Mooney-Rivlin Curve Fit

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.

This application is also used in Introduction to the Optimization Module.

Model Definition

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 applied force, F, and resulting extension, ΔL, of a true Mooney–Rivlin material is

(1)

where A0 is the original cross-section 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 Equation 1 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.

Results

Figure 1 shows measured stress and stress computed using the material properties that have been fitted to the measured data.

Figure 1: Measured (turquoise circles) and computed (blue line) engineering stresses. In addition, two other stress measures are plotted for comparison.

It should be noted that for large stretches, the values of the different stress measures will differ significantly. For a uniaxial stress state and an incompressible material like in this case, the true stress (Cauchy stress; stress per current area) is related to the engineering stress by

(2)

The 2nd Piola-Kirchhoff stress is defined as

(3)

It is thus necessary to pay attention to the stress measures actually used when a interpreting results from an actual simulation in which a material model for large stretches is used.

Optimization_Module/Parameter_Estimation/curve_fit_mooney_rivlin

Modeling Instructions

From the menu, choose .

New

In the window, click

Add Component

In the toolbar, click

and choose .

Global Definitions

Add the stretch parameter lambda which has been varied in the measured data. Also add the unknown material parameters as global model parameters.

Parameters 1

In the window, under click .

In the window for , locate the section.

In the table, enter the following settings:


Name	Expression	Value	Description
lambda	1	1	Stretch
C10	1[MPa]	1E6 Pa	Mooney-Rivlin parameter
C01	1[MPa]	1E6 Pa	Mooney-Rivlin parameter

Variables 1

In the toolbar, click

and choose .

Set up the assumed relationship between stretch and engineering stress as a variable expression in terms of lambda and the yet unknown C10 and C01.

In the window for , locate the section.

In the table, enter the following settings:


Name	Expression	Unit	Description
P	2(C10+C01/lambda)(lambda-1/lambda^2)	Pa	Engineering stress

Add Study

In the toolbar, click

to open the window.

Go to the window.

Find the subsection. In the tree, select .

Click in the window toolbar.

In the toolbar, click

to close the window.

Use a study step to set up control variables and select an optimization solver. The solver is particularly efficient for least-squares problems such as this one.

Study 1

Parameter Estimation

In the toolbar, click