Problem Set #3
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Problem Set #3
Due October 20
Problem 1
We showed in class that we can approximate the behavior of biologic materials by combining an infinite number of three parameter models. This concept was used to develop the final formulation for the Quasilinear Viscoelastic Model. Consider N Kelvin elements in series with a
spring:
For the previous problem set, you derived the relation for one Kelvin element in series with a spring.
(a) Show that the constitutive equation has the form:
(p0 + p1 D + … + pN DN) s = (E1 + µ 1 D) (E2 + µ2 D) … (EN + µN D) e
(b) Show that the stress relaxation function G(t) is also the sum of N exponential terms:
-t -t -t
G(t) = G0 + G1et1 + G2 et2 ... + GN etN
Are there simple expressions for G0 , Gi , ti in terms of the constants E0 , Ei , µi?
Problem 2
Stress relaxation tests were performed on two tendons. The stress relaxation tests consisted of a ramp phase applied over the first 0.1 seconds followed by relaxation for 600 seconds. Two stress relaxation data sets are provided. One data set is from an injured tendon treated with rehabilitation and one data set is from an injured tendon which was not treated.
(a) Perform curve-fits to the data sets to determine the material properties of each tissue. Fit the data to the QLV model (5 parameters), the three parameter model, a two parameter model (Kelvin or Maxwell), and a simple logarithmic equation. Present a table of results for the material properties and plot the results along with the curvefits.
NOTE: for the QLV curvefits, use the m.files provided. Type ‘QLV_GUI’ at the Matlab command prompt to run program. Click on ‘Help’ and read the descriptions for each button. To perform curvefits, click on the buttons in the following order: ‘Load data’, ‘Reduce data’ (by 10, this will reduce the amount of time it takes for the curvefit to complete), ‘Zoom’ (click once just past t=0 and then once just before equilibrium; this is required to cut out the ramp phase and to remove junk data from the end), ‘t0 correction’ (this will reset the time zero, to take into account the ramp phase length), ‘Ln fit’ or ‘QLV fit’ .
(b) How do the time constants compare for the different models? Is there an advantage to increasing the complexity of the model from two parameters → three parameters → five parameters? Is there an advantage to using one of the models compared to a simple algebraic equation (e.g., logarithmic)?
(c) Discuss the effect of the rehabilitation treatment on tendon healing.
2022-10-19