PSI - Issue 18

S.V. Slovikov et al. / Procedia Structural Integrity 18 (2019) 198–204 S.V. Slovikov and O.A.Staroverov / Structural Integrity Procedia 00 (2019) 000–000

200

3

2. Materials and methods The mechanical behavior of the abdominal region for simple form of a research is modelled by a thin axisymmetric elastic shell and is defined by the equation of Laplace (Tuktamyshev et al., 2013). The state of the shell can be considered momentless if the following conditions are met: the middle surface of the shell is smooth and simply connected, the load on the shell changes smoothly and continuously, the edges of the shell can move freely in the normal direction to the middle surface, the load on the edge of the shell is tangential to the middle surface. The description of the mechanical behavior of shells of this type is based on the theories of large strains of soft shells, considering the nonlinearity of the "stress-strain" dependencies of materials and not having restrictions on the strains. The essential difficulties of integrating the differential equations of the general nonlinear theory of soft shells help to bypass numerical calculation methods. The main ones are the finite-difference method and the finite element method. They are based on variational principles and allow mathematically determining stress and strain fields. Thus, using methods of continuum mechanics, replace soft tissues and meshes with an equivalent continuous medium, find the strain fields arising in the model of the abdominal region in the conditions of mesh implantation, solving differential equations implemented on the basis of the finite element method (Hernández-Gascón et al., 2011). Having diagrams of tissue strain, we can give an approximate simulation reflecting the results of using a specific surgical mesh. The main difficulty is the limited knowledge of the mechanical behavior of the recovered tissue of a specific person. Aponeurosis tissues are multilayer plates that have a pronounced orthotropy directed along the fibers. Plates placed one above the other in a diverse direction have a behavior close to isotropic, if we consider the area of the fabric described by a circle with a diameter of 5-15 cm. This conclusion can be drawn from the results of processing experimental data presented in the work by Grassel et al. (2005), where a more significant difference in the strain diagrams is observed for different ages and sexes than when selected in different directions and areas of the abdominal. Therefore, to assess the mechanical behavior of the mesh during implantation in the abdominal, it is possible to use a single dependence of the conditioned stress on the strain. When a person is at rest, intra-abdominal pressure is in the range from 1 kPa to 7 kPa, and in the process of vital activity it can increase briefly with laughter up to 9 kPa, cough 16 kPa, with extension of the body to 19 kPa (Tuktamyshev, 2016). Using finite element method for a thin-walled shell, strain fields were obtained for a model abdominal with a mesh implanted into it. In this case, nonlinear modeling was used. The behavior of the material was determined by the “conditional stress-strain” dependences. Dependencies “conditional stress-strain” obtained from experiments are set either using equation (1) or using a tabular representation of data (piecewise linear approximation with the required number of points).

Strain is defined as true (logarithmic) strain, since the values of displacements are large. The boundary conditions are set in such a way that the boundaries of the mesh zone u i m , v i

m and the zones of the

a , v

i a are equal.

aponeurosis tissues u i

, i i u u v v   a m a i i

m

,

(2)

The pressure is set uniformly over the entire surface of the simulated area. The conditional meridian (  11 ) and circumferential (  22 ) stress is set by force per unit length ( N 1 , N 2 ):

N

N

(3)

22 , 

,







1

2

11

h

h

where h is the conditional thickness of the shell. The meridian (  11 ) and circumferential (  22 ) strains are determined by the nonlinear dependencies f (  ) obtained from testing the meshes with some simplifying assumptions.