Directional Young's Modulus — NiTi Alloy

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

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.