1. Global Parameters
2. Body 1 Configuration
3. Body 2 Configuration
Hertzian Contact Footprint Summary
Major Axis (a)
Elliptical, semi-major axis from bisection on \(k=a/b\) via elliptic integrals:
\[a = \left(\frac{3F\,k^2\,\mathcal{E}(e)}{\pi E^*(A_H+B_H)}\right)^{\!1/3}\]Line contact: equals cylinder length \(L\).
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\[a = \left(\frac{3F\,k^2\,\mathcal{E}(e)}{\pi E^*(A_H+B_H)}\right)^{\!1/3}\]Line contact: equals cylinder length \(L\).
Minor Axis (b)
Elliptical: \(b = a/k\), aspect ratio \(k\geq1\).
Line contact half-width:
\[b = \sqrt{\frac{4F\,R^*}{\pi L E^*}}\]with \(R^* = \bigl(1/R_{1x}+1/R_{2x}\bigr)^{-1}\).
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Line contact half-width:
\[b = \sqrt{\frac{4F\,R^*}{\pi L E^*}}\]with \(R^* = \bigl(1/R_{1x}+1/R_{2x}\bigr)^{-1}\).
Max. Pressure (P₀)
Peak contact pressure at the footprint centre.
Elliptical: \[P_0 = \frac{3F}{2\pi ab}\]Line contact: \[P_0 = \frac{2F}{\pi b L}\]Boundary condition for all subsurface stress calculations.
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Elliptical: \[P_0 = \frac{3F}{2\pi ab}\]Line contact: \[P_0 = \frac{2F}{\pi b L}\]Boundary condition for all subsurface stress calculations.
Approach (\(\delta\))
Elastic approach; mutual convergence of remote points.
Elliptical: \[\delta = \frac{3F}{2\pi a E^*}\,\mathcal{K}(e)\]Line contact per body \(i\):
\[\delta_i = \frac{2F}{\pi L E'_i}\!\left[\ln\frac{2R^*}{b} - \frac{1-2\nu_i}{2}\right]\]
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Elliptical: \[\delta = \frac{3F}{2\pi a E^*}\,\mathcal{K}(e)\]Line contact per body \(i\):
\[\delta_i = \frac{2F}{\pi L E'_i}\!\left[\ln\frac{2R^*}{b} - \frac{1-2\nu_i}{2}\right]\]
Body 1 Stress Results
Surface stress (σz)
Normal stress perpendicular to the contact surface at \(z=0\):
\[ \sigma_z^{\mathrm{surf}} = -P_0 \] This is the maximum compressive normal stress at the contact surface.
\[ \sigma_z^{\mathrm{surf}} = -P_0 \] This is the maximum compressive normal stress at the contact surface.
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MPa
Surface stress (σx)
Surface stress in the \(x\)-direction:
\[ \sigma_x = -P_0 \left[ 2\nu + (1-2\nu) \frac{b}{a+b} \right] \]
\[ \sigma_x = -P_0 \left[ 2\nu + (1-2\nu) \frac{b}{a+b} \right] \]
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MPa
Surface stress (σy)
Surface stress in the \(y\)-direction:
\[ \sigma_y = -P_0 \left[ 2\nu + (1-2\nu) \frac{a}{a+b} \right] \] For circular contact: \[ \sigma_x = \sigma_y = -P_0 \frac{1+2\nu}{2} \]
\[ \sigma_y = -P_0 \left[ 2\nu + (1-2\nu) \frac{a}{a+b} \right] \] For circular contact: \[ \sigma_x = \sigma_y = -P_0 \frac{1+2\nu}{2} \]
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MPa
Max. shear stress (τ1)
Maximum shear stress according to the Tresca criterion:
\[ \tau_1 = \frac{\sigma_1-\sigma_3}{2} \]
\[ \tau_1 = \frac{\sigma_1-\sigma_3}{2} \]
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@ z =
0.00
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Max. octahedral stress (τoct)
Octahedral shear stress:
\[ \tau_{\mathrm{oct}} = \frac{1}{3} \sqrt{ (\sigma_1-\sigma_2)^2 + (\sigma_2-\sigma_3)^2 + (\sigma_3-\sigma_1)^2 } \]
\[ \tau_{\mathrm{oct}} = \frac{1}{3} \sqrt{ (\sigma_1-\sigma_2)^2 + (\sigma_2-\sigma_3)^2 + (\sigma_3-\sigma_1)^2 } \]
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MPa
@ z =
0.00
mm
Max. von Mises stress (σvm)
Von Mises equivalent stress:
\[ \sigma_{\mathrm{VM}} = \sqrt{ \frac{1}{2} \left[ (\sigma_x-\sigma_y)^2 + (\sigma_y-\sigma_z)^2 + (\sigma_z-\sigma_x)^2 \right] } \]
\[ \sigma_{\mathrm{VM}} = \sqrt{ \frac{1}{2} \left[ (\sigma_x-\sigma_y)^2 + (\sigma_y-\sigma_z)^2 + (\sigma_z-\sigma_x)^2 \right] } \]
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MPa
Max. orthogonal stress (τzx)
Reversing orthogonal shear in the \(xz\)-plane, evaluated at
\((x=a,\;y=0)\):
\[ \frac{a^2}{a^2+\lambda_1} + \frac{z^2}{\lambda_1} = 1 \]
\[ \frac{a^2}{a^2+\lambda_1} + \frac{z^2}{\lambda_1} = 1 \]
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MPa
@ z =
0.00
mm
Max. orthogonal stress (τyz)
Reversing orthogonal shear in the \(yz\)-plane, evaluated at
\((x=0,\;y=b)\):
\[ \frac{b^2}{b^2+\lambda_1} + \frac{z^2}{\lambda_1} = 1 \]
\[ \frac{b^2}{b^2+\lambda_1} + \frac{z^2}{\lambda_1} = 1 \]
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MPa
@ z =
0.00
mm
Body 2 Stress Results
Surface stress (σz)
Normal stress perpendicular to the contact surface at \(z=0\):
\[ \sigma_z^{\mathrm{surf}} = -P_0 \] This is the maximum compressive normal stress at the contact surface.
\[ \sigma_z^{\mathrm{surf}} = -P_0 \] This is the maximum compressive normal stress at the contact surface.
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MPa
Surface stress (σx)
Surface stress in the \(x\)-direction:
\[ \sigma_x = -P_0 \left[ 2\nu + (1-2\nu) \frac{b}{a+b} \right] \]
\[ \sigma_x = -P_0 \left[ 2\nu + (1-2\nu) \frac{b}{a+b} \right] \]
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MPa
Surface stress (σy)
Surface stress in the \(y\)-direction:
\[ \sigma_y = -P_0 \left[ 2\nu + (1-2\nu) \frac{a}{a+b} \right] \] For circular contact: \[ \sigma_x = \sigma_y = -P_0 \frac{1+2\nu}{2} \]
\[ \sigma_y = -P_0 \left[ 2\nu + (1-2\nu) \frac{a}{a+b} \right] \] For circular contact: \[ \sigma_x = \sigma_y = -P_0 \frac{1+2\nu}{2} \]
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MPa
Max. shear stress (τ1)
Maximum shear stress according to the Tresca criterion:
\[ \tau_1 = \frac{\sigma_1-\sigma_3}{2} \]
\[ \tau_1 = \frac{\sigma_1-\sigma_3}{2} \]
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MPa
@ z =
0.00
mm
Max. octahedral stress (τoct)
Octahedral shear stress:
\[ \tau_{\mathrm{oct}} = \frac{1}{3} \sqrt{ (\sigma_1-\sigma_2)^2 + (\sigma_2-\sigma_3)^2 + (\sigma_3-\sigma_1)^2 } \]
\[ \tau_{\mathrm{oct}} = \frac{1}{3} \sqrt{ (\sigma_1-\sigma_2)^2 + (\sigma_2-\sigma_3)^2 + (\sigma_3-\sigma_1)^2 } \]
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MPa
@ z =
0.00
mm
Max. von Mises stress (σvm)
Von Mises equivalent stress:
\[ \sigma_{\mathrm{VM}} = \sqrt{ \frac{1}{2} \left[ (\sigma_x-\sigma_y)^2 + (\sigma_y-\sigma_z)^2 + (\sigma_z-\sigma_x)^2 \right] } \]
\[ \sigma_{\mathrm{VM}} = \sqrt{ \frac{1}{2} \left[ (\sigma_x-\sigma_y)^2 + (\sigma_y-\sigma_z)^2 + (\sigma_z-\sigma_x)^2 \right] } \]
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MPa
Max. orthogonal stress (τzx)
Reversing orthogonal shear in the \(xz\)-plane, evaluated at
\((x=a,\;y=0)\):
\[ \frac{a^2}{a^2+\lambda_1} + \frac{z^2}{\lambda_1} = 1 \]
\[ \frac{a^2}{a^2+\lambda_1} + \frac{z^2}{\lambda_1} = 1 \]
0.0
MPa
@ z =
0.00
mm
Max. orthogonal stress (τyz)
Reversing orthogonal shear in the \(yz\)-plane, evaluated at
\((x=0,\;y=b)\):
\[ \frac{b^2}{b^2+\lambda_1} + \frac{z^2}{\lambda_1} = 1 \]
\[ \frac{b^2}{b^2+\lambda_1} + \frac{z^2}{\lambda_1} = 1 \]
0.0
MPa
@ z =
0.00
mm