Hello all

I have spent a large amount of time trying to undertstand the multiscale aggregating method (MAD) of Prof. Belytschko. Unfortunately, I still can not compltely understand the method.

Phu

### Do you have any specific

Hello ,

I might be able to help you. Do you have any specific question about the method or the general idea is not clear for you?

### Hello tabarraei, Firstly,

Hello tabarraei,

Firstly, thank you very much for your help. I think that I understand the general idea of the MAD. However to make sure that is indeed the case, please let me first present what I understand about this method.

Basically the MAD is a computational homogenization method applied for failure analysis. Since standard homogenization, which determines the response of a macroscopic point from the unit cell associated to this point, is based on the principle of scale separation. Unfortunately this principle is invalid in case of strain localization, so in the MAD, a unit cell is linked to a macroscopic area (not a point). Precisely the unit cell matches exactly one macro element.

For the unit cells, micro cracks are explicitly modelled (by XFEM but it is not the main point here) so the standard averaging for the macroscopic stress is altered for the micro displacement field is not continuous everywhere over the unit cell. Roughly the macro stress is normally given by

sigma_M = integration along the outer boundary of unit cell of t \otimes x + integration along the micro cracks [[u \otimes n]]

which apparently complicates the calculation of sigma_M. In the MAD method, the concept of perforated unit cell was proposed in the sense that the averaging procedure is performed over the domain excluding the microcracks. Thanks to this the macro stress is computed as usual

sigma_M = integration along the outer boundary of unit cell of t \otimes x

Up to now, everything is fine. You have the homogenized stress computed from micro boundary quantities. How about the macro crack? They are determined based on the so-called extended strain averaging theorem that reads

A <epsilon_m> = A epsilon_M +  integration along the micro cracks [[u \otimes n]]

(m and M denote micro and macro quantities, <.>designates the average of (.) and A is the unit cell area)

Rearrangemet of the above yields:

integration along the micro cracks [[u \otimes n]]= A (<epsilon_m> - epsilon_M)

The macro crack equivalent to the micro cracks is then determined by

integration along the macro crack [[u_M]] \otimes n_M= A (<epsilon_m> - epsilon_M) (*)

The (*) equation enables to compute the crack opening [[u_M]] and the crack normal n_M.

My questions are then

(1) The above reasoning of mine is correct or wrong?

(2) When a macro crack is initiated on a macro element?

(3) Assuming that (2) was answered, then (*) is used to compute [[u_M]] and n_M but how about the length of that crack? In the paper, Prof. Belytschko said something about crack nucleation with length l_c smaller than the macro length. l_c = ???

I am looking forward to hear from you. Again thanks a lot.