**5. Results obtained**

### **5.1. 1D simulations**

### *5.1.1. Compression loads response*

Responses from the evidence to compressive loads can be seen in Figure 7. Figure 7a shows the negative shift of the solid component that increases its value negatively with increasing loading time. The time initially is zero (0) for t = 0.67s and reaching figures of -15x10-4mmN-1s-1 at the time of maximum loading, t = 45s for x = 0.3. As the liquid flows, the behavior is similar to a linear elastic behavior, because the only component that supports the load is solid.

Figure 7b shows the decrease in pressure due to the fluid outlet presented by the compression of the tissue. For an initial time t = 0.67s the pressure exerted on the fluid in the tissue is 2 MPa, but as the permanence of the compressive, the pressure decreases and takes values close to 0 MPa, thereby external pressure is balanced, also zero.

**Figure 7.** Response of tissue to compression forces. a. Displacement of solid matrix. b. Changes in the fluid pressure in the tissue in presence of the displacement of the same.

### *5.1.2. Tensil loads response*

208 Injury and Skeletal Biomechanics

*4.3.2. Boundary conditions* 

*4.3.3. Loading conditions* 

frequency equivalent to 0.1 Hz

*5.1.1. Compression loads response* 

**5. Results obtained** 

**5.1. 1D simulations** 

Simulation was performed so as not to allow displacement at the bottom. Load was applied on the upper edge, allowing the fluid outlet only at the bottom of the tissue fragment, as shown in Figure 6a. For 2D, a condition of tissue confinement was simulated, as shown the Figure 6b, so as to present only flow at the bottom. The burden was placed at the top and lateral and bottom movement were restricted, similar conditions to those reported in several experimental studies, including that of Ateshian et al, 1997; Frijns, 2000 and Wu et al, 1999.

Calculations to simulate the applied load were performed from the data for the AC reported by Wu et al. For the 1D simulations, a load was applied of -2.033 N/m in compression, 2.033 N/m in tension and cyclic loading with frequency equal to 0.1 Hz. In 2D simulations a load was applied on the upper face of cartilage fragment corresponding to the value -4.0397e-6 N/m in compression, 4.0397e-6 N/m in tension. For cyclic loading, it was applied at a

Responses from the evidence to compressive loads can be seen in Figure 7. Figure 7a shows the negative shift of the solid component that increases its value negatively with increasing loading time. The time initially is zero (0) for t = 0.67s and reaching figures of -15x10-4mmN-1s-1 at the time of maximum loading, t = 45s for x = 0.3. As the liquid flows, the behavior is similar

Figure 7b shows the decrease in pressure due to the fluid outlet presented by the compression of the tissue. For an initial time t = 0.67s the pressure exerted on the fluid in the tissue is 2 MPa, but as the permanence of the compressive, the pressure decreases and takes

to a linear elastic behavior, because the only component that supports the load is solid.

values close to 0 MPa, thereby external pressure is balanced, also zero.

**Figure 6.** AC scheme for the confinement conditions in (a) 1D and (b) 2D.

Figure 8 summarizes the results obtained for tensile strength. Figure 8a shows the positive displacement of the solid component in response to tensile load imposed on tissue. The figure shows how the displacements are positive in presence of tension maintained over time. From an initial value of 0 mmN-1s-1 at t = 0.67s, displacement increases up to a maximum of 16x10-4 mmN-1s-1 at a maximum load time t = 45s.

**Figure 8.** Response of tissue to tensile forces. a. Displacement of solid component tissue upward. b. Change in the fluid pressure generated by the redistribution of fluid in the tissue.

Figure 8b shows the increase of fluid pressure generated by the entry thereof into the tissue due to the pressure difference between the internal and external environment (fluid suction). This pressure difference is caused by displacement of the solid component in the positive direction in response to imposed tensile load. It is shown that this progression of the increase of pressure manifests over time, creating a redistribution of the fluid in the tissue.

### 210 Injury and Skeletal Biomechanics

### *5.1.3. Oscillating loads response*

Responses obtained in the simulation of oscillating charges can be seen in Figure 9. Figure 9a shows the alternating displacement presented at each oscillation.

Mechanical Behavior of Articular Cartilage 211

**5.2. 2D simulations** 

*5.2.1. Compression loads response* 

Figure 10a corresponds to the displacement of the solid in the *y* axis for each time instant. The displacements are small in the first moments of the load and it can be seen that as the load increases, the time of displacement increases toward more negative values, demonstrating a greater deformation of the solid phase of the tissue. These displacements are produced by the fluid outlet in response to the maintained compression load and it is observed that the greatest displacement occurs in the upper layers of the tissue responsible for receiving the load

**Figure 10.** 2D - Response to compressive load. a. *y* displacements during 45 seconds. b. Behavior of the

Figure 10b shows the behavior of the fluid pressure at each instant of time. It is observed that the pressure decreases rapidly in all tissue layers reaching values close to 0 MPa in a very short time. This variation of fluid pressure corresponds to the decrease of the same in function of the load exerted over the time and poro-elastic tissue behavior which allows the

Results obtained from the tensile simulation are shown in Figure 11. Figure 11a shows the displacement of the solid in the *y* axis for different time instants. There are major shifts in the early stages of loading (t = 0.67s) and displacement decreases with increasing load time. It is further noted that with a sustained tensile load at a given time, the greater displacement or elongation occurs in the upper layers of the tissue where the stress is felt in the first instant, which is why the transfer of deformation is smaller at a greater depth of tissue. In this case the fluid pressures tend to increase because the tissue seeks to balance the inside and the outside

fluid pressure *p* due to its outflow in presence of the load.

fluid outlet.

*5.2.2. Tensil loads response* 

directly, while the displacement transference is less at a greater tissue depth.

**Figure 9.** Response to cyclic load. a. Alternate displacement of the solid component. b. Changes in the fluid pressure in oscillate form. c. Delay between the solid deformation and fluid pressure variation.

Initially the movements of the solid are small, close to 0 mmN-1s-1 for t = 0.67s. However, they increase over time reaching a value of 6x10-4 mmN-1s-1 or -6x10-4mmN -1s-1, at t = 45s, according to the tissue load, tension or compression respectively. Figure 9b shows the development of the process of oscillation in the fluid pressures in response to movement that the solid matrix undergoes upon perceiving the cyclic loading. Initially the pressure lowers and then makes an adjustment that increases, being in inverse phase with the evolution of the deformation of the solid phase of the tissue. I.e., once the solid is deformed in the positive direction, the pressure changes, becoming more negative and vice versa.

The obvious alternating deformation processes of the solid in the face of the application of cyclic loads shows that the displacements are caused by the loads exerted on the tissue and the mobilization of fluid from or into the interior as applied tension or compressive loads respectively. These loads, in turn, generate alternation with respect to each instant of time between the variation of the pressure pattern and the variation of the displacement pattern. However, this action is not in complete phase with displacement. One can appreciate the presence and delay of alternating processes described above, because the equation that represents the displacements is elliptical and corresponds to a displacement equation in space while the equation that represents the pressure corresponds to a parabolic equation and represents a much slower diffusion process than the process of displacement (See Fig. 9c).

### **5.2. 2D simulations**

210 Injury and Skeletal Biomechanics

9c).

*5.1.3. Oscillating loads response* 

Responses obtained in the simulation of oscillating charges can be seen in Figure 9. Figure

**Figure 9.** Response to cyclic load. a. Alternate displacement of the solid component. b. Changes in the fluid pressure in oscillate form. c. Delay between the solid deformation and fluid pressure variation.

Initially the movements of the solid are small, close to 0 mmN-1s-1 for t = 0.67s. However, they increase over time reaching a value of 6x10-4 mmN-1s-1 or -6x10-4mmN -1s-1, at t = 45s, according to the tissue load, tension or compression respectively. Figure 9b shows the development of the process of oscillation in the fluid pressures in response to movement that the solid matrix undergoes upon perceiving the cyclic loading. Initially the pressure lowers and then makes an adjustment that increases, being in inverse phase with the evolution of the deformation of the solid phase of the tissue. I.e., once the solid is deformed in the positive direction, the pressure changes, becoming more negative and vice versa.

The obvious alternating deformation processes of the solid in the face of the application of cyclic loads shows that the displacements are caused by the loads exerted on the tissue and the mobilization of fluid from or into the interior as applied tension or compressive loads respectively. These loads, in turn, generate alternation with respect to each instant of time between the variation of the pressure pattern and the variation of the displacement pattern. However, this action is not in complete phase with displacement. One can appreciate the presence and delay of alternating processes described above, because the equation that represents the displacements is elliptical and corresponds to a displacement equation in space while the equation that represents the pressure corresponds to a parabolic equation and represents a much slower diffusion process than the process of displacement (See Fig.

9a shows the alternating displacement presented at each oscillation.

### *5.2.1. Compression loads response*

Figure 10a corresponds to the displacement of the solid in the *y* axis for each time instant. The displacements are small in the first moments of the load and it can be seen that as the load increases, the time of displacement increases toward more negative values, demonstrating a greater deformation of the solid phase of the tissue. These displacements are produced by the fluid outlet in response to the maintained compression load and it is observed that the greatest displacement occurs in the upper layers of the tissue responsible for receiving the load directly, while the displacement transference is less at a greater tissue depth.

**Figure 10.** 2D - Response to compressive load. a. *y* displacements during 45 seconds. b. Behavior of the fluid pressure *p* due to its outflow in presence of the load.

Figure 10b shows the behavior of the fluid pressure at each instant of time. It is observed that the pressure decreases rapidly in all tissue layers reaching values close to 0 MPa in a very short time. This variation of fluid pressure corresponds to the decrease of the same in function of the load exerted over the time and poro-elastic tissue behavior which allows the fluid outlet.

### *5.2.2. Tensil loads response*

Results obtained from the tensile simulation are shown in Figure 11. Figure 11a shows the displacement of the solid in the *y* axis for different time instants. There are major shifts in the early stages of loading (t = 0.67s) and displacement decreases with increasing load time. It is further noted that with a sustained tensile load at a given time, the greater displacement or elongation occurs in the upper layers of the tissue where the stress is felt in the first instant, which is why the transfer of deformation is smaller at a greater depth of tissue. In this case the fluid pressures tend to increase because the tissue seeks to balance the inside and the outside environment. Then displacement occurs in response to stress and produces a reorganization of the fluid within the tissue, interfering with the variation in the pressure thereof.

Mechanical Behavior of Articular Cartilage 213

response to the compression of the tissue. This is due to the rapidity with which compressive and tensile loads are alternated, loads which prevent the behavior of cartilage during the cyclic loading from exhibiting the behavior that corresponds exclusively to the

**Figure 12.** 2D - Response to oscillate load. a. *y* displacements. b. Oscillation of the pressure *p* within the

There are several theories explaining the behavior of AC in the presence of load conditions, summarized in computational models that include the swelling process and the properties of the anisotropic structure of collagen. The most frequently used tests to determine the mechanical qualities of the AC are the confined compression, the unconfined compression, the indentation and the swelling tests (Wilson, et al., 2005), carried out using numerical

For purposes of meeting the stated objectives, we simulated a condition of confinement of the tissue that allows the flow at the bottom to restrict lateral and bottom movement. Conditions were similar to those reported in practical experiments as the papers presented by Ateshian et al., 1997; Frijns, 2000 and Wu et al., 1999; among others. The data obtained from the simulations confirm the theory of biphasic articular cartilage, first proposed by Mow et al., 1980; and supported by several authors as Haider et al., 2006; Wilson et al., 2005, Haider & Guilak , 2007; Meng et al., 2002; Wu et al., 1997, Terada et al., 1998, Donzelli et al.,

Results allow us to conclude that articular cartilage exhibits a displacement response of the solid component (matrix) and a variation in the pressure of fluid component due to the exit or entrance thereof, with decreases in pressure in response to compressive loads and increases at the same tensile loads. The displacement is caused by outflow of fluid in response to the maintained compressive load. However it is important to note that once the tissue reaches its maximum displacement, it behaves as a solid rather than as a poro-elastic material. From this point the fluid can't flow out of the tissue because the pressure is

case of tension or compression.

tissue according to the load.

**6. Final discussion** 

approximation tools.

1999 and Donzelli & Spilker, 1998; among others.

balanced with the external fluid begins to bear part of the load.

**Figure 11.** 2D-tensile load response. a. *y* displacements. b. . Behavior of the fluid pressure *p* due to its inflow in presence of the load.

Figure 11b shows the behavior of the fluid pressure *p* for different time instants. It is observed that the fluid pressure is increased by small amounts in response to sustained tensile load. This action is due to the compensation of the tissue in response to deformation in elongation by the solid phase of the tissue that requires a redistribution of fluid into the tissue. However, it is evident that at the end of the tensile load time (approximately at t = 27s), the tissue can't undergo greater deformation in elongation and hence the fluid pressure also tends to stabilize at the inside thereof, causing a steady pressure maintained close to zero which balances with the external pressure.

### *5.2.3. Oscillating loads response*

Figure 12a corresponds to the displacement of the solid in the *y* axis at different time instants. It is noted that the displacements or deformations occur alternately; at the times 1, 3 and 5 (t = 0.67s, t = 27s = 6.57s respectively) the displacements are made positive, behavior similar to that observed during exposure to stress loads. Conversely, at the times 2, 4 and 6 (t = 3.37s, t = 16.5 s and t = 45s respectively) the displacements are negative, consistent with the behavior exhibited by the matrix to compressive loads.

Figure 12b shows the behavior of the fluid pressure *p*. Similar to what happened with the deformations of the solid phase, the pressure oscillation in response to cyclic loading imposed on the tissue is evident. Note that at times 1, 3 and 5 (t = 0.67s, t = 6.57syt = 27s respectively) the pressure tends to decrease due to the redistribution of fluid in response to perceived stress loads. Because the loads are not maintained, the tissue can't compensate with fluid inlet from the outside which is why the pressure is not increased. Thus, at times 2, 4 and 6 (t = 3.37s, t = 16.5 s and t = 45s respectively), the pressure tends to increase in response to the compression of the tissue. This is due to the rapidity with which compressive and tensile loads are alternated, loads which prevent the behavior of cartilage during the cyclic loading from exhibiting the behavior that corresponds exclusively to the case of tension or compression.

**Figure 12.** 2D - Response to oscillate load. a. *y* displacements. b. Oscillation of the pressure *p* within the tissue according to the load.
