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

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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 thedisplacement of the midsurfaceplus therotation of the midsurface normal", "ris the undeformed shellmidsurface position”, "aisrotational displacement", "uisdisplacement vector on the midsurface", "arisdisplacement 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 vectoraandrare written directly in one piece, doesarrepresenta·rora×r, obviously it shouldn't be the latter, is this an operation? What is the exact meaning ofarhere? what is the difference betweenaandar? Using two letters to represent a vector is really misleading, canarbe replaced bya? Also why is this item multiplied byzagain, iszhere the local coordinate of the shell in the thickness direction, orzin the two-dimensional axisymmetric coordinate system (r,z).

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

3.Last but not least, bothuandarhave 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, anduandarcan be solved from these two equations? Could the second equation be written in the formf(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!



0 Replies Last Post 2024年3月13日 GMT+1 14:08
COMSOL Moderator

Hello SanDuo Xu

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