Estimation of Hertzian Contact Pressure during Nanoindentation

Material Testing laboratory

Experiment Procedure

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  1. Select the material for testing.
  2. Enter the indentation load and dwell time, then click Submit.
  3. Open the 2D view and click Indent to start the virtual indentation.
  4. Observe crack formation in Top View and 3D View.
  5. Use Measure Crack Length to view crack images and length table.
  6. Check the Result and Conclusion for final toughness values.

Simulation Area

Protective Enclosure Indenter Head Berkovich Indenter Optical Microscope Sample Stage Sample

Observations

Test Parameters

Strain Rate 0.05 s⁻¹
Load Rate 0.2 mN/s
Displacement Rate 10 nm/s
Thermal Drift ≤ 0.05 nm/s
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Hertzian Contact Formulae

Maximum Hertzian Pressure

p₀ = 3P 2πa² = 1.5 pₘ

Pressure Distribution

p(r) = p₀   ( 1 - r² a² )1/2

Contact Radius

a = ( 3PR 4E* )1/3

where, P = applied load and R = indenter tip radius


1 E* = 1 − νₛ² Eₛ + 1 − νᵢ² Eᵢ

where,

  • Eₛ, νₛ — Young’s modulus and Poisson’s ratio of specimen
  • Eᵢ, νᵢ — Young’s modulus and Poisson’s ratio of indenter
  • E* — reduced (effective) modulus
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Conclusions

  • Alumina, due to its higher stiffness, deforms less under load, producing a smaller contact area and consequently higher Hertzian contact pressure.
  • The elevated pressure generates larger subsurface stresses that promote crack initiation and brittle fracture rather than plastic flow.
  • Mild steel, being more compliant, spreads the load over a larger area, reducing contact pressure and encouraging plastic deformation.
  • Thus, Hertzian pressure helps predict wear mode: ductile (plastic flow) in mild steel and brittle (cracking/chipping) in alumina.
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Material Comparison Results

Property Mild Steel Alumina Remarks
Specimen modulus Eₛ (GPa) 210 380 Alumina is stiffer than MS
Poisson’s ratio νₛ 0.30 0.22 —
Max depth hₘₐₓ (nm) 42 32 —
Stiffness S = dP/dh (kN/m) 177 237 Alumina is stiffer than MS
Reduced modulus E* (GPa) 192 296 Alumina is stiffer than MS
Load P (mN) 5 5 constant
Tip radius R (µm) 5 5 constant
Contact radius a (µm) 0.46 0.40 Decreases with stiffness
Contact area A = πa² (m²) 6.66 × 10⁻¹³ 4.96 × 10⁻¹³ Decreases with stiffness
Mean pressure pₘ (GPa) 7.5 10.1 Increases with stiffness
Max pressure p₀ (GPa) 11.3 15.2 Increases with stiffness
Max shear stress τₘₐₓ (GPa) 3.5 4.7 Increases with stiffness

Do you Know estimation of H & E using p-h curve obtained during Nanoindentation