Question about Example "Dynamic Analysis of Clamped Beam"-Example from PDE Toolbox

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Frederik Rentzsch 2023-3-31

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回答： Alexis He 2023-4-14

In the example Dynamic Analysis of Clamped Beam of the PDE Toolbox documentation the c coefficients

c = [2*G + mu; 0; G; 0; G; mu; 0; G; 0; 2*G + mu];

are being used in the linear case. I would like to understand how these coefficients correspond to the underlying equations, hoping to gain some insight in how the vector c must be specified. Because the problem is 2D (N=2) and there are 10 entries in vector c, i suppose the 2N(2N+1)/2-element form from specifyCoefficients is being used? But I don't understand where the entries in c are coming from. I would appreciate, if someone could explain (roughly) why the vector c looks this way.

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Answer 1

Alexis He 2023-4-14

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The Lagrangian formulation of the equilibrium equations in index notation is :

For linear solution, we ignore the non-linear strain term in the Green-Lagrange strain tensor E and the first Piola-Kirchoff stress tensor P=F⋅S=F⋅C⋅E :

Thus, (1) will become :

For this 2-D problem, we have :

with using the constitutive relations in matrix form, we have :

replacing the

with

we have :

After refering to "tensor product" for 2-D systems", we have non-zero terms in tensor c：

P.S. :

means the c Coefficient of term

in i-th equation

The 2N(2N+1)/2-Element Column Vector representation of c and the matrix representation of c coefficient ( with a “flattening” of the N-by-N-by-2-by-2 tensor to a 2N-by-2N matrix when N=2 ) are ：