## Forum

### The material behaviour of corrugated shapes

1. Regarding to the fact that an orthotropic material has 3 planes of symmetry , a transversely isotropic has one axis of symmetry , and so on ... , which type we can assume for a corrugated shape ? with how many constants we can define this geometry ?  I know that most general is anisotropic , but regarding to the geometry , what can be the planes or axises of symmetry ?

2. The material symmetry you are talking about is 3D. It does not make that much sense to consider a corrugated shape as a 3D material, unless the corrugated shape repeating along all the three directions. Hence a corrugated shape is either inserted between two layers to make a sandwich structure or be a structure by itself. Its property is better characterized in terms of structural properties, such as ABD matrices for classical plate model etc.

3. The material symmetry you are talking about is 3D. It does not make that much sense to consider a corrugated shape as a 3D material, unless the corrugated shape repeating along all the three directions. Hence a corrugated shape is either inserted between two layers to make a sandwich structure or be a structure by itself. Its property is better characterized in terms of structural properties, such as ABD matrices for classical plate model etc.

4. if all of the structure be assumed together , not only the corrugated I mean , all the other parts that are in contact with eachother , then how would it be ? also I have seen the constituitive matrix between [N,M] matrix and [e,k] in this paper , http://www.enu.kz/repository/2010/AIAA-2010-2706.pdf page  4 , it's got a bit confusing , also the contact of the upper layers are taken into consideration , the matrix is mensioned as ABD matrix , but the relation is between the stresses and strains and material is assumed orthotropic , I think I am missing a point

what is I not assum a corrugated as a shell and as a solid ? what would be different in this assumption ?

5. That equation of the paper you pointed out is an ad hoc assumption. There is no reason why the stiffness matrix should look like that. The original material for all the parts could be orthotropic, but it does not mean the complete structure will be orthotropic. Assume corrugated shape as a solid will introduce significant error. For example, the corrugated shape could  have extension-bending coupling but this coupling cannot be represented by a solid.

6. so for the mentioned model , assuming with the contacts all together , what would be the appropriate model ? still ABD ? I mean if I solve the equation for Forces,moments - strains , I will have this matrix

how many constants do I need to define a model like in the paper ?

7. and may I ask why the solid model can not represent this coupling ?

8. which model to use depending what are the information you want to capture. If transverse shear is important, then you need the transverse shear stiffness matrix in addition to ABD matrix. If you follow a mechanics of material type of approach as used in the paper you pointed out, you need at least 6 load cases with corresponding six set of boundary conditions. Some of boundary conditions are very challenging. If you use my method, you don't have to struggle with the boundary conditions.

Regarding why you cannot model it as a solid, assuming you apply uniform pressure both ends, if it is modeled as an orthotropic solid, you will have only extension and not bending. However, the panel is not symmetric, bending will be introduced in the real structure under applied uniform pressure.

9. Thank you so much

regarding your method , in the paper , the analytical method is discussed not the numerical one , so how can I apply a numerical discussion and formulas in a FEM method ?

10. With a variational statement, you can always implement in FEM. How to do it can  be found in FEM textbooks.

11. May I please ask that should I know how the Psi function is defined like ?  I mean for calculation of the strain energy of the system , first , should I apply it on a single cell ? and then can I do this on a commercial FEM software , i mean when the variational statement changes , I think the methodology of solving it also changes , my question is how can I apply this change in a commercial software ?

12. As another additional question ,  I would like to ask why the solid elements can not represent the extension - bending coupling ?

13. commerical fea code cannot be used to solve for the equations we have derived in our paper. A special purpose finite element code is needed.

Regarding why you cannot model it as a solid, assuming you apply uniform pressure both ends, if it is modeled as an orthotropic solid, you will have only extension and not bending. However, the panel is not symmetric, bending will be introduced in the real structure under applied uniform pressure.

14. But for doing so , I feel like doing all the steps like meshing and so could be very time consuming , is it possible to do only the solution in the code and get the meshing information and so on in the software ? also with the time I  have got , I feel like it might be very time consuming

about the bending and coupling , I understood the condition , but what is the reason , that the solid can not introduce the orthotropic properties ?

15. The mesh can be done using commercial software. However, the solution must be coded separately. Even if your homogenize it as an orthotropic solid, still could not caputer extension-bending coupling. Just try to apply to uniform pressure to an othtoropic solid in one direction, you only get uniform compressive stress along that direction accorind to 3D elasticity, While in the original structure you get bending and extension, that is the stress distribution along the thickness is not uniform.

16. I think I need to know the function Psi from the equation 131 , to know how to calculate the energy function , cause in the equation 138 , the function Psi and it's derivative is required , I am just still not sure how to do it

17. what I have seen in assymptotic expansion for the displacement field is like :

u(k) = u0(x) + e u1k(x,y) + e^2 u2(x,y)

that I don't find the similarity with the eq. 131

18. Yes, they are not similar as our theory is different. Particularly if you want to homogenize it to be plate theory, you have to express it in terms plate kinematics.

19. Thank you very much

but my problem is not resolved , whther I want to model it like a plate or not , the elements of the equation 59 are not introduced precisely , so I do not know how to variate them to calculate the strain energy

20. One Question I have is  how do we define and use the term ABD matrix when like in your paper  we have a one layer shall and no fibers

what does the term ABD refer to ? when we don't have a transformation of stresses in one direction

I am somehow mixing the concepts in Laminates and shells