Elastic modulus of iron single crystals as a function of orientation and temperature

H. K. D. H. Bhadeshia

Analytical model calculating orientation-dependent Young's modulus E in single crystals and determining isotropic polycrystalline tensile and shear bounds (Voigt, Reuss, and Hill) as a function of temperature across BCC (α-Fe), FCC (γ-Fe), and hexagonal close-packed (ε-Fe) iron.

The Zener anisotropy [1948, Elastic Properties of Single Crystals, University of Chicago Press] is the ratio between the highest and lowest shear resistance achievable in a cubic lattice, applying equally to ferrite (BCC body-centred cubic) and austenite (FCC face-centred cubic). The most difficult shear is on the cube plane and cube direction in that plane, whereas it is easiest to shear on a diagonal plane {110} along a face diagonal ⟨110⟩:

A = 2c44 c11 − c12 = 2(s11 − s12) s44

For hexagonal close-packed (ε-iron), the transverse isotropy demands five independent elastic stiffnesses (c11, c12, c13, c33, c44). Elastic anisotropy in HCP is governed by two transverse ratios: Abasal = 2c44 / (c11 − c12) and the axial ratio c33 / c11.

We do not deal with the shear modulus G as a function of orientation because that requires the specification of two orthogonal vectors, the plane and shear direction. There is no unique value of G for a single plane or direction. For example, shearing on the (110) plane in cubic iron gives:

  • G = c44 if sheared along [001]
  • G = 1 2 (c11 − c12) if sheared along [110]

Framework

Orientation dependence: cubic versus hexagonal crystals

In cubic single crystals (BCC α and FCC γ), elasticity is described by three independent constants (c11, c12, c44). The tensile Young's modulus along a direction with direction cosines [l1, l2, l3] is governed by [Dieter, Mechanical Metallurgy, 3rd edition, p. 59]:

1 E = s11 − 2 ( s11 − s12 − 1 2 s44 ) Γ

where Γ = l12l22 + l22l32 + l32l12 spans from 0 (⟨100⟩) to 1/3 (⟨111⟩).

In hexagonal close-packed ε-iron (stable at high pressures > 10–13 GPa, e.g. Earth's core conditions, and retained in thin films), transverse isotropy in the basal plane dictates that the Young's modulus depends strictly on the angle θ between the tensile axis and the hexagonal c-axis [0001] via l3 = cos θ:

1 E(θ) = (1 − l32)2s11 + l34s33 + l32(1 − l32)(2s13 + s44)

The very large modulus of ε-iron at 100 GPa pressure is because this causes the unit cell volume to decrease by 35%, pushing the iron atoms deep into the repulsion core of their interatomic potential well.

  • Voigt W. Ueber die Beziehung zwischen den beiden Elasticitätsconstanten isotroper Körper. Annalen der Physik. 274 (1889) 573-587.
  • Reuß A. Berechnung der Fließgrenze von Mischkristallen auf Grund der Plastizitätsbedingung für Einkristalle. ZAMM. 9 (1929) 49-58.
  • Hill, R. The Elastic Behaviour of a Crystalline Aggregate. Proc. Phys. Soc. A, 65 (1952) 349–354.
  • Mao H.K. et al. Phonon density of states of iron up to 153 GPa. Science 292 (2001) 914-916; Sha X. & Cohen R.E. First-principles thermoelasticity of ε-Fe. Phys. Rev. B 81 (2010) 094105.

Polycrystalline shear

In an untextured polycrystal with equiaxed, randomly orientated grains, effective isotropic moduli are estimated assuming:

(a) Voigt when a macroscopically uniform shear strain εij is applied across the external boundary of the polycrystal, every individual crystalline grain undergoes the exact same strain (εijgrain = εij).

(b) Reuss postulated that applying a macroscopically uniform stress σij to the boundary causes every single crystal in the aggregate to support the exact same stress state:

σijgrain = σij

(c) Hill realised that real polycrystalline grains neither deform with strictly uniform strain nor experience strictly uniform stress; Hill reasoned that the true effective modulus must lie within the bounds.

  • Cubic Voigt and Reuss shear bounds:
    GVoigt = (c11 − c12 + 3c44) 5 ,   GReuss = 5 4(s11 − s12) + 3s44
  • Hexagonal (HCP) Voigt and Reuss shear bounds:
    GVoigt = 2c11 + c33 − c12 − 2c13 + 6c44 + 3(c11 − c12) 30 ,  
    GReuss = 15 2(7s11 − 5s12 − 4s13 + 2s33) + 6s44
  • Hill average shear and tensile moduli:
    GHill = GVoigt + GReuss 2 ,   EHill = 9KHillGHill 3KHill + GHill
  • Curie magnetic anomaly: Below TC = 1043 K in α-Fe, spontaneous ferromagnetic ordering provides anomalous stiffness, producing an inflection as temperature approaches this Curie temperature.
Iron allotrope
Set absolute temperature: 298 K (25 °C)
0 K 298 K (Room) 1043 K (Curie) 1200 K
Set discrete temperature:
Temperature-dependent elastic constants A = 2.41
c₁₁(T)
231.4 GPa
s₁₁ = 7.60 1/TPa
c₁₂(T)
134.7 GPa
s₁₂ = -2.79 1/TPa
c₄₄(T)
116.4 GPa
s₄₄ = 8.59 1/TPa
Zener anisotropy factor (2c₄₄ / (c₁₁ − c₁₂)): 2.41
Crystallographic tensile axis [1 1 1]
Orientation factor sweep (Γ): 0.3333
0 [100] 0.10 0.25 [110] 0.333 [111]
Principal orientation presets:
Normalised direction cosines [l₁, l₂, l₃]: [0.577, 0.577, 0.577]
Orientation factor (Γ): 0.3333 (maximum)
Isotropic polycrystalline bounds Random texture
Voigt upper tensile bound (EVoigt): 228.4 GPa
Reuss lower tensile bound (EReuss): 194.2 GPa
Hill average Young's modulus (EHill): 211.3 GPa
Voigt upper shear bound (GVoigt): 89.2 GPa
Reuss lower shear bound (GReuss): 73.8 GPa
Hill average shear modulus (GHill): 81.5 GPa
Bulk modulus (K) and Poisson's ratio (ν): K = 166.9 GPa, ν = 0.296
Single-crystal tensile Young's modulus
284.1 GPa
Direction: [1 1 1] at 298 K
Modulus along [1 0 0]
131.6 GPa
Most compliant axis
Modulus along [1 1 1]
284.1 GPa
Stiffest axis
Effective GHill
81.5 GPa
Isotropic shear
Moduli comparisons
Selected direction ([1 1 1]) 284.1 GPa
Effective polycrystalline tensile modulus (EHill) 211.3 GPa
Effective polycrystalline shear modulus (GHill) 81.5 GPa
Single crystal along [1 0 0] 131.6 GPa
Single-crystal Young's modulus against orientation factor
Orientation factor spans 0 to 1/3 at current temperature
Single-crystal E(Γ)
Polycrystal EHill
Current orientation
Polycrystalline shear modulus against temperature
Effective shear modulus GHill across temperature range
Hill shear average GHill(T)
Voigt / Reuss bounds
Current state

Literature temperature coefficients and single-crystal elastic constants

Experimental and first-principles benchmarks for iron allotropes across temperature regimes, stated strictly in SI units.

Allotrope / phase Temperature & pressure Stiffness constants (GPa) E[min] E[max] Effective EHill Effective GHill Reference
BCC α-iron (ferrite) 0 K, 0 GPa c₁₁=243.1, c₁₂=138.1, c₄₄=121.9 139.7 GPa [100] 297.8 GPa [111] 222.1 GPa 85.7 GPa Dever, J. Appl. Phys. 43 (1972) 3293
BCC α-iron (ferrite) 298 K, 0 GPa c₁₁=231.4, c₁₂=134.7, c₄₄=116.4 131.6 GPa [100] 284.1 GPa [111] 211.3 GPa 81.5 GPa Rayne & Chandrasekhar, Phys. Rev. 122 (1961) 1714
BCC α-iron (Curie point) 1043 K, 0 GPa c₁₁=161.2, c₁₂=114.8, c₄₄=82.3 65.7 GPa [100] 198.8 GPa [111] 135.5 GPa 50.8 GPa Dever (1972) magnetic anomaly
FCC γ-iron (austenite) 1185 K (A₃), 0 GPa c₁₁=154.0, c₁₂=122.0, c₄₄=77.0 50.5 GPa [100] 220.5 GPa [111] 135.2 GPa 49.8 GPa Kamm & Alers (1964); Ledbetter (1981)
FCC γ-iron (austenite) 1400 K, 0 GPa c₁₁=138.8, c₁₂=112.4, c₄₄=69.1 42.8 GPa [100] 198.2 GPa [111] 120.4 GPa 44.1 GPa Ledbetter (1981) review
HCP ε-iron (hexaferrum) 300 K, 100 GPa pressure c₁₁=852, c₁₂=420, c₁₃=362, c₃₃=928, c₄₄=248 612 GPa (⊥ c) 736 GPa (|| c) 682 GPa 268 GPa Mao et al., Science 292 (2001) 914; Sha & Cohen, PRB 81 (2010)