Directional Young's Modulus — Nb (Niobium)
Overview
The directional Young's modulus E(d) quantifies how material stiffness varies with loading direction. Each point in this 3D point cloud represents a direction vector d = (sin θ cos φ, sin θ sin φ, cos θ), with the radial distance from origin equal to E(d). The color mapping (icefire) shows stiffness magnitude—warmer colors indicate higher stiffness.
Material Properties
Material Type: Cubic crystal
Elastic Constants (GPa): c₁₁ = 240.2, c₁₂ = 125.6, c₄₄ = 28.2
Cubic crystals exhibit 8-lobed symmetric patterns reflecting cubic symmetry. Key directions: ⟨100⟩ (cube edges), ⟨110⟩ (face diagonals), ⟨111⟩ (body diagonals).
Computation Method
Starting from the 6×6 stiffness matrix C in Voigt notation: (1) compute compliance S = C⁻¹, (2) sample unit sphere uniformly (200×200 grid), (3) for each direction d, apply unit stress σ = d ⊗ d, (4) compute strain ε = Sσ, (5) evaluate E(d) = 1/(d·ε·d), (6) plot r(θ,φ) = E(θ,φ)·d as point cloud.
Interpreting Results
The shape reveals material anisotropy. For cubic materials, symmetries manifest as 8-lobed patterns. Rotate the plot to explore crystallographic directions (⟨100⟩, ⟨110⟩, ⟨111⟩) and observe how orientation affects structural response—critical for single-crystal or textured material design.