Soft tissue follows an exponential deformation behavior in uniaxial tension while it is in the physiologic or

Question:

Soft tissue follows an exponential deformation behavior in uniaxial tension while it is in the physiologic or normal range elongation. This can be expressed as

Where σ = stress, ε = strain, and E₀and α are material constants that are determined experimentally. To evaluate the two material constants, the above equation is differentiated with respect to ε. Using the above equation establishes the fundamental relationship for soft tissue

dσ/dε = E₀ +ασ

To evaluate E₀ and α, stress-strain data is plotted as dσ/dε versus σ and the intercept and slope of this plot are the two material constants, respectively,

In the following table is stress-strain data for heart chordate tecndineae (small tendons used to hold heart valves closed during contraction of the heart muscle; this data is from loading the tissue, while different curves are produced on unloading).

Calculate the derivative dσ/dε using finite differences. Plot the data and eliminate the data points near the zero points that appear not to follow the straight-line relationship. The error in this data comes from the inability of the instrumentation to read the small values in this region. Perform a regression analysis of the remaining data points to determine the values of E₀ and α.

Plot the stress versus strain data points along with the analytic curve expressed by the first equation. This will indicate how well the analytic curve matches the data.

Many times this does not work well because the value of E₀ is difficult to evaluate using this technique. To solve this problem E₀ is not used. A data point is selected (σ, ε) that is in the middle of the regression analysis range. These values are substituted into the first equation and a value for E₀/α is determined and substituted into the first equation, which becomes

Using this approach, experimental data that is well defined will produce a good match of the data points and the analytic curve. Use this new relationship and again plot the stress versus strain data points and this new analytic curve.