Tutorial Model — Curve Fitting
This tutorial model demonstrates how to use the Optimization Module to estimate unknown function parameters based on measured data. The two-parameter Mooney–Rivlin solid material model is used as an example, but the procedure is generally applicable when you need to fit a parameterized analytic function to measured data.
The two-parameter incompressible Mooney–Rivlin material model describes the local behavior of rubber-like materials. The model assumes that the local strain energy density in an incompressible solid is a simple function of local strain invariants.
In a standard tensile test, a rotationally symmetric test specimen is pulled in such a way that it extends in one direction and contracts symmetrically in the other two. For this case of uniaxial extension, the relationship between an applied force, F, and the resulting extension, ΔL, of a true Mooney–Rivlin material is
where A0 is the original cross-sectional area of the test specimen and L0 is its reference length. The constants C10 and C01 are material parameters that must be determined by fitting the above equation to the experimental data from the tensile test.
In practice, tensile test data is delivered in a form which is independent of the geometry of the test specimen used. There are multiple possible formats. The one used here contains corresponding measured values of engineering stress, Pi, representing force per unit reference area
and stretch, λi, representing relative elongation
The expected relationship between these variables for a Mooney–Rivlin material is
Given N pairs of measurements (λi,Pi), i = 1,…,N, the values of C10 and C01 that best fit the measured data are considered to be those which minimize the total squared error
The curve fitting problem is therefore identical to an optimization problem.