitions:
PðP 0 ; tÞ PY MðM0 ; tÞ in bending tests: SðSe ; tÞ ¼ MY
in compression tests: SðSe ; tÞ ¼
ð1aÞ ð1bÞ
Here Se represents the generalized elastic stress obtained by dividing the elastic response P0 by PY, or by dividing M0 by MY as relevant. The generalized relaxation function S(Se, t) can be normalized in turn over the elastic stress Se, to give the reduced relaxation function
GðSe ; tÞ ¼
SðSe ; tÞ Se
ð2Þ
which describes the fraction of the initial generalized stress that is absorbed by the specimen at time t from the initial application of the external load. Two parameters can be evaluated from the reduced relaxation function G(Se, t) to characterize the relaxation behavior:(a) the initial relaxation rate that is the slope of the reduced relaxation curve at time t = 0 (at the instant of application of the initial stress):
_ e Þ ¼ 1 dGðSe ; tÞ GðS Se dt t¼0
ð3Þ
(b) the steady relaxation value, that is the asymptotic value attained at the end of the relaxation process:
G1 ðSe Þ ¼ lim
t!1
SðSe ; tÞ Se
ð4Þ
3. Experimental results A set of preliminary tests was carried out for validating the experimental procedure. Repeated relaxation tests were carried out on the same specimens either within a few hours from calf slaughtering or after different periods of storage at 24 °C, and good reproducibility of the relevant relaxation curves were obtained for storage periods up to 960 h (an example is reported in Fig. 1a). In a similar way, we assessed that a wait period of 1 h was enough to complete viscoelastic recovery of the specimen material from the previous load history and to allow the repeatability of the relaxation response (Fig. 1b). Representative examples of relaxation curves of the trabecular bone from the present tests are reproduced in Figs. 2 and 3. Similar trends of the relaxation response were shown both in compression and in bending trials: under application of a constant deformation, either the internal force or the moment exhibited a continuous decrease from the initial value, until a steady regime value was attained within 300 s from the beginning of the test, and at 600 s the further decrease of the relaxation curve over the value measured at 300 s was never significant (p > 0.05 in each test). 3.1. Compression tests Once relaxation curves like the ones depicted in Fig. 2a, which report the decrease in the internal axial force absorbed by the specimen over time, were normalized with respect to the initial force P0, the resulting reduced relaxation curves did not match (Fig. 2b). The analysis of the relaxation behavior was carried out in terms of generalized stresses, and the influence of the initial stress was investigated by means of paired t-tests between groups of normalized curves relative to different values of Se. Since the relaxation process reached a steady regime after about 300 s, the fraction of the initial stress at t = 300 s was taken _ e Þ and the steady relaxation value G1 ðSe Þ were comparable between relaxation as G1 ðSe Þ. Both the initial relaxation rate GðS tests performed at medium or high values of Se, but the influence of the elastic variable became significant when the comparison was made between high and low values of the stress: the initial relaxation rate increased and the steady relaxation value decreased on decreasing the initial stress (Fig. 4).
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a
-1000
-900
Axial Load [N]
-800
-700
-600
-500
control 288 h at -24°C
72 h at -24°C
96 h at -24°C
960 h at -24°C
-400 0
100
200
300
time [s]
b
-800
control after 1 h recovery
Axial Load [N]
-750
-700
-650
-600 0
100
200
300
time [s] Fig. 1. Preliminary tests: (a) stress relaxation curves relevant to the same specimen tested within 6 h from calf slaughtering (control) and after storage at 24 °C for 72, 96, 288 and 960 h overlap; (b) reproducibility of the relaxation curve shown by the virgin test specimen (control) is attained after 1 h recovery in saline at +37 °C temperature.
3.2. Bending tests The relaxation curves obtained in the bending tests exhibited the same features as found in the compression tests. Once relaxation curves as shown in Fig. 3a, which report the current value of the internal bending moment at the midspan of the specimen, were normalized over the preset value of the moment, the resulting reduced relaxation curves diverge (Fig. 3b). _ e Þ and the steady relaxation value G1 ðSe Þ, evaluated again at Like in compression tests, both the initial relaxation rate GðS t = 300 s, were comparable between the relaxation tests at the highest values of the generalized elastic stress (Se = 0.38 vs. Se = 0.50), but the dependence on the elastic stress was not negligible at lower levels, producing an increase in the initial relaxation rate and a decrease in the steady relaxation value at decreasing of Se (Fig. 5). 4. Reduced relaxation function Both in compression and in bending experiments, the relaxation of trabecular bone was affected by the magnitude of the initial stress, Se. The amount of relaxation reduced in proportion to the increase in the elastic stress, showing a decrease in the initial relaxation rate and an increase in the steady relaxation value. However, the influence of the generalized elastic stress Se became less significant as the stress was increased, and it vanished at Se values larger than 0.3. A mathematical expression for the reduced relaxation function consistent with the experimental data was therefore developed. The basic assumptions upon which the model was established are:
V. Quaglini et al. / Mechanics Research Communications 36 (2009) 275–283
a
279
-800
Axial Load [N]
-600
-400
-200
0 0
100
200
300
time [s]
Normalized relaxation [-]
b
1.00
P0 = 727 N 0.95
P0 = 577 N P0 = 445 N P0 = 320 N
0.90 P0 = 168 N 0.85 0
100
200
300
time [s] Fig. 2. (a) Relaxation curves in compression tests at different values of the preset axial load. (b) After normalization over the initial value of axial load, the reduced stress relaxation curves do not match.
(a) the relaxation mechanism consists of two different processes, the first one prevailing in the short-term period, and the second one prevailing in the long-term; (b) both relaxation processes are described by exponential functions with characteristic times s1 and s2, respectively; (c) since the initial relaxation rate, which rules the short-term dynamics, is dependent on the initial value of the generalized stress, the characteristic time s1 is a function of the stress variable Se; (d) since the relaxation exhausted within 300 s, whichever was the initial stress at time t = 0, the long-term dynamics is independent of the elastic variable, and a constant value of the characteristic time s2 is taken; (e) the steady value of relaxation is a function of the elastic variable Se. So the following mathematical expression was proposed for the normalized relaxation function:
GðSe ; tÞ ¼ A0 ðSe Þ þ A1 ðSe Þ exp
t t þ A2 ðSe Þ exp s1 ðSe Þ s2
ð5Þ
where s1 and s2 represent the characteristic times of the two relaxation processes, A1 and A2 are the weights of such processes, and A0 is the asymptotic relaxation value. In order to fulfill the condition that at time t = 0 s the stress does coincide with the elastic value, a further requirement is
GðSe ; 0Þ ¼ A0 ðSe Þ þ A1 ðSe Þ þ A2 ðSe Þ ¼ 1
ð6Þ
The reduced relaxation curves at the different preset values of stress for all the specimens were curve-fitted by using Eq. (5). Since in the three-point bending test configuration the actual distribution of stresses is not uniform within the specimen, parameters A0, A1 and A2 had not a straightforward physical interpretation, so we focused our attention on the characteristic
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Bending moment [Nmm]
a
160
120
80
40
0 0
100
200
300
time [s]
Normalized moment [-]
b
1
M0 = 150 Nmm 0.9
M0 = 112.5
M0 = 75 Nmm 0.8 M0 = 37.5 Nmm
0.7 0
100
200
300
time [s] Fig. 3. (a) Relaxation curves in the three-point bending tests at different values of the bending moment at the midspan cross-section of the specimen. (b) After normalization over the preset value of bending moment, the normalized relaxation curves diverge.
times only. According to our assumption, a good fit was obtained by assuming a constant value s2 = 95 s, whichever the specimen and the elastic stress value. On the contrary, the value of s1 was affected from the preset stress, ranging between 2.18 and 4.25 s. To describe the dependence of s1 on the elastic variable Se, an exponential expression was used
s1 ðSe Þ ¼ a expðbSe Þ
ð7Þ
The values of the material parameters a and b were identified for each donor sample and are reported in Tables 1 and 2, along with the relevant correlation factor. Despite the experimental uncertainty and the variability among different specimens due to tissue non homogeneity, good correlation was found between the experimental values of s1 and the values predicted by Eq. (7), as a function of the elastic variable Se: the correlation factor ranged between 0.91 and 0.97 whichever the test condition. Ignoring sample B #6, a good agreement was found between compression and bending groups, with statistically comparable values of a and b parameters. 5. Discussion In the present study, the viscoelastic behavior of trabecular bone was investigated either in compression or in three-point bending test. In order to directly compare the behavior of the tissue in either test configurations, a generalized stress was defined as the ratio between the relaxed value of the internal force or moment and the corresponding value at the limit of elasticity. The relaxation tests were carried out at different values of the initial generalized stress Se, which represents the ‘‘elastic” stress absorbed by the specimen before relaxation occurs.
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-1
initial relaxation rate [s ]
a
0.015
0.012
0.009
0.006
(*)
0.003
(*)
(*)
(*)
(**)
(**)
(**)
0.000 0.1
0.2
0.3
0.4
0.5
b
1.00
steady relaxation [-]
generalized elastic stress Se
0.95
0.90
(*)
0.85
(*)
(*)
(*)
(***)
(***)
(***)
0.3
0.4
0.80 0.1
0.2
0.5
generalized elastic stress Se _ e Þ; (b) steady relaxation value G1 ðSe Þ determined Fig. 4. Relaxation tests in compression. Average value ± standard deviation of (a) initial relaxation rate GðS at different values of elastic stress Se on 11 specimens of trabecular bone: () different from Se = 0.1 at p < 0.01; () different from Se = 0.2 at p < 0.01; () different from Se = 0.2 at p < 0.05.
In opposition to the behavior of cortical bone tissue (Lakes and Katz, 1979b), and in contrast with the assumptions made by Deligianni et al. (1994) and Guedes et al. (2006), the results showed that the viscoelastic behavior of trabecular bone is nonlinear with respect to the elastic variable, since the magnitude of relaxation affected the initial stress: the higher the initial stress, the slower the relaxation rate and the smaller the decrease in stress at the steady regime. A possible explanation of the discrepancy between our results and the literature is that the influence of the initial load on the reduced relaxation curve becomes significant only at values lower than 30% of the limit of elasticity, while the other studies were restricted only to high values of initial load. To model the experimental reduced relaxation data, an analytical expression, Eq. (5), was proposed. This expression is borrowed from analytical relaxation models proposed for the cortical bone (Sasaki et al., 1993; Iyo et al., 2004, 2006), in that it assumes that the relaxation of trabecular bone has two distinct stages, one relevant to the short-term, and the other to the long-term dynamics. Each relaxation mechanism is modeled by an exponential function with its own characteristic time. According to the exponential model, the stress relaxation process is considered to be governed by the viscoelastic properties of the matrix collagen fibers within the bone structure (Sasaki et al, 1993). It should be noted that even though for most of the living tissue the assumption of the independence of stress relaxation on elastic variables is usually made (Fung, 1993), the influence of the initial load on the magnitude of stress relaxation was shown for some soft collagenous tissues (Garcia Sestafe et al., 1994; Johnson et al., 1996). However, according to the evidence that whichever the initial stress, the relaxation curve attained a constant asymptotic value in about 300 s, we assumed a constant value of the long-term characteristic time s2 among the whole specimens, independent of the elastic stress and the load configuration. On the contrary, the short-term time s1 was affected by the initial stress, and an exponential function was proposed on an empirical basis to model such dependence. The characteristic times of the relaxation function were comparable among the individual donors and between the two groups tested either in bending or in compression. Therefore, the nonlinear relaxation mechanism suggested in the present
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-1
initial relaxation rate [s ]
a
0.020
0.015
0.010
(*)
0.005
(*)
(*)
(**)
(***)
0.38
0.50
0.000 0.13
0.25
generalized elastic stress Se
b
1.00
steady relaxation [-]
0.95
0.90
0.85
0.80
(*)
(*)
(*)
(**)
(***)
0.38
0.50
0.75 0.13
0.25
generalized elastic stress Se _ e Þ; (b) steady relaxation value G1 ðSe Þ determined at Fig. 5. Relaxation tests in bending. Average value ± standard deviation of (a) initial relaxation rate GðS different values of elastic stress Se on 13 specimens of trabecular bone: () different from Se = 0.13 at p < 0.01; () different from Se = 0.25 at p < 0.05; () different from Se = 0.25 at p < 0.01.
Table 1 Coefficients a and b and correlation coefficient q of the exponential regression fit, Eq. (7), of characteristic time s1 in compression tests. Five relaxation tests were carried out for each specimen (N: number of specimens from each donor) and used for curve-fitting. Donor
N
a (s)
b ()
q
B B B B B
4 2 2 2 1
3.8673 4.2363 3.4303 3.3017 2.7370
0.0087 0.0084 0.0060 0.0069 0.0079
0.906 0.963 0.911 0.928 0.919
#1 #2 #3 #4 #5
Table 2 Coefficients a and b of Eq. (6) and correlation coefficient q of the exponential regression fit, Eq. (7), of characteristic time s1 in the three-point bending test. Four relaxation tests were carried out for each specimen (N: number of specimens from each donor) and used for curve-fitting. Donor
N
a (s)
b ()
q
B B B B B B
2 2 3 2 2 2
2.1639 3.0948 3.0128 2.9563 3.7412 3.5852
0.0198 0.0087 0.0111 0.0104 0.0056 0.0036
0.912 0.973 0.906 0.911 0.961 0.907
#6 #7 #8 #9 #10 #11
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study actually represents an intrinsic process of trabecular bone, which is not affected by individual tissue variability or load configuration. The present report belongs to a wider work that is still in progress aiming at delineating the viscoelastic features of trabecular bone. At present, only the dependence of the characteristic times of the relaxation function s1 and s2 on the elastic variable has been modeled. Future works will extend the investigation to the whole coefficients of Eq. (5). References Bowman, S.M., Keaveny, T.M., Gibson, L.J., Hayes, W.C., McMahon, T.A., 1994. Compressive creep behavior of bovine trabecular bone. J. Biomech. 27, 301– 310. Caler, W.E., Carter, D.R., 1989. Bone creep-fatigue damage accumulation. J. Biomech. 22, 625–635. Deligianni, D.D., Maris, A., Missirlis, Y.F., 1994. Stress relaxation behaviour of trabecular bone specimens. J. Biomech. 27, 1469–1476. Fondrk, M., Bahniuk, E., Davy, D.T., Michaels, C., 1988. Some viscoplastic characteristics of bovine and human cortical bone. J. Biomech. 21, 623–630. Fung, Y.C., 1993. Biomechanics. Mechanical Properties of Living Tissues, second ed. Springer-Verlag, New York. Garcia Sestafe, J.V., Garcia Paez, J.M., Carrera San Martin, A., Jorge-Herrera, E., Navidad, R., Candela, I., Castillo-Olivares, J.L., 1994. Description of the mathematical law that defines the relaxation of bovine pericardium subjected to stress. J. Biomed. Mater. Res. 28, 755–760. Guedes, R.M., Simões, J.A., Morais, J.L., 2006. Viscoelastic behaviour and failure of bovine cancellous bone under constant strain rate. J. Biomech. 39, 49–60. Iyo, T., Maki, Y., Sasaki, N., Nakata, M., 2004. Anisotropic viscoelastic properties of cortical bone. J. Biomech. 37, 1433–1437. Iyo, T., Sasaki, N., Maki, Y., Nakata, M., 2006. Mathematical description of stress relaxation of bovine femoral cortical bone. Biorheology 43, 117–132. Johnson, G.A., Liversay, G.A., Woo, S.L.-Y., Rajagopal, K.R., 1996. A single integral finite strain viscoelastic model of ligaments and tendons. ASME J. Biomech. Eng. 118, 221–226. Lakes, R.S., Katz, J.L., 1979a. Viscoelastic properties of wet cortical bone – II. Relaxation mechanisms. J. Biomech. 12, 679–687. Lakes, R.S., Katz, J.L., 1979b. Viscoelastic properties of wet cortical bone – III. A nonlinear constitutive equation. J. Biomech. 12, 689–698. Lakes, R.S., Katz, J.L., Sternstein, S., 1979. Viscoelastic properties of wet cortical bone – I. Torsional and biaxial studies. J. Biomech. 12, 657–678. Sasaki, N., Nakayama, Y., Yoshikawa, M., Enyo, A., 1993. Stress relaxation function of bone and bone collagen. J. Biomech. 26, 1366–1376. Schoenfeld, C.M., Lautenschlager, E.P., Meyer Jr., P.R., 1974. Mechanical properties of human cancellous bone in the femoral head. Med. Biol. Eng. 12, 313– 317. Vahey, J.W., Lewis, J.L., Vanderby, R., 1987. Elastic moduli, yield stress, and ultimate stress of cancellous bone in the canine proximal femur. J. Biomech. 20, 29–33. Yamashita, J., Furman, B.R., Rawls, H.R., Wang, X., Agrawal, C.M., 2001. The use of Dynamic mechanical analysis to assess the viscoelastic properties of human cortical bone. J. Biomed. Mater. Res. (Appl. Biomater.) 58B, 47–53.
Contents lists available at ScienceDirect
Mechanics Research Communications journal homepage: www.elsevier.com/locate/mechrescom
Nonlinear stress relaxation of trabecular bone Virginio Quaglini *, Valentina La Russa, Stefano Corneo Department of Structural Engineering, Politecnico of Milano, Piazza Leonardo da Vinci 32 – 20133 Milano, Italy
a r t i c l e
i n f o
Article history: Received 9 September 2008 Available online 21 November 2008
Keywords: Viscoelasticity Stress relaxation Nonlinearity Trabecular bone
a b s t r a c t The viscoelastic behavior of bovine trabecular bone is investigated by means of relaxation tests carried out either in compression or in bending configurations. The experimental curves show that the stress relaxation process has two distinct stages: a long-term process depending only on the time variable, and a short-term process whose magnitude is affected from elastic stresses. An analytical expression describing the normalized relaxation curves is proposed, and the relevant characteristic times are identified by fitting the experimental data. The characteristic times are found to be material properties not depending on the particular loading configuration. Ó 2008 Elsevier Ltd. All rights reserved.
1. Introduction In the last forty years, the viscoelasticity of cortical bone has been the subject of a number of experimental and theoretical studies. Creep tests (Caler and Carter, 1989; Bowman et al., 1994; Fondrk et al., 1988), relaxation tests (Sasaki et al., 1993; Iyo et al., 2004) and dynamic–mechanical analyses (Lakes et al., 1979; Yamashita et al., 2001) were carried out using either human or bovine models, in tension, compression, bending and in torsion. The viscoelastic features of bone were explained as a result of several processes including thermoelastic coupling, piezoelectric coupling, motion of fluids in canals and interstitia of bone and intrinsic viscoelasticity of collagen fibers (Lakes and Katz, 1979a). The main outcomes can be summarized in the following points: (1) bone viscoelasticity can be handled within the framework of the quasi-linear viscoelasticity (a description of the theory can be found in Fung, 1993): the ‘‘elastic” stress–strain relationship is generally nonlinear, but the memory is independent of the magnitude of stress or strain (Lakes and Katz, 1979b); (2) the stress relaxation process can be split into two stages, one relevant to the short-term behavior, and the second one to the long-term dynamics; for both processes, analytical expressions based on exponential-law functions were proposed (Sasaki et al., 1993; Iyo et al., 2004, 2006); (3) the short-term relaxation process is likely to be isotropic, while the long-term dynamics is affected by the orientation of the applied strain. Although drawn for the cortical bone, these findings in part apply to trabecular bone as well. Deligianni et al. (1994) performed stress relaxation tests at different values of the initial stress in the range between 0.33 and 0.66 of the ultimate stress, and confirmed both the assumptions about the separation of stress and time variables within the relaxation function, and the anisotropy of the relaxation properties. Guedes et al. (2006) proposed power-law expressions for the relaxation function, and
* Corresponding author. Tel.: +39 02 23994248; fax: +39 02 23994286. E-mail address: [email protected] (V. Quaglini). 0093-6413/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.mechrescom.2008.10.012
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showed that these expressions well match the exponential model proposed by Iyo et al. (2004) in the short-term process, but diverge over large times. In this study, we investigated the viscoelastic behavior of trabecular bone by means of stress relaxation tests carried out either in compression or in bending load configuration at different values of the initial stress in a bovine model. The aim was to assess the reliability of the quasi-linear viscoelastic model within a wide range of stress conditions, and to develop a suitable analytical expression for the stress relaxation function consistent with the experimental data. 2. Materials and methods 2.1. Test specimens The tests were performed on specimens of trabecular bone harvested from bovine femoral heads. Eleven donor animals (age ranging from 18 to 24 months) taken from a slaughter were included into the protocol. Each animal was identified with a label as B #X, where X = 1, 2, 3, . . . After removing of fat, muscles and other soft tissues, bovine femurs were placed in a custom vice and the heads were resected with the help of a die along a plane perpendicular to the femoral neck’s axis. Two distinct types of specimens were machined: cylindrical specimens (height 20 mm, diameter 10 mm) and beamshaped specimens (cross-section 5 5 mm, length approximately 40 mm). The specimens were extracted from the core of the femoral heads, with their long axes perpendicular to the head’s resection plane, in order to minimize the influences of non homogeneity (see, for example, Schoenfeld et al. (1974) and Vahey et al. (1987)) and anisotropy (Deligianni et al., 1994) of the tissue, so only one or two specimens were obtained from each femur, for a total of 42 specimens (21 cylindrical specimens and 21 beam specimens). The specimens were defatted from the bone marrow according to a standard procedure consisting of immersion in 1:3 solution of water and chloride, followed by three repetitions of ultrasonic cleaning in demineralized water for 5 min and subsequent air-drying for 2 min. After preparation, all specimens were stored frozen at 24 °C. Prior to testing, the specimens were thawed in physiological saline and conditioned at ambient temperature for at least 1 h. 2.2. Experimental protocol The tests were carried out on an MTS Synergie 200 H (MTS, Minneapolis, MN) testing machine equipped with a 1000 kN load cell. A custom environmental chamber comprising a thermostatic bath filled with saline solution was mounted on the testing machine, allowing control of the degree of moisture and the temperature (37 ± 1 °C) of the specimen throughout the test. Relaxation tests were performed either in compression on the cylindrical specimens or in the three-point bending configuration on the beam-shaped specimens. In order to harmonize the testing protocol and the procedure for data analysis between the two distinct load configurations, a suitable notation was introduced. Let P and M indicate the internal force and moment produced within the specimen from the external load, namely, the axial force in the compression test (P) and the bending moment at the midspan cross-section in the three-point bending test (M). Preliminary relaxation tests either in compression or in bending were carried out to assess the suitability of the storage procedure to preserve the viscoelastic properties of the trabecular bone specimens over time, and to determine the minimum period required by the bone tissue for complete recovery from the previously applied stress history. The cylindrical specimens were mechanically preconditioned by execution of 10 loading cycles between 0 and 72 N, which corresponded to about 60% of the compressive load at the limit of elasticity (PY = 1412 ± 101.6 N) previously determined. The preconditioned specimens were assayed in stress relaxation tests consisting in the application of a compressive strain at the constant rate of 0.9 mm/mm min1, until a specified preset load was attained; the strain was then held constant and the decrease in load recorded over a period of 600 s. Eleven cylindrical specimens were submitted for the execution of the stress relaxation test at five different values of the internal force (P0 = 120, 240, 360, 480 and 600 N), for a total of 55 tests. The beam-shaped specimens were tested in a three-point bending configuration, relying on two supports with a span width of 15 mm, according to a similar procedure. A set of preliminary tests was carried out on dummy specimens, and the bending moment at the midspan of the specimen was determined at the limit of elasticity (MY = 361.1 ± 53.6 N mm). Mechanical preconditioning was carried out by loading the specimen so that the midspan bending moment ranged between 0 and 180 N mm, the upper boundary corresponding again to about 60% of the yield limit. In the stress relaxation tests, the specimens were loaded at their midspan at a constant rate of 2 mm/mm min1 until a preset value of bending moment was attained; the deflection was then held constant for 600 s, and the relevant decrease in the internal moment at the midspan recorded. Thirteen beam-shaped specimens were tested each at four different values of the preset moment (M0 = 37.5, 75, 112.5 and 150 N mm) for a total of 52 tests. Throughout the experiments the specimens were kept immersed in physiological saline at 37 ± 1 °C within the environmental chamber on the testing machine. After each test, the specimen was removed from the testing bench and left to recover from the previous loading history for a period of 1 h in saline solution.
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2.3. Data analysis The response of a viscoelastic solid in a relaxation test depends on both the initial load and on the time elapsed from the beginning of the test, so it is possible to describe the history of the internal force P, (or the internal moment M as relevant), by means of an analytical function of both P0 (or M0) and t as independent variables, where P0 (M0) indicates the elastic value of the internal load (moment) applied at the beginning of the test (t = 0 s), before relaxation occurs. Under the condition P(t) 6 PY (M(t) 6 MY), where PY (MY) represents the value of the internal force (internal moment) at the limit of elasticity, we can introduce either for cylindric specimens tested in compression or for beam specimens tested in bending, a generalized stress variable S independent of the shape and size of the specimen according to the following definitions:
PðP 0 ; tÞ PY MðM0 ; tÞ in bending tests: SðSe ; tÞ ¼ MY
in compression tests: SðSe ; tÞ ¼
ð1aÞ ð1bÞ
Here Se represents the generalized elastic stress obtained by dividing the elastic response P0 by PY, or by dividing M0 by MY as relevant. The generalized relaxation function S(Se, t) can be normalized in turn over the elastic stress Se, to give the reduced relaxation function
GðSe ; tÞ ¼
SðSe ; tÞ Se
ð2Þ
which describes the fraction of the initial generalized stress that is absorbed by the specimen at time t from the initial application of the external load. Two parameters can be evaluated from the reduced relaxation function G(Se, t) to characterize the relaxation behavior:(a) the initial relaxation rate that is the slope of the reduced relaxation curve at time t = 0 (at the instant of application of the initial stress):
_ e Þ ¼ 1 dGðSe ; tÞ GðS Se dt t¼0
ð3Þ
(b) the steady relaxation value, that is the asymptotic value attained at the end of the relaxation process:
G1 ðSe Þ ¼ lim
t!1
SðSe ; tÞ Se
ð4Þ
3. Experimental results A set of preliminary tests was carried out for validating the experimental procedure. Repeated relaxation tests were carried out on the same specimens either within a few hours from calf slaughtering or after different periods of storage at 24 °C, and good reproducibility of the relevant relaxation curves were obtained for storage periods up to 960 h (an example is reported in Fig. 1a). In a similar way, we assessed that a wait period of 1 h was enough to complete viscoelastic recovery of the specimen material from the previous load history and to allow the repeatability of the relaxation response (Fig. 1b). Representative examples of relaxation curves of the trabecular bone from the present tests are reproduced in Figs. 2 and 3. Similar trends of the relaxation response were shown both in compression and in bending trials: under application of a constant deformation, either the internal force or the moment exhibited a continuous decrease from the initial value, until a steady regime value was attained within 300 s from the beginning of the test, and at 600 s the further decrease of the relaxation curve over the value measured at 300 s was never significant (p > 0.05 in each test). 3.1. Compression tests Once relaxation curves like the ones depicted in Fig. 2a, which report the decrease in the internal axial force absorbed by the specimen over time, were normalized with respect to the initial force P0, the resulting reduced relaxation curves did not match (Fig. 2b). The analysis of the relaxation behavior was carried out in terms of generalized stresses, and the influence of the initial stress was investigated by means of paired t-tests between groups of normalized curves relative to different values of Se. Since the relaxation process reached a steady regime after about 300 s, the fraction of the initial stress at t = 300 s was taken _ e Þ and the steady relaxation value G1 ðSe Þ were comparable between relaxation as G1 ðSe Þ. Both the initial relaxation rate GðS tests performed at medium or high values of Se, but the influence of the elastic variable became significant when the comparison was made between high and low values of the stress: the initial relaxation rate increased and the steady relaxation value decreased on decreasing the initial stress (Fig. 4).
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a
-1000
-900
Axial Load [N]
-800
-700
-600
-500
control 288 h at -24°C
72 h at -24°C
96 h at -24°C
960 h at -24°C
-400 0
100
200
300
time [s]
b
-800
control after 1 h recovery
Axial Load [N]
-750
-700
-650
-600 0
100
200
300
time [s] Fig. 1. Preliminary tests: (a) stress relaxation curves relevant to the same specimen tested within 6 h from calf slaughtering (control) and after storage at 24 °C for 72, 96, 288 and 960 h overlap; (b) reproducibility of the relaxation curve shown by the virgin test specimen (control) is attained after 1 h recovery in saline at +37 °C temperature.
3.2. Bending tests The relaxation curves obtained in the bending tests exhibited the same features as found in the compression tests. Once relaxation curves as shown in Fig. 3a, which report the current value of the internal bending moment at the midspan of the specimen, were normalized over the preset value of the moment, the resulting reduced relaxation curves diverge (Fig. 3b). _ e Þ and the steady relaxation value G1 ðSe Þ, evaluated again at Like in compression tests, both the initial relaxation rate GðS t = 300 s, were comparable between the relaxation tests at the highest values of the generalized elastic stress (Se = 0.38 vs. Se = 0.50), but the dependence on the elastic stress was not negligible at lower levels, producing an increase in the initial relaxation rate and a decrease in the steady relaxation value at decreasing of Se (Fig. 5). 4. Reduced relaxation function Both in compression and in bending experiments, the relaxation of trabecular bone was affected by the magnitude of the initial stress, Se. The amount of relaxation reduced in proportion to the increase in the elastic stress, showing a decrease in the initial relaxation rate and an increase in the steady relaxation value. However, the influence of the generalized elastic stress Se became less significant as the stress was increased, and it vanished at Se values larger than 0.3. A mathematical expression for the reduced relaxation function consistent with the experimental data was therefore developed. The basic assumptions upon which the model was established are:
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a
279
-800
Axial Load [N]
-600
-400
-200
0 0
100
200
300
time [s]
Normalized relaxation [-]
b
1.00
P0 = 727 N 0.95
P0 = 577 N P0 = 445 N P0 = 320 N
0.90 P0 = 168 N 0.85 0
100
200
300
time [s] Fig. 2. (a) Relaxation curves in compression tests at different values of the preset axial load. (b) After normalization over the initial value of axial load, the reduced stress relaxation curves do not match.
(a) the relaxation mechanism consists of two different processes, the first one prevailing in the short-term period, and the second one prevailing in the long-term; (b) both relaxation processes are described by exponential functions with characteristic times s1 and s2, respectively; (c) since the initial relaxation rate, which rules the short-term dynamics, is dependent on the initial value of the generalized stress, the characteristic time s1 is a function of the stress variable Se; (d) since the relaxation exhausted within 300 s, whichever was the initial stress at time t = 0, the long-term dynamics is independent of the elastic variable, and a constant value of the characteristic time s2 is taken; (e) the steady value of relaxation is a function of the elastic variable Se. So the following mathematical expression was proposed for the normalized relaxation function:
GðSe ; tÞ ¼ A0 ðSe Þ þ A1 ðSe Þ exp
t t þ A2 ðSe Þ exp s1 ðSe Þ s2
ð5Þ
where s1 and s2 represent the characteristic times of the two relaxation processes, A1 and A2 are the weights of such processes, and A0 is the asymptotic relaxation value. In order to fulfill the condition that at time t = 0 s the stress does coincide with the elastic value, a further requirement is
GðSe ; 0Þ ¼ A0 ðSe Þ þ A1 ðSe Þ þ A2 ðSe Þ ¼ 1
ð6Þ
The reduced relaxation curves at the different preset values of stress for all the specimens were curve-fitted by using Eq. (5). Since in the three-point bending test configuration the actual distribution of stresses is not uniform within the specimen, parameters A0, A1 and A2 had not a straightforward physical interpretation, so we focused our attention on the characteristic
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Bending moment [Nmm]
a
160
120
80
40
0 0
100
200
300
time [s]
Normalized moment [-]
b
1
M0 = 150 Nmm 0.9
M0 = 112.5
M0 = 75 Nmm 0.8 M0 = 37.5 Nmm
0.7 0
100
200
300
time [s] Fig. 3. (a) Relaxation curves in the three-point bending tests at different values of the bending moment at the midspan cross-section of the specimen. (b) After normalization over the preset value of bending moment, the normalized relaxation curves diverge.
times only. According to our assumption, a good fit was obtained by assuming a constant value s2 = 95 s, whichever the specimen and the elastic stress value. On the contrary, the value of s1 was affected from the preset stress, ranging between 2.18 and 4.25 s. To describe the dependence of s1 on the elastic variable Se, an exponential expression was used
s1 ðSe Þ ¼ a expðbSe Þ
ð7Þ
The values of the material parameters a and b were identified for each donor sample and are reported in Tables 1 and 2, along with the relevant correlation factor. Despite the experimental uncertainty and the variability among different specimens due to tissue non homogeneity, good correlation was found between the experimental values of s1 and the values predicted by Eq. (7), as a function of the elastic variable Se: the correlation factor ranged between 0.91 and 0.97 whichever the test condition. Ignoring sample B #6, a good agreement was found between compression and bending groups, with statistically comparable values of a and b parameters. 5. Discussion In the present study, the viscoelastic behavior of trabecular bone was investigated either in compression or in three-point bending test. In order to directly compare the behavior of the tissue in either test configurations, a generalized stress was defined as the ratio between the relaxed value of the internal force or moment and the corresponding value at the limit of elasticity. The relaxation tests were carried out at different values of the initial generalized stress Se, which represents the ‘‘elastic” stress absorbed by the specimen before relaxation occurs.
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-1
initial relaxation rate [s ]
a
0.015
0.012
0.009
0.006
(*)
0.003
(*)
(*)
(*)
(**)
(**)
(**)
0.000 0.1
0.2
0.3
0.4
0.5
b
1.00
steady relaxation [-]
generalized elastic stress Se
0.95
0.90
(*)
0.85
(*)
(*)
(*)
(***)
(***)
(***)
0.3
0.4
0.80 0.1
0.2
0.5
generalized elastic stress Se _ e Þ; (b) steady relaxation value G1 ðSe Þ determined Fig. 4. Relaxation tests in compression. Average value ± standard deviation of (a) initial relaxation rate GðS at different values of elastic stress Se on 11 specimens of trabecular bone: () different from Se = 0.1 at p < 0.01; () different from Se = 0.2 at p < 0.01; () different from Se = 0.2 at p < 0.05.
In opposition to the behavior of cortical bone tissue (Lakes and Katz, 1979b), and in contrast with the assumptions made by Deligianni et al. (1994) and Guedes et al. (2006), the results showed that the viscoelastic behavior of trabecular bone is nonlinear with respect to the elastic variable, since the magnitude of relaxation affected the initial stress: the higher the initial stress, the slower the relaxation rate and the smaller the decrease in stress at the steady regime. A possible explanation of the discrepancy between our results and the literature is that the influence of the initial load on the reduced relaxation curve becomes significant only at values lower than 30% of the limit of elasticity, while the other studies were restricted only to high values of initial load. To model the experimental reduced relaxation data, an analytical expression, Eq. (5), was proposed. This expression is borrowed from analytical relaxation models proposed for the cortical bone (Sasaki et al., 1993; Iyo et al., 2004, 2006), in that it assumes that the relaxation of trabecular bone has two distinct stages, one relevant to the short-term, and the other to the long-term dynamics. Each relaxation mechanism is modeled by an exponential function with its own characteristic time. According to the exponential model, the stress relaxation process is considered to be governed by the viscoelastic properties of the matrix collagen fibers within the bone structure (Sasaki et al, 1993). It should be noted that even though for most of the living tissue the assumption of the independence of stress relaxation on elastic variables is usually made (Fung, 1993), the influence of the initial load on the magnitude of stress relaxation was shown for some soft collagenous tissues (Garcia Sestafe et al., 1994; Johnson et al., 1996). However, according to the evidence that whichever the initial stress, the relaxation curve attained a constant asymptotic value in about 300 s, we assumed a constant value of the long-term characteristic time s2 among the whole specimens, independent of the elastic stress and the load configuration. On the contrary, the short-term time s1 was affected by the initial stress, and an exponential function was proposed on an empirical basis to model such dependence. The characteristic times of the relaxation function were comparable among the individual donors and between the two groups tested either in bending or in compression. Therefore, the nonlinear relaxation mechanism suggested in the present
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-1
initial relaxation rate [s ]
a
0.020
0.015
0.010
(*)
0.005
(*)
(*)
(**)
(***)
0.38
0.50
0.000 0.13
0.25
generalized elastic stress Se
b
1.00
steady relaxation [-]
0.95
0.90
0.85
0.80
(*)
(*)
(*)
(**)
(***)
0.38
0.50
0.75 0.13
0.25
generalized elastic stress Se _ e Þ; (b) steady relaxation value G1 ðSe Þ determined at Fig. 5. Relaxation tests in bending. Average value ± standard deviation of (a) initial relaxation rate GðS different values of elastic stress Se on 13 specimens of trabecular bone: () different from Se = 0.13 at p < 0.01; () different from Se = 0.25 at p < 0.05; () different from Se = 0.25 at p < 0.01.
Table 1 Coefficients a and b and correlation coefficient q of the exponential regression fit, Eq. (7), of characteristic time s1 in compression tests. Five relaxation tests were carried out for each specimen (N: number of specimens from each donor) and used for curve-fitting. Donor
N
a (s)
b ()
q
B B B B B
4 2 2 2 1
3.8673 4.2363 3.4303 3.3017 2.7370
0.0087 0.0084 0.0060 0.0069 0.0079
0.906 0.963 0.911 0.928 0.919
#1 #2 #3 #4 #5
Table 2 Coefficients a and b of Eq. (6) and correlation coefficient q of the exponential regression fit, Eq. (7), of characteristic time s1 in the three-point bending test. Four relaxation tests were carried out for each specimen (N: number of specimens from each donor) and used for curve-fitting. Donor
N
a (s)
b ()
q
B B B B B B
2 2 3 2 2 2
2.1639 3.0948 3.0128 2.9563 3.7412 3.5852
0.0198 0.0087 0.0111 0.0104 0.0056 0.0036
0.912 0.973 0.906 0.911 0.961 0.907
#6 #7 #8 #9 #10 #11
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study actually represents an intrinsic process of trabecular bone, which is not affected by individual tissue variability or load configuration. The present report belongs to a wider work that is still in progress aiming at delineating the viscoelastic features of trabecular bone. At present, only the dependence of the characteristic times of the relaxation function s1 and s2 on the elastic variable has been modeled. Future works will extend the investigation to the whole coefficients of Eq. (5). References Bowman, S.M., Keaveny, T.M., Gibson, L.J., Hayes, W.C., McMahon, T.A., 1994. Compressive creep behavior of bovine trabecular bone. J. Biomech. 27, 301– 310. Caler, W.E., Carter, D.R., 1989. Bone creep-fatigue damage accumulation. J. Biomech. 22, 625–635. Deligianni, D.D., Maris, A., Missirlis, Y.F., 1994. Stress relaxation behaviour of trabecular bone specimens. J. Biomech. 27, 1469–1476. Fondrk, M., Bahniuk, E., Davy, D.T., Michaels, C., 1988. Some viscoplastic characteristics of bovine and human cortical bone. J. Biomech. 21, 623–630. Fung, Y.C., 1993. Biomechanics. Mechanical Properties of Living Tissues, second ed. Springer-Verlag, New York. Garcia Sestafe, J.V., Garcia Paez, J.M., Carrera San Martin, A., Jorge-Herrera, E., Navidad, R., Candela, I., Castillo-Olivares, J.L., 1994. Description of the mathematical law that defines the relaxation of bovine pericardium subjected to stress. J. Biomed. Mater. Res. 28, 755–760. Guedes, R.M., Simões, J.A., Morais, J.L., 2006. Viscoelastic behaviour and failure of bovine cancellous bone under constant strain rate. J. Biomech. 39, 49–60. Iyo, T., Maki, Y., Sasaki, N., Nakata, M., 2004. Anisotropic viscoelastic properties of cortical bone. J. Biomech. 37, 1433–1437. Iyo, T., Sasaki, N., Maki, Y., Nakata, M., 2006. Mathematical description of stress relaxation of bovine femoral cortical bone. Biorheology 43, 117–132. Johnson, G.A., Liversay, G.A., Woo, S.L.-Y., Rajagopal, K.R., 1996. A single integral finite strain viscoelastic model of ligaments and tendons. ASME J. Biomech. Eng. 118, 221–226. Lakes, R.S., Katz, J.L., 1979a. Viscoelastic properties of wet cortical bone – II. Relaxation mechanisms. J. Biomech. 12, 679–687. Lakes, R.S., Katz, J.L., 1979b. Viscoelastic properties of wet cortical bone – III. A nonlinear constitutive equation. J. Biomech. 12, 689–698. Lakes, R.S., Katz, J.L., Sternstein, S., 1979. Viscoelastic properties of wet cortical bone – I. Torsional and biaxial studies. J. Biomech. 12, 657–678. Sasaki, N., Nakayama, Y., Yoshikawa, M., Enyo, A., 1993. Stress relaxation function of bone and bone collagen. J. Biomech. 26, 1366–1376. Schoenfeld, C.M., Lautenschlager, E.P., Meyer Jr., P.R., 1974. Mechanical properties of human cancellous bone in the femoral head. Med. Biol. Eng. 12, 313– 317. Vahey, J.W., Lewis, J.L., Vanderby, R., 1987. Elastic moduli, yield stress, and ultimate stress of cancellous bone in the canine proximal femur. J. Biomech. 20, 29–33. Yamashita, J., Furman, B.R., Rawls, H.R., Wang, X., Agrawal, C.M., 2001. The use of Dynamic mechanical analysis to assess the viscoelastic properties of human cortical bone. J. Biomed. Mater. Res. (Appl. Biomater.) 58B, 47–53.