Deformation Energy: Exploring Linear & Nonlinear Elasticity

[Päällikkö]

This is a bit technical and I'm not exactly sure how to formulate my problem, but I'll try my best. I've been reading Landau & Lifgarbagez - Theory of Elasticity, and got not further than page 10, where they derive the free energy of deformation.

So, what they do is that they assume linear elasticity and Cartesian coordinates. Starting from the volumetric integral of the stress tensor times a variation in the displacement vector, they are able to show that the work related to this small change is, roughly speaking, minus the stress tensor times a variation of the strain tensor.

Ok, so what I'd like to know is if this result applies to nonlinear elasticity as well. I've seen claims that it does, citing that the proof is more or less elementary. I can't quite immediately see how the L&L approach could be generalized, though. Help anyone? Ideally I'd like to see a result without the assumption of Cartesian coordinates, but any pointers would be nice.

I've written below the version found in L&L:

$$\int_V \delta R dV = \int_S \sigma_{ik} \delta u_i df_k - \int_V \sigma_{ik}\frac{\partial\delta u_i}{\partial x_k} dV$$
Take V infinite and assume that stress vanishes at infinity. Noticing that the stress tensor is symmetric we may write
$$\int_V \delta R dV = -\frac{1}{2}\int_V \sigma_{ik} \delta \left(\frac{\partial u_i}{\partial x_k} + \frac{\partial u_k}{\partial x_i}\right) dV = -\int_V \sigma_{ik}\delta u_{ik} dV$$

Thus, they conclude, $$\delta R = - \sigma_{ik}\delta u_{ik}$$.

This I don't think would work for the nonlinear case, as it's missing the term
$$\frac{\partial u_l}{\partial x_i}\frac{\partial u_l}{\partial x_k}$$


Any help would be much appreciated.

 

[eljose79]

Thank you for sharing your question with us. It seems like you are trying to understand how the result derived in L&L for linear elasticity can be extended to nonlinear elasticity. I am happy to help guide you in the right direction.

Firstly, it is important to note that the derivation in L&L assumes a small deformation and linear elasticity. In this case, the stress-strain relationship can be written as \sigma_{ik} = \lambda\delta_{ik}\epsilon_{ll} + 2\mu\epsilon_{ik}, where \lambda and \mu are the Lamé parameters, \delta_{ik} is the Kronecker delta, and \epsilon_{ik} is the strain tensor.

In the case of nonlinear elasticity, the stress-strain relationship becomes more complex and can no longer be written in this simple form. However, the fundamental principles of energy conservation and work done by external forces still apply.

To extend the L&L derivation to nonlinear elasticity, one must consider the total potential energy of the system, which includes both the elastic energy and the external work done by forces. This can be written as U = U_{el} + W_{ext}, where U_{el} is the elastic energy and W_{ext} is the work done by external forces.

Using the principle of virtual work, one can show that the variation in the total potential energy is equal to the virtual work done by external forces. This can be written as \delta U = \delta W_{ext}. From here, the derivation follows a similar path as in L&L, with the addition of the nonlinear terms in the stress-strain relationship.

I hope this helps to clarify things for you. If you need more specific guidance or would like to discuss this further, please do not hesitate to reach out.

 

[charng]

I understand your confusion and frustration with trying to understand the free energy of deformation in nonlinear elasticity. It is a complex topic, and even the most experienced scientists can struggle with it at times. However, I am happy to provide some insight and clarification on the matter.

Firstly, the result derived in L&L for linear elasticity can be generalized to nonlinear elasticity, but it may require a different approach. In linear elasticity, the stress-strain relationship is linear, meaning that the stress is directly proportional to the strain. However, in nonlinear elasticity, the stress-strain relationship is nonlinear, meaning that the stress is not directly proportional to the strain. This makes the derivation more complicated, but not impossible.

One way to approach the problem in nonlinear elasticity is to use a variational principle, similar to the one used in linear elasticity. This involves finding a functional that represents the total energy of the system, and then minimizing it with respect to the displacement field. This will yield the equations of equilibrium, which can then be used to derive the free energy of deformation.

In this approach, the term you mentioned, \frac{\partial u_l}{\partial x_i}\frac{\partial u_l}{\partial x_k}, is taken into account in the derivation. It represents the contribution of the nonlinear terms in the stress-strain relationship. So, in essence, the result in L&L is just a simplified version of the more general result that takes into account nonlinear effects.

As for the assumption of Cartesian coordinates, it is possible to generalize the result to other coordinate systems, but it may require a more advanced mathematical approach. It may also depend on the specific problem and geometry being studied.

In summary, the result derived in L&L for linear elasticity can be generalized to nonlinear elasticity, but it may require a different approach and take into account additional terms. I hope this helps clarify your understanding and provides some direction for further exploration.