You have already accepted Star's answer, so you may not even see this answer. But in fact, there is no linear part of such a curve. A perfectly elastic material should show an initially linear behavior, true. So is this force-displacement curve representative of a perfectly linearly elastic material?
But now let us look more carefuly at that initial portion.
So merely by viewing that portion where you think the curve is "linear", we see it does not look at all linear.
mdl = fittype('a*F^2 + b*F','indep','F');
mdlest = fit(F(K),D(K),mdl,'start',[1 1])
a = -3.617e+07 (-3.623e+07, -3.611e+07)
b = 8.393e+06 (8.389e+06, 8.398e+06)
I chose a model with no constant term, since we know that when the force is zero, so must be the displacement.
What is important to see in that fit is the fit is remarkably good over that region, and that the quadratic coefficient is actually more non-zero than the linear coefficient.
And that leaves me with the claim this curve is not that of an perfectly elastic material (within bounds before it undergoes plastic deformation) but that of a nonlinearly elastic material. There is essentially NO linear section of the start of that curve.
At best, you may decide to approximate it with a linear section, but that would mistake the true behavior seen here. As well, if this really is the force-displacement curve (I wonder how much you smoothed the curve) I would look to see if the material undergoes plastic deformation, so after you stretch it by a small amount, and then release it, does that mateiral return to the original rest length?
Finally, there is a distinct possibility that whatever tool you used to "smooth" this curve was used inappropriately, causing the curve to APPEAR to be so strongly quadratic. I cannot know for certain if this is true, but it seems highly likely, as that section of the curve is clearly very strongly a quadratic polynomial. If that is the case, then your smoothing of the data made it impossible to truly understand the behavior of this material.