Mechanical Modeling of Aligned Connective Tissue

The stress in the elastic element is assumed to be linear and equal to the product of the elastic modulus (E) and the strain (e), and the stress in the viscous element is given by the product of the viscosity (p) and the strain rate (dddt) as given by Equation (8.3). 

If the elastic modulus is obtained from the slope of the elastic stress-strain curve, then we can evaluate the first term on the right-hand side in Equation (8.3) from experimental data elastic stress-strain curves. The second term on the right-hand side in Equation (8.3) can be evaluated from the product of the strain rate, which is set in a constant strain-rate experiment, and the viscosity. As we discussed in Chapter 3, the viscosity of a macromolecule is related to the shape factor v, therefore we can evaluate the second term on the right-hand side of Equation (8.3) from the product of the shape factor and the strain rate. 

In the simplest case the stress required to strain a viscoelastic material to a particular strain is the sum of an elastic term and a viscous term. This is somewhat more complicated for most tissues but this thought process can be used to understand the behavior of tendon after the crimp is straightened. Below we use this approach to model the behavior of tendon. 

Tendon contains dense ECM composed primarily of aligned collagen fibers. The modeling of this tissue is simpler than other ECMs because the analysis requires considering the mechanism of collagen fibril deformation, which has been studied in great depth. On a molecular basis the initial part 

From the stress value at a particular strain found on the viscous stress-strain curve (see the second term found in Equation (8.3)) we can calculate the viscosity from the stress if we know the strain rate. Because 

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