Keywords: biological tissue, micro-contacts and interface, thermo-mechanics.

Generally cells and tissues have low strength at temperatures near 0°C because the cooling to the freezing temperature makes irreversible damage. This is developed in the still modes so principally: i) ice crystal mechanical action, ii) modified ph in cellular liquids, iii) solute concentration growth and others. When the volume increase, follows the low-strength lines, the ice formed cannot be spherical but has a seed form. From the mechanics point of view the physics of the problem leads to arrangements in the frame of a non-simple contact mechanics because here we have a free-boundary (ice-line) against a soft biological boundary. Thus, when a soft solid is squeezed rapidly against another solid, with no-smooth surface, a large amount of elastic energy may, initially, be stored in the local (asperity induced) deformation field at the interface. We develop in this paper by means of two steps namely, the first one concerning the moving freezing line upto the contact and the second one the contact effects. Biological metabolism in living cells dramatically diminishes at low temperatures and this is one fact that permits the long-term preservation of living cells for either scientific research or many medical applications. The rapid and intense freezing of tissue, as done in cryosurgical procedures and cryopreservation, produces a localized sharply demarcated wound. As the temperature falls, cell metabolism progressively fails, and if continued sufficiently long, the cell is so adversely affected by hypothermia that death may result even though the cell was not subjected to freezing temperatures. Again, at the freezing state water is crystallized, and this has more serious consequences than the earlier cooling. As the process continues, ice crystal grow, cells shrink and membranes and cells constituents are damaged. The simplest approach, to describe bio-heat transfer, is to consider living tissue in terms of an effective homogeneous continuum where thermal interaction between cellular tissue and blood is treated as a distributed heat sink. Within the framework of this problem the bio-heat equation assumes a particular form and propagation of the freezing front is based in terms of the free boundary problem of the Stefan type.

Then our problem becomes the determination of the law of the propagation of the freezing front of the liquid. To do this, we consider operating the 0°C temperature on the plane x=0, while inside the cell the temperature remains as the environment. Mechanically speaking we consider an elasticity plane problem, as well as a strip case. It is explained how to determine in the plane region the stress field satisfying the equilibrium equations when body forces can be negligible. We approach the problem with Airy functions and resolved by applying the Fourier transform.

Successively a coupled problem has been considered because the interaction among the freezing front and cell membrane is given. We distinguish two particular cases namely, the smooth diffuse contact one, and the sharp contact one. In other words the key-concept of our problem may be carried-out as well as the Stefan's problem through proper specifications. We modelled the coupled problems in the elastic half-space framework, considering a load function derived by the contact between the cell membrane and external surface of the freezing front. Initially we solve over the general framework by means of a Hankel transforms and successively we have the results in two ways, namely sharp and smooth contacts. For these, we establish the penetration displacements and the resulting stresses.

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