Multiscale Coupled Analysis
Theory: effective properties from microstructure
Overview
When analysing composite parts, do you model each fibre?
That isn't practical. Carbon fibres are about 7 μm across, so even a 2 mm plate has over 200 through its thickness; meshing a whole part fibre by fibre would need an enormous number of elements. Instead an equivalent smeared material (homogeneous, anisotropic) is used, with constants derived from the microstructure — by simple formulas or by finite-element analysis of a small representative volume (RVE) of fibres and resin. Multiscale coupling solves the part and RVEs back and forth; the method is chosen by the accuracy needed.
Simple homogenisation formulas
V_f and V_m are fibre and matrix volume fractions; E_f1 and E_f2 fibre axial and transverse moduli; E_m matrix modulus. Halpin–Tsai ξ is 2 for transverse stiffness with circular fibres. Values: carbon fibre E_f1 = 230 GPa, E_f2 = 15 GPa; epoxy E_m = 3.5 GPa.
Voigt and Reuss bounds
For the average stiffness of mixtures, Voigt (1889) averaged assuming equal strain in each constituent and Reuss (1929) assuming equal stress. Hill later showed the true value lies between them, making them upper and lower bounds. In the 1960s Hashin and Shtrikman derived tighter bounds and Halpin and Tsai gave convenient formulas matching experiments. RVE numerical analysis now dominates, but these formulas remain sanity checks for results.
Worked examples
Example 1: effective moduli of unidirectional CFRP
| Fibre fraction | Axial E1 | Transverse E2 (series) | Transverse E2 (Halpin–Tsai) | Transverse E2 (parallel, upper) |
|---|---|---|---|---|
| 40% | 94.1 GPa | 5.05 GPa | 6.28 GPa | 8.1 GPa |
| 60% | 139.4 GPa | 6.48 GPa | 8.30 GPa | 10.4 GPa |
Example 2: wrongly assuming isotropic fibres (60% fibre)
| Fibre transverse modulus | E2 (series) | E2 (Halpin–Tsai) |
|---|---|---|
| 15 GPa (correct anisotropy) | 6.48 GPa | 8.30 GPa |
| 230 GPa (assumed isotropic) | 8.55 GPa | 17.61 GPa |
Thermal conductivity (fibre axial 10, transverse 1, resin 0.2 W/mK; 60% fibre): 6.08 W/mK along fibres, 0.38 transverse in series and 0.47 W/mK by Maxwell-Garnett.
Axial stiffness is the same whatever the formula, yet transverse varies this much with formula and assumption.
Along the fibres, fibre and resin stretch equally, so the simple stiffness average (rule of mixtures) is nearly right. Transversely, load passes alternately through fibre and resin, so the soft resin dominates, and fibre arrangement and stress concentration in the resin matter. The series formula is a lower bound and the parallel an upper bound, with the true value between; Halpin–Tsai is an intermediate estimate fitted to experiments. Carbon fibre itself is anisotropic, so ignoring that doubles the transverse estimate. If transverse loads or heat matter in the part, solve an RVE of the fibre arrangement by finite elements and compare with measurements to set material constants.
Multiscale workflow
- Gather constituent properties (fibre, matrix, interface) including anisotropy.
- Use simple formulas for bounds and estimates.
- Analyse an RVE of the fibre arrangement with periodic boundary conditions.
- Use effective constants in the part analysis, returning local strains to RVEs for local stress checks if needed.
- Validate constants against coupon tests.
“A part that cracked transversely”
In one CFRP part, cracks appeared earlier than expected in testing where load acted across the fibres. The analysis had treated fibres as isotropic, roughly doubling transverse stiffness, misjudging load sharing and resin stress. Recomputing effective constants with an RVE including fibre anisotropy, and checking local resin stress inside the RVE, explained the crack location and load.
Common mistakes
Mistakes and fixes
| Mistake | Effect | Fix |
|---|---|---|
| Assuming isotropic fibres | Transverse overestimated | Include fibre anisotropy |
| Rule of mixtures transversely | Stiffness greatly overestimated | Series or Halpin–Tsai |
| Improper RVE boundaries | Biased constants | Periodic boundaries |
| Ignoring local stress | Matrix failure missed | Check back in the RVE |
| Isotropic conductivity | Wrong temperatures | Give directional values |
I'd like to learn related topics.
Related pages include model order reduction, data-driven multiphysics, data transfer between non-matching meshes, staggered coupling and convergence of coupled solves.
Related Topics
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