You can get a good description of RVE analysis and its boundary conditions (homogeneous displacement, homogeneous traction, pbc) from https://cdmhub.org/resources/1036. The real pbc is that u_i-\epsilon_{ij}x_j should be equal on the corresponding edges. This type of boundary conditions can be applied using coupled equations constraints. This requires that one creates a mesh with corresponding nodes on periodic edges. For real, complex microstructures, this could be a challenge. For 2D RVE, another issue with RVE analysis is that you can only get in-plane properties. Actually you should need 3D properties even if your macroscopic problem can be approximate as plane strain or plane stress problem. 

A recently developed general-purpose multiscale constitutive modeling code called SwiftComp based on Mechanics of Structure Genome, can be used for micromechanics which is essentially a RVE analysis for dummies. The user does not have to specify the boundary conditions and periodic mesh is not required. All the user has to do is to provide the finite element mesh as input. Another unique feature of SwiftComp is that it can compute complete set of 3D properties from a 2D RVE. All the properties are computed within one analysis and thus it is at least six times more efficient than traditional 3D RVE analysis. The code can be launched freely in the cloud at https://cdmhub.org/resources/scstandard or used as a plugin for ABAQUS or ANSYS. You are welcome to try it to see the difference from the RVE analysis you are familiar with.

https://cdmhub.org/resources/scstandard

The publications below should provide answers. Cheers.

Article Overall behaviour of heterogeneous elastoviscoplastic materi...

I just realized that I can add one publication link. Here is the 2nd one:

Article Approach to micro-macro modeling of heterogeneous materials

Basically, the periodic boundary (i.e. u(0)=u(a)) is to ensure overall compatibility between micro and macro levels ( i.e. the assumption that the macro is nothing but repeated micro RVE's). it is better suited for somewhat dense fillers within a somehow uniform matrix. Homogeneous boundary (u(a)=0) is more or less better suited for cases when the perturbation in stress or strain fields is said to vanish at the ends of or outside the RVE boundary (more or less), thus it is better used for lightly packed materials.

Dear Mr. Onur Guvenc, thanks a lot for the references. I was able to implement PBC with the help of it. :)

Dear Mr. Mahmud. thanks for the differentiation between the two. I had read somewhere that Homogenous boundary conditions result into a condition of "plane remains plane" kind of situation for boundaries. Can you be a bit more elaborative what do you mean by stress-strain perturbations?. Or may be you can refer me to some good literature/book/article for to read on different kinds of boundary conditions which people have been uisng in micromechanics. Thanks anyways.

I have some problem on obtaining the APDL Ansys to set the periodic boundary condition. May I request a APDL Commands to obtain a 2D RVE of unidirectional fibrous composite. 

You can get a good description of RVE analysis and its boundary conditions (homogeneous displacement, homogeneous traction, pbc) from https://cdmhub.org/resources/1036. The real pbc is that u_i-\epsilon_{ij}x_j should be equal on the corresponding edges. This type of boundary conditions can be applied using coupled equations constraints. This requires that one creates a mesh with corresponding nodes on periodic edges. For real, complex microstructures, this could be a challenge. For 2D RVE, another issue with RVE analysis is that you can only get in-plane properties. Actually you should need 3D properties even if your macroscopic problem can be approximate as plane strain or plane stress problem. 

A recently developed general-purpose multiscale constitutive modeling code called SwiftComp based on Mechanics of Structure Genome, can be used for micromechanics which is essentially a RVE analysis for dummies. The user does not have to specify the boundary conditions and periodic mesh is not required. All the user has to do is to provide the finite element mesh as input. Another unique feature of SwiftComp is that it can compute complete set of 3D properties from a 2D RVE. All the properties are computed within one analysis and thus it is at least six times more efficient than traditional 3D RVE analysis. The code can be launched freely in the cloud at https://cdmhub.org/resources/scstandard or used as a plugin for ABAQUS or ANSYS. You are welcome to try it to see the difference from the RVE analysis you are familiar with.

https://cdmhub.org/resources/scstandard

Thanks Professor Yu for sharing such a great article (https://cdmhub.org/resources/1036). 

Hi Khizar Rouf,

Could you please share with PBC materials notes of Prof. Wenbin yu

You can download it yourself through signuping on cdmHUB.