Microstructural model of the viscoelastic behaviour of biological tissues

A model of viscoelastic material, whose elastic and dissipative potentials are build on a microscopic restructuring, is proposed. The continuum body is viewed as a collection of ‘elementary structures’ whose borders’ deformation is governed by the gradient of deformation F. Each elementary structure is constituted of a central cell floating in a viscous matrix to mimic, in a simple way, the micro-structure of biological tissues. The application of an external load leads to the restructuring of the inner geometry of the elementary structure by stretching or contracting the fibers present in cell and in matrix, and by running the flow of the viscous fluid around the central cell. Simulations are restricted to the two-dimensionnal unidirectional traction test. In this case, the inner configuration of the elementary structure is completely characterized by one size length of the cell, let say

c

1

. The law governing the time evolution of the internal variable

c

1

is determined solving an ordinary differential equation, resulting from the compensation between elastic and viscous forces within the elementary structure

f

(c

1

+

g

(c

1

,c

1

)=0), where f derives from the elastic potential, and g derives from the dissipative potential. The time dependence is transmitted to the macroscopic behaviour of material. A simulation test applying an instantaneous strain is done and the relaxation properties of the material are discussed.

Figure 1

Traction test and material microstructure.

Figure 2

Response of the microstructure to an instantaneous strain loading.

Figure 3

Response of the macro-stress to an instantaneous strain loading (relaxation phenomenon).