How to understand the meaning of "ar" in the shell's governing equation?

Posted Mar 13, 2024, 9:08 a.m. EDT Structural Mechanics, Physics Interfaces Version 5.6 0 Replies

SanDuo Xu

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Dear all, I have performed a simulation of a flat shell under 2D axisymmetry and I have some doubts about the shell's governing equations. A schematic diagram of shell deformation is shown in Fig.1.The governing equations for shell interface can be seen in Fig.2.

The user manual states："the deformation field is described by the displacement of the midsurface plus the rotation of the midsurface normal", "r is the undeformed shell midsurface position”, "a is rotational displacement", "u is displacement vector on the midsurface", "ar is displacement of shell normals".

1.In the first equation，it seems that the second term of the equation represents the rotational acceleration of the shell, but I don't understand why the vector a and r are written directly in one piece, does ar represent a·r or a×r, obviously it shouldn't be the latter, is this an operation? What is the exact meaning of ar here? what is the difference between a and ar? Using two letters to represent a vector is really misleading, can ar be replaced by a? Also why is this item multiplied by z again, is z here the local coordinate of the shell in the thickness direction, or z in the two-dimensional axisymmetric coordinate system (r,z).

2.In the second equation, what does the σ_z=0, -1<=z<=1 mean? Is it that the Cauchy stress σ is 0 in thickness directions?

3.Last but not least, both u and ar have a component in (r,z), so there are a total of four unknowns. So the number of equations is less than the number of unknown quantities, and u and ar can be solved from these two equations? Could the second equation be written in the form f(u, ar)? are some equations missing?

4.Is the understanding of the terms on the right-hand side of the equation correct: the first item is stress, the second is volumetric force, the third is bending moment. The above questions are very important to me, thank you very much for answering them!

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0 Replies Last Post Mar 13, 2024, 9:08 a.m. EDT

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Hello SanDuo Xu

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