Numerical modelling of soft tissues requires gathering of experimental data of biomechanical properties of these bodies. In conducting studies on the properties of soft tissue, on large samples of research material (pork liver), Hu and Desai (2005) assumed that the tissue is a material that is incompressible, homogenous and isotropic.
Assuming the load force of the cubic sample as F, the elongation coefficient as X and the initial contact surface on the cube Ao, stresses a according to Cauchy were written in the form:
F
– = k, (8.104)
Ao
while strains e in the form:
s = lnQ. (8.105)
By dividing the experimental load-strain curve (Fig. 8.20) into small subregions, the authors noted that the dependence of force on movement is linear. For each small subregion, Si corresponds to each Fi value; therefore, ei corresponds to each ai value. Based on the assumption of linear courses of the curve in subregions, it has been assumed that the LEM (Local Elastic Modulus) is expressed by the following equation:
Ei = -—— fori = 1, 2, 3 (8.106)
si ei-1
For large samples subjected to compression, it was assumed that there were no stresses on the front and side walls of the sample. Piola-Kirchhoff’s stresses tensor is connected with the stresses tensor in the chosen point by the formula:
P = SF-T. (8.107)
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Fig. 8.20 Experimental curves of the stiffness of soft tissue (Hu and Desai 2005) |
In this way, Piola-Kirchhoff’s stresses tensor has been written in the form:
P = – pF—T + 2m1F – 2m2B—1F—T, (8.108)
where
p Lagrange operator corresponding to limitations of incompressions,
Det(F) =1, B = FFT—Cauchy-Green tensor,
where IB = trace (B) and IIB = (trace(B)2 – trace(B)2)/2 are the main constants with respect to B.
For incompressible and isotropic materials, the energy of elastic strains U is a function of the main constants with respect to B, hence U = U(IB, IIB). Based on this, Piola-Kirchhoff’s stresses tensor P fulfils the equation:
Div (P) = 0.
The equations
P = —pF-T + 2x1F – 2m2B—1F—T and Div(P) = 0, (8.111)
show that the constitutive equation for stresses can be determined when the function of energy strains is known.
To describe the properties of soft tissue, Hu and Desai (2005) used the models of Ogden and Mooney-Rivlin discussed earlier. On this basis, in the system ABAQUS, a numerical model was built by preparing adequate 2D meshes in a flat state of stresses and a flat state of strains. The bone of tissue was modelled as an elastic material consisting of N elements. It was also assumed that the coefficient of friction between the sample and the plate of pressurer will be equal to zero. For modelling, separate flat state of strains and flat state of stresses was used. A four-node CPS4-type element was used in a flat state of stresses, and a four-node CPE4-type element was used in the flat state of strains. The movements of the sample were used as input data, while the reaction forces were used as a comparison with the results of experimental studies. The total strain of the sample (more than 25 % of the initial amount) was divided into 30 subregions (each subregion demonstrated 1 % of elongation increase). For each subregion j (j = 1, 2…) of the force – movement curve, the current fragment of the strain was imported in order to calculate the next linear region. Data import was conducted with the use of the standard functionality of the system ABAQUS. Also the value of Poisson’s coefficient equal to 0.3 was assumed, as well as the initial LEM value equal to E1j, in order to begin simulation. Then, the experimentally measured values of displacements A^exp were applied to the nodes of the numerical model. For calculations,
the finite elements method was used in order to determine the reaction AFfem. The results of numerical calculations AFfem were compared with the results of experimental studies AFexp. The value of the linear elasticity modulus was updated based on the equation:
Df exp
E+1j = Ej DFFEM for U = 1.2. 3— (8:112)
until AFfem in the new iteration was not similar to the experimentally measured value AFexp. The similarity evaluation criterion was the similarity coefficient expressed by the equation:
The corresponding value of the local elasticity modulus has been recorded as Elem. The first index “i” describing the linear elasticity modulus E^ is the number of the iteration in each successive step in the subregion, in which LEM was calculated. The second index “j” determines the number of the subregion. The geometry of deformations and mesh for the subregion (j) were imported to the subregion model (j + 1), and then the calculation process was run from the beginning until the entire process of iteration was completed. The procedure of iterations has been shown in Fig. 8.21.
On the basis of the analysis, it was demonstrated that both in the flat state of stresses and in the flat state of strains, identical results were obtained. Therefore, this did not significantly affect the quality of the results of numerical calculations. Slight differences in elem values were also obtained by comparing the numerical method with the experimental method. For the obtained dependencies elem = f (AA), approximation was conducted using polynomials of the 4th degree. Table 8.4 summarises the calculated values of material parameters of Mooney-Rivlin and Ogden’s models. It turns out that Ogden’s model describes the characteristics of soft tissue much better than Mooney-Rivlin’s model. However, this rule is correct only for the quasi-static analysis.
When using furniture, with which the user comes into contact directly, pressure forces, caused by maintaining his position, cannot cause limitations in blood circulation at the surface. By averaging the pressure in the arterial system, we obtain a value of around 100 mmHg (13.32 kPa), in the capillaries around 25 mmHg (3.33 kPa), and in the final part of the venous system, it amounts on average to around 10 mmHg (1.33 kPa) (Guzik 2001). According to Krutul (2004) in places where the bones push onto tissues harder, pressure increases and the lumen of blood vessels is reduced, which leads to the damage of skin tissue. The stresses of the external surface are 3-5 times smaller than internal stresses that occur as a result. Therefore, the limit value of the pressure amounting to 32 mmHg (4.26 kPa) (closing the lumen of capillaries) must be appropriately reduced and range from
Fig. 8.21 The method of calculating the local elastic modulus (LEM) of soft tissue according to Hu and Desai (2005)
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Table 8.4 Material parameters of different tissues in Mooney-Rivlin and Ogden’s model according to Hu and Desai (2005)
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6.4 mmHg (0.85 kPa) to 10.6 mmHg (1.41 kPa). Each pressure greater than these values may result in closing the light of veins, then of arteries, which slows down the flow of blood or stops its circulation, causing local ischaemia. If being in a seated position lasts a long time, it may be the cause of so-called pins and needles of
the lower limbs. Another negative effect of the improper choice of seat can be the compression of the spine muscle and gluteal muscle by the sacral vertebra and sciatica, which results in an increase of stiffness of the muscles where the bone meets the muscle. The studies of Gefen et al. (2005) demonstrated that the stiffening of muscle tissue occurs in a living system of muscles exposed to pressure of 35 kPa for 35 min or longer, and in the same muscles that have been exposed to pressure of 70 kPa for 15 min or longer. By simulating using numerical methods immobility in
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Table 8.5 Elastic properties of human body soft tissues
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a seated position on a hard base, it was found that after four hours of maintaining this position, it causes compressive stresses in 50-60 % of the cross section of the muscle at a level of 35 kPa or more. It was also demonstrated that the intensity of injury damaging cells increases during the first 30 min of immobile sitting, which is the cause of the formation of pressure pain.
Therefore, with the numerical simulation of the effect of a base on the human body, it is important to define the mechanical parameters of the human body’s tissue. These values, based on the literature, are provided in Table 8.5.
