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Modeling material nonlinearity in a cantilever beam
Posted Nov 14, 2023, 4:25 p.m. EST Acoustics & Vibrations, Materials, Results & Visualization, Studies & Solvers, Structural Mechanics, Equation-Based Modeling, Modeling Workflow, Material Models 1 Reply
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Hello!,
I am trying to model a canteliver beam having nonlinear material properties (by introducing a polynomial Youngs modulus) and want to study its frequency response curves. The beam that I am modeling is in micrometer and I have taken "beam" as my physics module and carried out frequency domain study and eigenfrequency study in the linear condition (Youngs modulus as a constant). Carrying out this, I was able to get the frequency range in which I was getting the peak beam amplitude. But the frequencies that I got are in 10^7 Hz. I am not sure whether that is possible for a beam that small. Secondly, assuming its correct, I introduced the nonlinearity in the Youngs modulus (as a plynomial in terms of strain). Then I tried doing the time dependent study to get the frequency response as COMSOL does not support frequency domain and Eigen frequency study in the nonlienar condition. When applying values for the time domain study, the time period, times step and stop time when calculated are very small (in terms of 10^-8s) (since the frequencies were very large) and I am not sure if its correct to consider those values. Again assuming to be all correct, when I applied time domain study, it did not go through.
Can I know how do I ge the frequency response curve where the peak is leaning towards one of its side?
Regards, Shashank
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A couple of things to note if you are going this path:
- Note the difference between tangent and secant modulus. Either
or 
. The latter definition applies in COMSOL. - In a beam, the strain (in bending) varies linearly over the cross section. But in the formulation of a beam element in the Beam interface, there is no explicit variation of the strain. The bending moment is directly computed from
where Young's modulus is assumed to be constant over the cross section and the moment of inertia, I, implicitly contains the strain distribution. This means that even if you know 
for the material, you still need to convert it to an effective value for bending by integrating over the cross section using the fundamental assumptions of beam kinematics.
By "the peak is leaning towards one of its side", I guess that you mean that the system is essentially a Duffing oscillator https://en.wikipedia.org/wiki/Duffing_equation?
Such an analysis should, in principle, be possible. But you need to solve quite a large number of long time histories, one for each frequency. It takes many cycles until a stable cycle is reached, as long as the system is not heavily damped.
-------------------Henrik Sönnerlind
COMSOL