d U1

d y2

d U2

d y2

Now the effective deformation from the zero stress state to the current state has to satisfy dz = NM dy

which then gives the effective strain matrix as

where Z = NTN defines the residual strain matrix Z and relates the zero stress state to the reference state; the residual strain Z is what we want to calculate.

Small Strain Approximation

Let us now compute the expressions for small strains. Let

Z = Sij + 2zij, M = Stj - Ui,j, | zij | Ui,j |«1, (10.24)

where Sij = 1 if i = j, Sij = 0 if i = j and we use the comma notation to denote differentiation. In this case

But the infinitesimal strain, eij is related to the effective strain matrix, MT ZM by

MtZM = Sij - 2eij ^ eij = ±(Ui,j + UjA) - zij = ej - zij. (10.26)

So, in this small strain approximation situation when both residual and actual strains are small, the effective strain is equal to the actual strain, here denoted by eij, minus the effective strain, zij. We use this in our wound healing model equations below.

10.7 Matrix Secretion and Degradation

Before we can write the model equations we have to consider the addition and removal of new matrix fibres; again we use averages instead of using the various fibre densities to derive the relevant equations.

Loss of fibres affects the stress but not the residual strain if we assume that it affects all fibres in the same way. If all fibres were added in a relaxed state and secretion and degradation occurred continually, eventually the tissue matrix would become totally relaxed, a situation not observed in skin and scar tissue. So, we hypothesise that as the matrix density increases a larger and larger proportion of new collagen is added to the preexisting fibres. We assume that collagen not added to existing fibres forms new unstressed fibres.

Recalling the discussion above we can think of the effective strain as being represented by the ellipse dyTMTZM dy = 1 with its principal axes in the directions of the eigenvectors of MT ZM and of length 1 /^/Ai. The effect of adding new fibres is to relax the eigenvalues of the effective strain towards 1: the eigenvectors remain the same. We now derive the change in residual strain as we add new fibres, a crucial step in the model development.

Let the effective strain at time t be MTZ(t)M with eigenvalues Ai (t) and assume that

0 0

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