Should we use Hooke's Law (that linearly relates stresses to strains) if the stiffness of body is changing during deformation?

Assume I have a slender cantilever beam, fixed at one end and force applied on the other. Now, during the deformation, we know that the beam's stiffness is going to change and the force-displacement response is not going to be linear. In this case, Large Strain theory is works.

On the other hand, if I have a very short and thick cantilever beam fixed at one end and force applied on the other, so during the deformation, it can be assumed that the beam's stiffness is somewhat constant and so is displacement reponse to force (Ofcourse, if I don't increase the force to astronomical levels). Here, Small Strain Theory works.

My question is, can I use the generalized Hooke's Law (which linearly relates the stresses to strains, in order to obtain stresses from strains) for both of these cases or not? Is Hooke's Law valid for both of these situations, and if not, then for which situation it is valid for?

P.S: Assume the body doesn't encounter any plasticity at all

• I think you shall explore the nonlinear-elastic material and hyperelastic material as both deal with property changes during deformation. Note both are not considered the "Hookean Material" though behave elastically.
– r13
Nov 6 '21 at 18:46

It seems to me that you are confusing the macro behavior of the structure (Large displacement) with the microscopic behaviour of the material (generalised Hooke's law).

Whether or not you can use Hooke's law is depended only upon if the material is with a range of strains/stresses that the relation between stress and strain is proportional. Different material have different ranges of strain that this applies.

For example, you might use a hyperelastic material for a beam that does not exhibit this linearity between Stress and strain even at small strains. figure: Hyperelastic material stress strain (tendon) ( source Alison Hubbel)

so IMHO, the answer to you question is that it depends on the material you are using. In the majority of cases (i.e. for the most common engineering structural materials, under strains not in the placticity region), yes you can. But that is not always the case.

• For a ductile material under linear elastic range, so you are advising that we can use Hooke's Law, even if the material is undergoing changes in stiffness during deformation. So do you mean to assert that the non linear force-displacement relation (which some people refer to as Hooke's Law) has no association with the linear stress-strain relation (which the same people also refer to as Hooke's Law) and these are two different concepts and phenomenons? Nov 5 '21 at 11:21
• I'm not sure that "non linear force-displacement relation (which some people refer to as Hooke's Law)" is correct. To my understanding Hooke's law is about proportional (linear) force displacement relationship. Nov 5 '21 at 12:32
• I think you didn't understand what I meant. I am saying that we should not make any resemblance between the force-displacement relation Hooke's law and stress-strain relation Hooke's law. Because as you said, the former is on a macroscopic scale, while latter is on the microscopic scale. Even when the force-displacement relation becomes non-linear (when stiffness starts to change) during deformation and Hooke's Law is no longer valid, then still the stress-strain relation will keep on following the Hooke's law (unless it enters the plasticity region). What do you think? Nov 5 '21 at 13:57
• I think we are in agreement. I assume that by change of stiffness you mean both the change of cross-sectional properties (which is macroscopic and has nothing to do with generalised Hooke's law) and the change in the stress-strain slope (microscopic and is related to the generalised Hooke's law). Nov 5 '21 at 14:01
• I don't understand op's comment. In the expression, E = stress/strain = (F/A)*(L/displacement) or E = s/e = (F/A)*(L/D), if stress/strain (s/e) is linear, which of the rest terms is/are non-linear variable?
– r13
Nov 6 '21 at 15:41

If there is a non-linear material or even a linear material but under nonlinear stress-strain behavior we can not apply hooks law.

A simple example is your cantilever beam stressed beyond hooks law into elastic-plastic range.

where may be part of the section near the neutral axis is still in the elastic range but the top and bottom are in the plastic range.

Of course, we can not use the hooks law here.