Axisymmetric Rigid Indenter Contact Suite
Contact inputs
Compression and downward displacement are reported as positive.
Contact result
Linear-elastic half-space estimate
The ideal sharp punch edge has an inverse-square-root pressure singularity.
Normalized pressure distribution
Normalized surface displacement
How to use
- Select Flat punch, Cone, or Flat-tipped cone.
- Enter reduced modulus E* in MPa. For a rigid indenter on one elastic body, use E/(1−ν²).
- Choose whether indentation depth or normal load is the prescribed quantity.
- Enter the prescribed depth in µm or load in N.
- Enter the profile geometry: punch radius a, cone slope θ measured from the surface, or flat-tip radius b.
- Enter radial position r in mm to evaluate local pressure and downward surface displacement.
- Select Calculate; the form also updates as values change. Use Reset example to restore the flat-punch example.
- Read the canonical load result, secondary contact metrics, and normalized pressure/displacement plots. “Singular” identifies an ideal sharp-feature limit rather than a finite physical stress.
- Correct any range warning before using the result. Do not use this model for plastic, adhesive, rough, coated, frictional, finite-thickness, or strongly non-axisymmetric contact.
Equations used
The rigid indenter presses normally into a homogeneous isotropic linear-elastic half-space. Compression, pressure, indentation, and downward displacement are positive. The circular contact radius is a, radial coordinate is r, cone slope from the undeformed surface is θ, flat-tip radius is b, and reduced modulus is E*.
Symbols and units
The equations use SI working units. E* is reduced modulus and E is Young’s modulus. ν is Poisson’s ratio and is dimensionless. SI working-unit convention: E* and E are in Pa; FN is in N; d, a, b, r, x, f, g, and w are in m; kN is in N/m; p̄ and p(r) are in Pa; θ and φ are in radians; ν and g′ are dimensionless. FN is compressive normal load; d is indentation depth, a is contact radius (and the prescribed punch radius for a flat punch), b is flat-tip radius, and r is evaluation radius.
kN is normal contact stiffness in N/m. p̄ is mean contact pressure and p(r) is local normal pressure, both in Pa. w(r) is downward surface displacement in m. θ is cone slope measured from the undeformed surface. φ is the flat-tipped-cone contact angle parameter defined by φ=acos(b/a). θ and φ are in radians in the equations and are dimensionless angular measures.
For the flat-tipped cone, f(r) is the indenter profile height in m, x is the radial integration variable in m, g(x) is the transformed profile function in m, and g′(x) is its derivative with respect to x and is dimensionless. The pressure and outside-displacement integrals use deterministic endpoint-transformed quadrature.
Inputs and results are displayed in MPa, N, mm, µm, degrees, MN/m, and MPa as labeled; the implementation converts them to SI before evaluation.
Rigid indenter: E* = E/(1 − ν²)
All modes: kN = dFN/dd = 2E*a; p̄ = FN/(πa²)Flat cylindrical punch
FN = 2E*ad
p(r) = E*d/[π√(a² − r²)], 0 ≤ r < a
w(r) = d, r ≤ a
w(r) = (2d/π) asin(a/r), r > aThe fixed sharp edge r=a has an inverse-square-root pressure singularity.
Complete cone
d = (π/2)a tan θ
FN = (π/2)E*a² tan θ = (2/π)E*d²/tan θ
p(r) = (E*tan θ/2) acosh(a/r), 0 < r ≤ a
w(r) = d − r tan θ, r ≤ a
w(r) = tan θ[a asin(a/r) − r + √(r² − a²)], r > aThe ideal apex r=0 has a logarithmic pressure singularity.
Flat-tipped cone
f(r)=0 for r≤b; f(r)=(r−b)tan θ for r>b
φ = acos(b/a)
d = a tan θ φ
FN = E*a² tan θ[φ + sin φ cos φ]
g(x)=0 for x≤b; g(x)=x tan θ acos(b/x) for x>b
p(r)=(E*/π)∫max(r,b)a g′(x)/√(x²−r²) dx
g′(x)=tan θ[acos(b/x)+b/√(x²−b²)]
w(r)=d−f(r), r≤a
w(r)=(2/π)∫0a[d−g(x)]/√(r²−x²) dx, r>aThe pressure and outside-displacement integrals are evaluated deterministically by endpoint-transformed quadrature. The sharp profile transition r=b is singular.
Assumptions and limits
- Rigid axisymmetric indenter; frictionless, non-adhesive normal contact.
- Homogeneous isotropic linear elasticity, small strain, and a small cone slope.
- The contact radius must be small relative to body thickness and lateral dimensions.
- Ideal sharp features generate mathematical pressure singularities; real tip radii, yielding, roughness, or coatings regularize them.
- No plasticity, viscoelasticity, finite-thickness, residual-stress, thermal, or dynamic correction is included.