ИСТИНА

To simulate the mechanical interaction of the needle and brain tissue, it is proposed to idealize the geometric shapes of the skull and brain tissues by representing bone tissue as a hyperelastic hollow ball, the damping layer as a spherical layer of non-Newtonian fluid, and brain tissue as a viscoelastic material filling the spherical region. It is assumed that there are residual stresses in the material of the bone tissue resulting from its growth. The needle is modeled as an elastic hollow cylindrical flexible rod, partially embedded into a viscoelastic material that simulates brain tissue. The controlled force and moment are applied at the end of the rod. There is no direct contact of the needle with the hyperelastic hollow ball representing the bone tissue (it is assumed that the diameter of the previously prepared hole for the needle is large enough), however, bone tissue interacts with needle through a viscoelastic material filling the internal volume. The insertion of the needle is modeled as sliding with friction along a channel whose walls compress the needle. The compression force varies along the axis of the embedded part of the needle and changes in time. The magnitude of the compression forces is determined from the solution of the initial-boundary-value problem. The compression stiffness of the rod is assumed to be infinite, i.e. its deformation is reduced only to bending. Along the axis, the rod moves like an absolutely rigid body. The interaction of a viscoelastic material and a needle is modeled in the linear Winkler approximation as a dynamic system “beam-viscoelastic base” with a time-variable interaction zone length. The compression forces determine the force of resistance to penetration. Two statements of the penetration problem are considered. In the first statement the penetration law is considered to be given. From the solution of the dynamic bending problem, the compression forces of the embedded part can be determined. This allows one to find the distribution of the tangential forces on its surface and, accordingly, the resulting resistance to penetration. The longitudinal force (variable in time) is determined from equilibrium equations with account of given penetration law. In the second statement the longitudinal force is considered to be given. The law of penetration is determined from solving the inverse problem iteratively.