How to determine exact point of tangency? I have fitted my stress-strain data with $y=ax^3+bx^2+cx+d$ and also added tangent lines as shown in figure below. I am interested to see the deviation of the fit from the linearity. I am not aware of any theory that can be used to find the tangent point (the point of deviation of the fit from the linearity). Hence, based on my visual inspection I somewhat decided my tangent point at added vertical lines. This method is of course raising question of reliability. So I am looking for suggestions how I can quantitively determine the points of deviation for all four data sets. 
Your advice/suggestion would be highly appreciated!

Question update: The figure shows that there is a linear relationship at the beginning, but at some point (point of deviation from linearity), the linear relationship disappears and non-linear relationship prevails. I'm looking for a way to determine this point of deviation from linearity quantitatively, not as done by visual inspection. How do I do that?
(Sorry, not sure what should be the appropriate tag for this post!)   
 A: If you're asking about where the point of inflexion is on the cubic, it's where $\frac{d^2y}{dx^2}=6ax+2b$ is zero, which occurs when $x = -b/3a$.

However a fitted model doesn't give you the exact (population) values of $a$ and $b$, just noisy estimates, so you are left to estimate that point. 
This is a very similar problem to finding the turning point on a quadratic fit, which has been discussed in a number of posts here. Some of these posts will have some useful information, particularly if you want a confidence interval for the point of inflexion.
For example, see here and a related question here.
Finding the turning point in a logistic model is another common problem on CV which has a number of relevant posts. These can be found with a suitable search.
--
It's not clear that cubics are a good choice of model for this situation, unless you really expect the relationship to reach a maximum and then come back down again as Strain increases further:

If you don't expect that, I'd suggest considering functions that more nearly accord with the physical expectations of the behaviour.
(Perhaps the aforementioned logistic model may be of some relevance to you, I don't know.)

Response to updated question:

The figure shows that there is a linear relationship at the beginning, but at some point (point of deviation from linearity), the linear relationship disappears and non-linear relationship prevails. I'm looking for a way to determine this point of deviation from linearity quantitatively, not as done by visual inspection. How do I do that?

If you fit, say a nonlinear logistic model, the inflexion point in the logistic is a parameter of the model. If you fit some other model it will depend on the model.

If you really think it's linear for an initial period (rather than just close to linear), you might fit a function that is literally linear and then joins up to a function that has no curvature at the join point, and the same slope, akin to the left end of a natural cubic spline where the knot is a parameter, but instead with a function on the right that flattens out/asymptotes.
