Characteristic Length in Composite Damage Models

What Is It?

The characteristic length is a length scale associated with each finite element that is used in fracture-energy-based damage evolution to regularise the mesh dependency. It relates the element's energy dissipation to the fracture energy per unit crack area. The characteristic length converts the fracture energy (per unit area, in J/m²) to the energy per element (in J), ensuring that the total energy dissipated is correct regardless of element size.

Why It Matters

The characteristic length is the key parameter that makes fracture-energy-based regularisation work. Without it, the fracture energy cannot be correctly related to the element-level energy dissipation. The definition and computation of the characteristic length vary between solvers, and understanding the specific implementation is essential for correct regularisation. An incorrectly defined characteristic length defeats the purpose of the regularisation and produces mesh-dependent results.

Element Dimension

The characteristic length is typically based on the element dimension. For 2D elements, it may be the square root of the element area. For 3D elements, the cube root of the volume. For shell elements, it may be based on the in-plane element dimension. Some solvers use the element side length, others use a more complex function of the element geometry. The specific definition affects how the regularisation works and should be understood for the solver being used.

Characteristic length definitions (examples):

2D element:  l_c = √(A_element)  (square root of area)
3D element:  l_c = ∛(V_element)  (cube root of volume)
Shell element: l_c = √(A_element)  (in-plane)

Some solvers use the element side length
or a more complex function of geometry.

The specific definition is solver-dependent —
document and understand it.

Energy Regularisation

The characteristic length regularises the energy by relating the fracture energy G_c (per unit area) to the energy dissipated per element. The failure displacement is computed as δ_fail = 2 × G_c / (σ_init × l_c). For a smaller element (smaller l_c), the failure displacement is larger — the element softens over a longer displacement, dissipating more energy per unit volume. Because the element is smaller, the total energy (volume × energy density) is the same as for a larger element. This is how the regularisation achieves mesh objectivity.

Energy regularisation with characteristic length:

δ_fail = 2 × G_c / (σ_init × l_c)

For smaller element (smaller l_c):
  δ_fail is LARGER → element softens over more displacement
  Energy density is HIGHER (more energy per unit volume)
  But element volume is SMALLER
  Total energy = volume × energy density = SAME

For larger element (larger l_c):
  δ_fail is SMALLER → element softens over less displacement
  Energy density is LOWER
  But element volume is LARGER
  Total energy = SAME

This is how mesh objectivity is achieved.

Implementation Considerations

Several implementation considerations affect the characteristic length. The definition varies between solvers — some use the square root of area, others use a different function. For elements with high aspect ratios, the characteristic length may not be representative of the damage direction — a long thin element may have a characteristic length that does not match the crack direction. For cohesive elements, the characteristic length is typically the element thickness (the interface separation direction). The implementation should be verified with single-element tests.

• Definition varies between solvers — understand the specific implementation
• High aspect ratio elements — characteristic length may not match crack direction
• Cohesive elements — characteristic length is typically the element thickness
• Verify with single-element tests — check that energy = G_c / l_c

Effect on Mesh Dependency

The characteristic length directly affects mesh dependency. When correctly implemented, it eliminates mesh dependency — the results are objective. When incorrectly implemented or not used, the results remain mesh-dependent. The verification is straightforward: run the analysis with two mesh densities and compare. If the characteristic length is working correctly, the results should be similar. If not, the results will differ, and the implementation should be investigated.

The characteristic length directly controls mesh objectivity. Correct implementation eliminates mesh dependency. Verify by running two mesh densities — results should be similar. If they differ, investigate the implementation.

Practical Guidance

For practical analysis, the following guidance applies. Use a solver that supports fracture-energy-based evolution with a characteristic length — most commercial codes do. Understand the specific definition of l_c in the solver. Use elements with reasonable aspect ratios — avoid highly distorted elements where l_c may not be representative. Verify mesh objectivity with two mesh densities. Report the characteristic length definition and the mesh objectivity verification in the analysis documentation.

• Use a solver that supports fracture-energy-based evolution with characteristic length
• Understand the specific l_c definition in the solver
• Use reasonable element aspect ratios — avoid highly distorted elements
• Verify mesh objectivity with two mesh densities
• Report l_c definition and objectivity verification in documentation

Key Takeaways

• Characteristic length l_c relates element volume to fracture energy per unit area
• Typically based on element dimension — √area (2D), ∛volume (3D), or side length
• Failure displacement: δ_fail = 2G_c/(σ_init × l_c) — larger for smaller elements
• Achieves mesh objectivity — total energy is the same regardless of element size
• Definition is solver-dependent — understand, verify and document the implementation

Engineering judgement — what can change the conclusion

For Characteristic Length in Composite Damage Models, the harmonised review should concentrate on regularising damage so dissipated energy is mesh-objective while the characteristic dimension remains physically consistent with the element formulation and crack-band assumption. The engineering value comes from identifying the assumptions that can move the governing margin or failure mode, then testing those assumptions deliberately rather than adding complexity indiscriminately. Where simplified and high-fidelity methods coexist, the simpler method should be used as an independent trend or magnitude check so that agreement is based on physics rather than shared modelling assumptions.