3D Elastic Body Analysis
3D Elastic Body Analysis: Theoretical Foundations
Fundamentals of 3D Elasticity Theory
Professor, in the end, FEM structural analysis is solving "3D elasticity theory," right?
Yes. Plane stress, plane strain, axisymmetric, shell, beam... all structural elements are special cases of 3D elasticity theory. 3D elasticity theory is the foundation of everything, the "full model" without approximations.
Governing Equations
Please tell me the governing equations of 3D elasticity theory.
There are three fundamental equations.
1. Equilibrium Equations (Force Balance)
2. Strain-Displacement Relations (Compatibility Conditions)
3. Constitutive Law (Hooke's Law)
6 stress components ($\sigma_x, \sigma_y, \sigma_z, \tau_{xy}, \tau_{yz}, \tau_{xz}$), 6 strain components, 3 displacement components. That's 15 equations for 15 unknowns, right?
Perfect understanding. Consolidating these 15 equations with displacement as the only unknown gives the Navier equations (Lamé–Navier equations):
Here $\lambda, \mu$ are Lamé constants. $\mu = G$ (shear modulus), $\lambda = \frac{E\nu}{(1+\nu)(1-2\nu)}$.
D Matrix for Isotropic Elastic Materials
What about the $[D]$ matrix used in FEM?
3D constitutive law for isotropic elastic materials (Voigt notation):
So this 6×6 matrix is the foundation of everything. The $[D]$ for plane stress or plane strain is a contraction of this.
Correct. From the 3D $[D]$, plane stress contracts under the condition $\sigma_z = 0$, and plane strain under the condition $\varepsilon_z = 0$. Shells and beams are also derived from the 3D $[D]$ based on their respective assumptions.
Anisotropic Materials
What about non-isotropic materials?
For general anisotropic elastic materials, $[D]$ has 21 independent constants (6×6 symmetric matrix). However, in practice, the following special cases are common:
| Material Symmetry | Independent Constants | Examples |
|---|---|---|
| Isotropic | 2 ($E, \nu$) | Steel, Aluminum |
| Transversely Isotropic | 5 | Unidirectional reinforced CFRP layer, sedimentary soil |
| Orthotropic | 9 | Wood, woven CFRP |
| General Anisotropy | 21 | Crystals |
Is CFRP (carbon fiber composite) transversely isotropic?
A single layer of unidirectional material (UD material) is transversely isotropic. It is symmetric about the fiber direction and the plane orthogonal to it. When multiple layers are laminated, the whole becomes orthotropic or exhibits more complex symmetry.
Scenarios Where 3D Analysis is Essential
What are scenarios where 2D approximations cannot be used and 3D analysis is necessary?
There are surprisingly many problems that "cannot be solved without 3D," aren't there?
With the improvement of computer performance, 3D analysis has become commonplace. However, solving in 3D does not guarantee correctness. If mesh quality, boundary conditions, and material models are not correct, you'll get garbage even in 3D.
Summary
Let me organize the theory of 3D elastic body analysis.
Key Points:
- Three Fundamental Equations — Equilibrium, compatibility, constitutive law. Total 15 unknowns, 15 equations.
- The $[D]$ Matrix is the Foundation of Everything — 2D elements are contracted versions of this.
- 2 Constants for Isotropic, Up to 21 Constants for Anisotropic
- Scenarios Where 3D Analysis is Essential — 3D stress concentration, through-thickness stress, contact, complex shapes.
- 3D ≠ Accurate — The quality of input governs the results.
All FEM element theories boil down to 3D elasticity theory. Understanding this, it seems I can naturally judge which element to use.
Exactly. 3D elasticity theory is the "trunk," and each element type is a "branch." If you understand the trunk, you naturally know how to choose the branches.
Governing Equations of 3D Elasticity
The equilibrium equations for three-dimensional elastic bodies were independently derived by Navier and Cauchy in the 1820s. The generalized Hooke's law (σ=Cε, where C is the fourth-order elasticity tensor) connecting the 9-component stress tensor σij and the 6-component strain tensor εij has a maximum of 21 independent components C11 to C66. For isotropic materials, this reduces to the two parameters of Lamé constants λ and μ, which can be converted using Young's modulus E=μ(3λ+2μ)/(λ+μ).
Computational Methods for 3D Elastic Body Analysis
3D Solid Elements
What kinds of 3D solid elements are there?
There are three basic shapes: hexahedron (hex), tetrahedron (tet), and pentahedron (wedge/prism).
| Element | 1st Order | 2nd Order | Accuracy | Mesh Generation |
|---|---|---|---|---|
| Tetrahedron | TET4 (4 nodes) | TET10 (10 nodes) | TET4: Low / TET10: High | Easy automatic meshing |
| Hexahedron | HEX8 (8 nodes) | HEX20 (20 nodes) | HEX8: Medium / HEX20: Very High | Difficult automatic meshing |
| Pentahedron | WEDGE6 | WEDGE15 | Medium to High | Used for transition between hex and tet |
Is TET4 low accuracy?
TET4 is a constant strain element (the 3D version of CST). Strain is constant within the element, so it cannot represent stress gradients. TET4 should not be used in practice. TET10 (quadratic tetrahedron) provides sufficient accuracy.
But automatic meshing generates TET4 more easily, right?
That's true, but TET4 for accurate...
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