ABSTRACT
Recently, the Osteoporosis victims increase in the senior citizen. Therefore, the danger of the stress fracture due to the decrease in bone strength is pointed out. Especially, the damage accumulation behavior of the bone in the cyclic load becomes a problem for the fatigue of the bone. Moreover, it is necessary to consider the viscoelastic property for the prediction of the fatigue behavior, because the bone is a viscoelastic material. In this study, our objective is to estimate the fatigue behavior that considers the viscoelasticity behavior and the damage accumulation behavior as damage mechanics. The viscoelastic properties were revealed with the strain rate tests, and we calculated the 3-mode Maxwell parameters with these results. The damage accumulation behaviors were measured with the Acoustic Emission method. Using these results, we could estimate the fatigue behavior of cortical bone with the viscoelastic parameter and cumulative AE energy.
Key words : Cortical bone, Fatigue, Viscoelasticity, Acoustic emission, Strain rate dependency, 3-mode Maxwell model
Introduction
Recently, the Osteoporosis victims increase in the senior citizen. Therefore, the danger of the stress fracture due to the decrease in bone strength is pointed out. Especially, the damage accumulation behavior of the bone in the cyclic load becomes a problem for the fatigue of the bone. Moreover, it is necessary to consider the viscoelastic property for the prediction of the fatigue behavior, because the bone is a viscoelastic material. The bone is a specifically shaped and structured organ, which is made of bone tissue. The bone tissue, in turn, is mainly made of bone material (comprising mineral, collagen, and water) and bone cells. The cells provide the bone with the ability to respond to mechanical loading, but in the end, it is the predominant loading that determines the bone shape and macroscopic structure – not the cells. In recent years, many researchers seek to find how to give the effective mechanical stimulation to the bone at low amplitude and broad frequency strain. Flieger et al. (1998), Tanaka et al. (2003) reported the results of 0 Hz ~ 50 Hz[1, 2]. Furthermore, Hsieh et al. (1999)[3] tried to characterize the mechanical strain in the rat tiba or ulna during applied loading at frequencies of 1, 2, 5, 10 and 20 Hz. And other researchers investigated the mechanical character [4 - 11]. However, few researchers could estimate with viscoelastic parameter and damage mechanics depended on test time. In this study, our objective is to estimate the fatigue behavior that considers the viscoelasticity behavior and the damage accumulation behavior as damage mechanics.
Calculation of the viscoelastic parameters by static compression testing
On the static strain rate test, we can describe strain £(t) and stress o(t) as follows;
where, the subscript t shows the static strain rate testing and R is the strain rate. Then, these two equations were Laplace transformed, following equations were obtained.
Equations (3) and (4) were substituted for the following eq. (5).
Equation (5) showed the pseudo-elastic. Then, we got eq. (6).
When eq. (7) was substituted for eq. (6), eq. (8) was obtained.
where, the Er shows the relaxation modulus. Equation (8) was inverse Laplace transformed, then eq. (9) was obtained.
Using the strain and stress of the static compression testing, we can calculate the relaxation modulus.
To calculate the viscoelastic parameter, we use the 3-mode Maxwell model. The 3-mode Maxwell model was described as follows;
where, ke and k are elastic constants and t is the relaxation time. Then, eq. (10) was Laplace transformed and shown in eq. (11).
By the boundary conditions of relaxation testing, the equations as follows were obtained.
The Laplace transformed equations were eq. (14) and eq. (15).
Equations (14) and (15) were substituted for eq. (11), we got eq. (16)
Then, eq. (16) was inverse Laplace transformed, eq. (17) was obtained.
In this study, the 3 types of strain rate testing were carried out. We can calculate the relaxation modulus from these results by using eq. (9), and the viscoelastic parameters were calculated using eq. (17).
Experimental Procedure
Bone Specimens
Bovine cortical bone was used as a material investigated in this study. There are two types of cortical bones with different microstructure; i.e. Haversian and plexiform bones. Plexiform bone was used in this study. Figure 1 shows a microstructure of bovine plexiform bone and a schematic geometry of dyaphisis. It is observed in the microstructure perpendicular to the longitudinal axis (Fig. 1 (a)) that lamellar structure is developed along the circumferential axis. A number of lacunae were also observed among lamellae, which would contribute to fracture behavior as flaws under loading. Since lamellae are aligned along the longitudinal axis, elastic properties of bone are orthogonal. Specimens with the size of 5 mm x 5 mm x 12.5 mm were cut from the diaphysis of bovine femoral bone, which has the tubular structure as shown in Fig. 1(b). The symbols ‘x1′, ‘x2′ and ‘x3′ refer to the radial, tangential and longitudinal directions, respectively. In the following tests, compressive load was applied along x3 axis as living bodies. The specimens were kept freezing at -20°C, then cut under flowing water and kept wet just before the mechanical tests.
(a) Optical micrograph of cross section Fig. 1 Cross section of bovine bone
(b) Schematic representation of dyaphisis (Plexiform bone) and geometry of dyaphisis
Static Compression Test
Before the cyclic compression test, static compression tests of bovine cortical bone (plexiform bone) were carried out in the air at room temperature to calculate the viscoelastic parameter. Uniaxial compressive load was applied along the longitudinal (x3) axis of the bone specimens under 3 types constant crosshead speed of 1.0, 0.1 and 0.01 mm/min. until the applied load reached ~10MPa. Testing systems of static compression test and fatigue test are shown in Fig. 2, schematically. During the tests, longitudinal strain was measured using a strain gage attached directly on the specimen. To discuss the fatigue behavior after the static compression tests, the specimens must not have damages. To detect the materials’ damage, we used Acoustic Emission (AE) Systems. A wide band AE sensor (NF; AE-900M) was attached on the specimen. The surfaces of specimens were wiped before gluing the strain gage and AE sensors with cyano-acrylate adhesives and moistened immediately after the attachment. Detected AE signals were amplified by a pre-amplifier (Gain; 60 dB) through the band pass filter with a terminal of the a pre-amplifier). Amplified AE signals, load and strain were then recorded by an AE analyzer (AMSY-5; Vallen). The mechanical noise between the specimen and loading plate was restrained using Teflon sheets on the plates.
Fig. 2 Schematic diagram of static compression test system
Cyclic Compression
After the static compression test for calculating the viscoelastic parameter, cyclic compression tests were performed in the air at room temperature. The bone specimen was subjected to sinusoidal load of 3 Hz along the longitudinal (x3) axis of the bone. Stress ratio (minimum stress divided by maximum stress) was 0.05, and mean stress was 50MPa. The damage accumulation in bone under cyclic loading was monitored by AE measurement. The system was almost same as the static compression test (Fig. 2). In order to keep the specimens as wet as possible, surfaces of the specimen were moistened by the wet cotton.
Experimental Results
Static Compression Test
To calculate the viscoelastic parameter, the static compression tests were carried out with 3 types of strain rate. Figure 3 shows a typical result of stress-strain relationship for the static compression test. As shown in this figure, the stress increased with the strain rate increase. This is because the viscoelastic behavior made these change between different strain rates. Using these results, (9) and (17), we calculated the viscoelastic parameters, Ee, Ey and t, and the results are shown in table 1. During static compression tests, AE measurement was carried out to detect the mechanical damage. However, we could not detect the damage and microdamage by AE measurement. Therefore, we use same specimens for the fatigue tests.
Fig. 3 The results of static compression tests with 3 types strain rate test
Table 1 Viscoelastic parameters
|
Ee (MPa) |
7000 |
|
Ei (MPa) |
800 |
|
t |
100 |
Cyclic Compression Test
Figure 4 shows a typical result of strain-time relationship and the behavior of cumulative AE energy for the fatigue compression test. This strain shows the mean strain during fatigue test. As shown in this figure, the mean strain decreased little by little, and they vibrated from 2000 sec. to 3500 sec. It assumed that this vibration was caused by some environmental factors. In this paper, we ignored this vibration; therefore, we calculated the smooth curve of mean strain.
Figure 5 shows the smoothed mean strain. As shown in this result, smoothed mean strain decreased little by little. It was considered that this decrease had two affected factors. One is damage accumulation, and the other is viscoelastic behavior as creep deformation. In the condition of fatigue test, there is constant mean stress about 80MPa, and the stress worked as creep load. To divide the effect of two factors on fatigue tests, we estimate the creep behavior under the mean stress. The estimated data was shown in fig. 5. This curve shows only the viscoelastic behavior, that is, this curve shows the result of fatigue test without the damage accumulation. On the other hand, cumulative AE energy shows only the damage accumulation, since the AE sounds are generated by some cracking. Therefore, we tried to make the curve that shows the difference between estimated and experimental data, and it is shown in fig. 6. As shown in fig. 6, the curves, which indicate the difference data, and smoothed cumulative AE energy show good agreement without initial stage. When we use this relationship between difference and cumulative AE energy, we can estimate the mean strain during fatigue test.
Fig. 4 Mean strain and cumulative AE energy during fatigue test
Fig. 5 Smoothed mean strain and cumulative AE energy and estimated strain curve during fatigue test
Fig. 6 Difference between experimental and estimated strain data and smoothed cumulative AE energy
Conclusions
The static and cyclic compression tests of bovine cortical (plexiform) bone were carried out in this study. The viscoelastic parameters (3-mode Maxwell model and Boltzmann’s superposition principles) of bovine cortical bone were calculated with the results of the static compression tests with 3-types of strain rate. Damage accumulation during the fatigue test was monitored by AE measurement. The difference between experimental data and estimated data without damage information and smoothed cumulative AE energy were shown good agreement. Using the viscoelastic parameters and cumulative AE energy, we can estimate the mean strain during fatigue test.
