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Numerical Modelling of Fibre Reorientation in Soft Tissue

This contribution deals with the development of a hyperelastic and thermodynamically consistent model for soft tissue that is able to describe the reorientation of collagen fibres during a deformation process. To be specific, we are examining only the continuum scale of reorientation of the collagen fibres under constant mass. A uniform distribution of fibres in the tissue matrix is assumed. We assume a uniform distribution of fibrils (fibres as well) in the matrix. Their dimension is measured in  m, therefore at continuum level we consider only the orientation at continuum points. We are taking into consideration two independent groups of the collagen fibres. The framework of the formulation is inspired by a material model developed in [1]. In the formulation of the anisotropic behaviour of soft tissue we use a popular form for the strain energy function that can be found e.g. in [2, 3], among others. The anisotropic response is described by isotropic expressions modelling the matrix and two groups of the fibres separately. Thus, the strain energy function is decomposed into parts which are related to the matrix, and to the collagen fibres. The collagen architectures are assumed to tend to align with the stress field within the tissue. The initial fibres orientations are defined by structural tensors that are reoriented according to the loading by means of the rotation tensor. The change of the rotation tensor is described using a well known exponential map procedure. The reorientation function is introduced which is assumed to be a linear function of the thermodynamical force derived from the dissipation inequality given by the second law of thermodynamics. However, a more accurate reorientation function should be proposed in the future work which should be verified with appropriate experimental results. Efficiency of the proposed formulation is demonstrated using a uniaxially loaded soft tissue strip, Figure 1. The strip is loaded in the y direction on the upper edge, while at the lower edge it has suppressed displacements in the y-direction. The collagen fibres have initial position of 15° with respect to the y axis. Two calculations are performed, with and without reorientation mechanism, Figure 1 a) and b). As well, the change of the fibre orientation by applying the load is shown by figure 2. In future work the function is to be verified with experimental results as to its appropriate form.

soft tissue; collagen fibres; reorientation; large strain

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