# Determination of the domain of nonlinearity in a Neo-Hook solid model (Finite elements)

For a FEM simulation of a Neo-Hook solid model, how do we know we are in the "regime" of nonlinearity of the solid? In other words, how do I know the hyperelastic material law is really used, and not just a needless additional computational cost because it can be modeled with a linear elastic Hook law. Following these Wikipedia figures for example, does this mean the nonlinearity really happens when we see a nonlinear relationship between $$\lambda_{11}$$ and $$P_{11}$$?

For my case/simulation, I chose the element with the biggest stress values, and I calculated the Lagrange Green strain tensor $$\pmb{E}$$ there, I computed the eigendecomposition to get the biggest eigenvalue $$E_{11}$$ and its associated eigenvector $$\pmb{v}_{11}$$. I then compute the stretch as : $$\lambda_{11} = \sqrt{1+2E_{11}}$$. For the principal stress I compute : $$P_{11} = \pmb{v}_{11} \cdot (\pmb{P}\pmb{v}_{11})$$ where $$\pmb{P}$$ is the first Piola-Kirchoff stress. Should I expect to see a nonlinear relationship here? Is my definition of these quantities correct? In my case, I get something linear: I used $$W(\pmb{F}) = \frac{\mu_s}{2}\, (|\pmb{F}|^2 - 3 - 2\, \log(J))$$ with $$\mu = 1923076$$.

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• It looks like you’re still in the linear region, just to understand better, how are you running this model? Also maybe plotting stress $(\mathbf{\frac{\partial W}{\partial E}})$ and strain $(\mathbf{E})$ components directly might help in seeing where the nonlinearity is really present in the material model. May 25 at 20:42
• Your $\lambda_{11}$ is close to one. Which means it is close to linear elastic and hence you are seeing a linear response. Make $\lambda_{11}$ much bigger - say 1.2, or 2 or 5 or 10. Then see what you get. Also, you might try plotting the stress-strain curve without using FEM. Try deriving a (non-linear) equation for the stress-strain curve and solve it using fsolve or something equivalent.
– NNN
May 26 at 12:31
• @NNN, would you expand your comment into an answer? 2 days ago

Since your $$\lambda_{11}$$ is close to unity, it means that your response will be close to linear, because your strain is small. You have not provided enough strain for the non-linearity to kick-in.
I would plot over a larger range of $$\lambda_{11}$$, say from 1 to 5. That should be enough to see the non-linearity.
Also, before doing FEM, I would derive a non-linear relationship between $$\lambda_{11}$$ and $$P_{11}$$, solve the resulting equations with some non-linear solver and plot the results. And then compare with FEM.