318
A. Neam¸ tu Popescu et al.
Creep diagrams can be simulated using the Simulink model in MATLAB, and the
constitutive equation is
c 1 c 2
k 2
¨
d + c 1 ˙
d = F 0 , t < t 1 .
(8)
If the force F 0 is removed at time t 1 the constitutive equation of the recovery
behavior of the Burgers model can be obtained from (8)
c 1 c 2
k 2
¨
d + c 1 ˙
d = 0, t > t 1
(9)
3 Numerical Analysis of Creep and Creep Recovery
The simulation of spinal ligament behavior and the acquisition of creep and creep
recovery after traction are performed using the numerical analysis software given
by the SIMULINK module in MATLAB. Two cases are analyzed. In the first case
the constant force is applied in a time t 1 interval, after which the traction force is
removed for a relaxation phase for a time t 2 higher than t 1 time. The total creep time
and creep recovery time is 40 s. In the second case the traction force is applied for a
short time t 1 , after which for a shorter time t the traction force is removed. Then this
cycle is restarted. Intermittent drive of the traction force is obtained, and a certain
number of creep and creep recovery cycles are obtained.
The scheme for simulating creep and creep recovery behavior of the Burgers
model is done by following (8) and (9) and is shown in Fig. 2.
The elastic behavior of the ligaments causes them to return to their initial length
after the force is removed. Therefore, in the simulation, it was intended to observe
how the two elements that characterize the damping influence on the remanent length
of the ligament after the removal of the force. For this purpose, four curves were
obtained with the applied force, the elasticity and the damping constants were the
input elements and the elongation of the ligament was the output.
In the paper [9] the biomechanical properties of six types of human lumbar
spine ligaments were determined. Based on these results, the following data were
considered for simulations:
• for curve 1 (cyan line): F = 10 N, k 1 = 1 N/mm, c 1 = 20 Ns/mm, k 2 = 10 N/mm,
c 2 = 40 Ns/mm;
• for curve 2 (red line): F = 10 N, k 1 = 10 N/mm, c 1 = 30 Ns/mm, k 2 = 10 N/mm,
c 2 = 30 Ns/mm;
• for curve 3 (green line): F = 10 N, k 1 = 10 N/mm, c 1 = 35 Ns/mm, k 2 = 10 N/mm,
c 2 = 45 Ns/mm;
• for curve 4 (blue line) F = 10 N, k 1 = 10 N/mm, c 1 = 50 Ns/mm, k 2 = 10 N/mm,
c 2 = 50 Ns/mm.
A. Neam¸ tu Popescu et al.
Creep diagrams can be simulated using the Simulink model in MATLAB, and the
constitutive equation is
c 1 c 2
k 2
¨
d + c 1 ˙
d = F 0 , t < t 1 .
(8)
If the force F 0 is removed at time t 1 the constitutive equation of the recovery
behavior of the Burgers model can be obtained from (8)
c 1 c 2
k 2
¨
d + c 1 ˙
d = 0, t > t 1
(9)
3 Numerical Analysis of Creep and Creep Recovery
The simulation of spinal ligament behavior and the acquisition of creep and creep
recovery after traction are performed using the numerical analysis software given
by the SIMULINK module in MATLAB. Two cases are analyzed. In the first case
the constant force is applied in a time t 1 interval, after which the traction force is
removed for a relaxation phase for a time t 2 higher than t 1 time. The total creep time
and creep recovery time is 40 s. In the second case the traction force is applied for a
short time t 1 , after which for a shorter time t the traction force is removed. Then this
cycle is restarted. Intermittent drive of the traction force is obtained, and a certain
number of creep and creep recovery cycles are obtained.
The scheme for simulating creep and creep recovery behavior of the Burgers
model is done by following (8) and (9) and is shown in Fig. 2.
The elastic behavior of the ligaments causes them to return to their initial length
after the force is removed. Therefore, in the simulation, it was intended to observe
how the two elements that characterize the damping influence on the remanent length
of the ligament after the removal of the force. For this purpose, four curves were
obtained with the applied force, the elasticity and the damping constants were the
input elements and the elongation of the ligament was the output.
In the paper [9] the biomechanical properties of six types of human lumbar
spine ligaments were determined. Based on these results, the following data were
considered for simulations:
• for curve 1 (cyan line): F = 10 N, k 1 = 1 N/mm, c 1 = 20 Ns/mm, k 2 = 10 N/mm,
c 2 = 40 Ns/mm;
• for curve 2 (red line): F = 10 N, k 1 = 10 N/mm, c 1 = 30 Ns/mm, k 2 = 10 N/mm,
c 2 = 30 Ns/mm;
• for curve 3 (green line): F = 10 N, k 1 = 10 N/mm, c 1 = 35 Ns/mm, k 2 = 10 N/mm,
c 2 = 45 Ns/mm;
• for curve 4 (blue line) F = 10 N, k 1 = 10 N/mm, c 1 = 50 Ns/mm, k 2 = 10 N/mm,
c 2 = 50 Ns/mm.