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Prof. Yu's Research Group in the Cloud

RVE acceptable size to be counted as Micro

Just a general question about Homogenization , when we are selecting the RVE , how much should it be smaller than the Macro dimensions of the model , to be counted micro , I mean only that one part of the model is being repeated is enough to define the periodicity or it should be very smaller in all dimensions ?

1. A micromechanics theory contains three interdependent elements: RVE (or the micromechanics domain), governing equations for the RVE, and boundary conditions. For linear elastic problem, if RVE can be chosen in such a way that all its three dimensions are much smaller than the corresponding macro dimension, then 3D elasticity should be used for the gonvering equations, periodic boundary conditions. For some structures, such as sandwich plates with a corrugated core with corrugation in two directions, then one dimension of the RVE is the same as the plate (plate thickness) and two other dimensions are much smaller than the in-plane macro dimensions. Of course, at least one dimension of RVE should be much smaller than the corresponding than the macro dimension. Otherwise, homogenization cannot be used.

2. Thank you so much

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I have another doubt that Ã‚Â , for a material that is periodic in x direction , for 2 nodes in the x direction , we write the uA1 - uB1 = E11 * L , that defines the tension , then a uA2-uB2= E12 * L is introduced that actually means a rotation , when we apply these two PBC s together , it is like a prestressed structure under tension and also rotation , rotation due to the second equation Ã‚Â I don't know if this second equation is representing something correct , cause then I don't know we can apply forces afterwards , but these PBC s act like a force and the rotation thing , creates a condition that I think if I want to apply a tension to this structure , this rotation should not happen , could you please give me some hints , thank you very much

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3. Boundary condition applied such a way is itself a load condition. For example uA1 - uB1 = E11 * L corresponding to extension. No additional load is needed. This is so-called strain controlled boundary condition. You need to apply it one by one and get the stress tensor correspondingly. For example you applyÃ‚Â uA1 - uB1 = E11 * L, then constrain one more point to remove rigid body motion, compute <sigma_ij>, Ã‚Â this will give you the first column of your six by six stiffness matrix.

4. Another doubt I have is that to calculate the average stress Ã‚Â in homogenization the V and dv (material volume and element volume ) are the ones before or after deformation ?