1 Introduction
2 Shell models and finite element method
2.1 The 3D shell problem
2.2 Kirchhoff–Love model
2.3 Reissner–Mindlin model
2.4 Finite element method
3 Code verification
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strain tensor \(\epsilon _{ij}\),
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derivatives of the strain tensor \(\epsilon _{ij,k}\),
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covariant base vectors \(\mathbf {G}_{i}\),
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derivatives of the covariant base vectors \(\mathbf {G}_{i,j}\).
4 Verification examples
Field A | Field B | |
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\(u^1 =\)
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\(\theta ^1\)
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\(\sin (\pi \theta ^1) \cos (\pi \theta ^2)\)
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\(u^2 =\)
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\(\theta ^2\)
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\(\cos (\pi \theta ^1) \sin (\pi \theta ^2)\)
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\(u^3 =\)
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\(\theta ^1\theta ^2\)
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\(\sin (\pi \theta ^1 \theta ^2)\)
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\(v^1 =\)
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\(\theta ^1\theta ^2\)
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\(\sin (\pi \theta ^1 \theta ^2)\)
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\(v^2 =\)
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\(\theta ^1\theta ^2\)
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\(\sin (\pi \theta ^1 \theta ^2)\)
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Field A | Field B | Field C | |
---|---|---|---|
\(u^1 =\)
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\(\theta ^1\)
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\(f(\theta ^1,0.56) f(\theta ^2,0.65)\)
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\(\sin (\pi \theta ^1) \cos (\pi \theta ^2)\)
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\(u^2 =\)
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\(\theta ^2\)
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\(f(\theta ^1,0.56) f(\theta ^2,0.65)\)
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\(\sin (\pi \theta ^1) \cos (\pi \theta ^2)\)
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\(u^3 =\)
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\(\theta ^1\theta ^2\)
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\(f(\theta ^1,0.56) f(\theta ^2,0.65)\)
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\(\sin (\pi \theta ^1) \cos (\pi \theta ^2)\)
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\(\overline{\mathfrak {g}}_1\)
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\(\overline{\mathfrak {g}}_2\)
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\(\overline{\mathfrak {g}}_3\)
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\(\overline{\mathfrak {g}}_4\)
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---|---|---|---|---|
\(x_1 =\)
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\(\theta ^1\)
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\(\theta ^1+(\theta ^2)^2\)
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\(\cos (\theta ^1)\)
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\(\frac{2\theta ^1\theta ^2+2\theta ^1-\theta ^2}{3}\)
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\(x_2 =\)
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\(\theta ^2\)
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\(\theta ^2+(\theta ^1)^2\)
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\(\theta ^2\)
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\(\frac{-\theta ^1\theta ^2+2\theta ^1+2\theta ^2}{3}\)
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\(x_3 =\)
| 0 | 0 |
\(\sin (\theta ^1)\)
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\(\frac{2\theta ^1\theta ^2- \theta ^1+2\theta ^2}{3}\)
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