Where Can I Find Additional Material on the 1957 Eshelby Inclusion Problem?

[Koubaros]

I was wondering if any1 knew of extra material (besides the original paper that is) on the classical Eshelby inclusion problem as it is formulated in his paper in 1957. Any help or directions would be appreciated.

 

[PerennialII]

You got his later papers ? There were at least a couple of other papers which followed the 57 paper, perhaps not as 'new' as the 57 one, but added material nonetheless.

 

[Koubaros]

Breakdown

Unfortunately I do not have the later papers but it seems like that I cannot completely comprehend the logic of the problem, as it is described in the Introductory part of the paper and would need a little input-breakdown of the configuration and the steps he follows in order to solve it.

 

[PerennialII]

I went through the papers for my thesis a few years back (needed to work on the principles of the energy momentum tensor), I'll go through my drawer and dig them up.

 

[Koubaros]

Thanx

Thats a big thanks PerennialII. By the way, what was your thesis on?

 

[PerennialII]

Ok, I found 4 dusty papers by Eshelby from my shelf -- actually a review-like paper from the 70s seems to be the "clearest" of them all.

Essentially what Eshelby tried (and in my mind succeeded better than anyone before or after) was to develop a methodology suited for analyzing the behavior ('via the developed generalized 'force' concept) of different kind of material and geometrical 'irregularities' such as lattice defects and microscopic/macroscopic cracks. The term 'force' is seen to be applied quite often in context of the energy momentum tensor, but overall what we're talking about is essentially an energy release rate, i.e. the parameter used to characterize the state of e.g. cracks is the rate of energy variation (overall, the idea is to understand and characterize the behavior of defects, essentially e.g. a material interface can be qualified as a defect in this respect, cracks are naturally obvious ones). The idea behind this thinking is that the rate of energy variation, if you take for example fracture mechanics as an example, is a parameter seen suited for characterization of cracking/loading phenomena, and Eshelby's energy-momentum tensor is a tool suited very well for such analysis (actually, modern continuum theories of this field relie to date on the tensor). The basic concept of why such an energy related parameter were to be useful can be derived from 1st law of thermodynamics, ground breakers from Eshelby's timeframe in this respect were for example Cherepanov and Rice (even though papers from the 80s-90s have a bit more 'complete' theories).

There are quite a bit of ways to come up with the energy momentum tensor, Eshelby has his own approach which I see as kind of a mix-up of calculus of variations and potential energy treatment, perhaps the most elegant method is to approach it directly using thermodynamics. Eshelby goes a bit on his own way by defining a function "L", a density function (Lagrangian kind) by

$$L = L (u_i , u_{i,j}, x_m)$$

which contains the displacement field, its first derivative and the spatial coordinate. Integrating the density function over a region of space and finding the stationary value the Euler's equations result (local equilibrium conditions):

$$\frac{\partial}{\partial x_i}\frac{\partial L}{\partial u_{i,j}}-\frac{\partial L}{\partial u_i} = 0$$

Simplifying using chain rule leads to:

$$\frac{\partial L}{\partial x_l}=\frac{\partial}{\partial x_i}( \frac{\partial L}{\partial u_{i,j}} u_{i,l}-L \delta_{lj}) = \frac{\partial P_{lj}}{\partial x_i}$$

where $$P_{lj}$$ is the energy momentum tensor. Typically the density function is taken as the strain energy density in order not to make things too complex. Doing this results in

$$P_{lj}=W \delta_{lj}-p_{ij} u_{i,l}$$

where

$$p_{ij}=\frac{\partial W}{\partial u_{i,j}}$$

The form given above is directly implemented in the J-integral, which has an interpretation as the energy required for separation of crack faces or material removal. What sort of examples I've seen Eshelby give himself have typically been such that the energy momentum tensor can be used to quantify the generalized force and the change in the total energy of a defect containing system when the defect is subjected to a displacement field, in such a case the generalized force acting on the defect can be attained by integrating the energy momentum tensor over the surface surrounding the defect. Eshelby has applied this concept to evaluate 'forces' acting on interfaces such as microstructural phenomena, the martensitic transformation being one of the examples. Anyhow, the simplest interpretations can be given for cracks as was above.

Dissection to follow ? The specifics of the problem can be quite tricky, especially since the physical interpretation ain't all that easy to build. In that respect understanding the tensor as a component in local thermodynamical equilibrium helps one to get a grasp on it.

Btw ... I did my thesis (msc) on usage of path-independent integrals in characterization of local fracture phenomena ... so it's relatively close to this field.

 

[Gouse Pasha]

I need help in solving plate with an elastic inclusion. I was searching for Eshelby's papers can you send me those papers. My interest is to find stress distribution.