PSI - Issue 35

Galina Eremina et al. / Procedia Structural Integrity 35 (2022) 115–123

118

4

Galina Eremina et al.,/ Structural Integrity Procedia 00 (2021) 000–000

macroscale (Eremina et al., 2019). The material of the endplate is of great importance for its unique combinations of high porosity and low permeability. Cancellous and cartilage tissues have different elastic characteristics at different scales (Wolfram et al., 2010). Thus, experiments on compression (macroscale) show the value of the elastic modulus for cancellous tissue about 100 MPa (Garo et al., 2009, Ogurkowska et al., 2020). With micro and nano-indentation (the penetration depth of the indenter corresponds to the size of the matrix cells), the obtained values for the elastic modulus reaches 10– 15 GPa for spongy tissue (Haj-Ali et al., 2017) and 7–12 GPa for cartilaginous tissue (Dall'Ara et al., 2013). The annulus has a low modulus of elasticity and ultimate strength (Shan et al., 2019). When modeling the mechanical behavior of these tissues using the poroelastic model (Argobi et al., 1996, Malandrino et al., 2009), the limiting values of the elastic modulus for solid matrix (grains) can be taken from the data presented in the above-mentioned literature.

Table 1. Poroelastic parameters of lumber spinal tissues.

ȡ , kg/m 3

k , m 2 3e-19 3e-19 7e-18 1e-16 1e-19

Type of tissues

Ks, GPa

E , MPa

, MPa

Ȟ

ș

ı 4 –

Annulus Nucleus

1060 1060 1000 1850

10 10 12 15 17

2.5 1.5

0.2 0.3

0.8 0.8 0.8

Endplate (cartilage)

10

0.46

10

Cortical

10000

0.3 0.2

0.04

100

Cancellous

700

100

0.7

10

2.2.2. Loading and boundary conditions In the case of uniaxial tension/compression, the mechanical load was applied by setting the same velocity in the vertical direction to the upper layer of the automata while fixing the automata of the lower layer of the sample. At the initial stage, the velocity of the automata of the upper layer gradually increased according to a sinusoidal law from 0 to í v m/s, and then remained constant. During indentation, the movement of the indenter was simulated by setting the velocity in the vertical direction to the upper layer of the indenter. To simulate the increasing force acting on the indenter along the Z-axis in the experiment, the velocity of the automata of the upper layer gradually increased according to a sinusoidal law from 0 to í v m/s and then remained constant until the indenter was immersed at a given penetration depth. Loading velocity was set equal to zero for some time to simulate the hold effect. When unloading, the indenter velocity increased gradually from 0 to + v m/s and then kept on moving upwards at the same velocity. This scheme was used to eliminate artificial dynamic effects and to ensure a smooth and fast transition of the sample deformation process to a quasi-stationary mode. The value of v was chosen depending on experimental data from the literature (see subsection 2.3.2). For the automata of the upper and lower layers of the sample, horizontal displacements were allowed, and the lateral surfaces of the sample were free. 2.3. Results of simulation 2.3.1. Verification The models were verified by convergence analysis of the effective elastic modulus (slope of stress-strain curve) and strength (maximum of stress-strain curve). The convergence analysis was carried out by changing the specified discretization of the samples. The value of the output parameter of the model samples with an increase in the number of automata ( n) at constant size of the sample ( l ) should approach a limit value. The compressive/tensile moduli and strengths of all the samples were determined as slopes and maxima of the loading curves, correspondingly. The relative deviations of the obtained ultimate strength ı i for all specimens from the specified value ı 0 are shown in Fig. 2, a. The relative deviations of the calculated elastic moduli E i for all samples from Young’s modulus E 0 are shown in Fig. 2, b. The analysis result showed that samples with a base size of at least 30 Â n can be considered as the representative ones (deviation of elastic modulus did not exceed 5 %,