Hertzian Contact Stress | General Calculator for Two Bodies in Contact

	Hertzian Contact: General Calculator
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Assumptions

1. Surfaces in contact are perfectly smooth.
2. Material limits are not exceeded.
3. Materials are homogeneous.
4. No frictional forces within the contact area.
5. Primary plane is the XZ-plane of the body's coordinate system.
6. Secondary plane is the YZ-plane of the body's coordinate system.
7. Radii are positive if the center of curvature lies within the given body.
8. As the bodies approach sphericity, the approximate and exact solutions converge.
9. When the bodies are far from being spherical, the calculations yield conservative results, by up to 30%.

Geometry

R₁		mm	Radius of body 1 in the primary plane

R₁^'		mm	Radius of body 1 in the secondary plane

R₂		mm	Radius of body 2 in the primary plane

R₂^'		mm	Radius of body 2 in the secondary plane

Φ		°	Angle between the primary planes of bodies 1 and 2

Applied Load

F		N	Preload Force

Material Properties
Body 1:
Mat:			Material
Type:		-	Material Type (Ductile or Brittle)
E		GPa	Modulus of Elasticity
ν		-	Poisson's ratio
σ_a		MPa	Allowable stress (tensile or flexural)


Body 2:
Mat:			Material
Type:		-	Material Type (Ductile or Brittle)
E		GPa	Modulus of Elasticity
ν		-	Poisson's ratio
σ_a		MPa	Allowable stress (tensile or flexural)
Equivalent and Allowable Properties

E_e		GPa	Effective Young's Modulus
D_e		mm	Equivalent diameter for the two bodies
P_f		MPa	Hertzian contact pressure at failure
Coefficients

cos(θ)		-	Cosine of the auxillary angle theta
α		-	transcendental function of the auxilliary angle, expressed in terms of elliptic integrals
β		-	transcendental function of the auxilliary angle, expressed in terms of elliptic integrals
λ		-	transcendental function of the auxilliary angle, expressed in terms of elliptic integrals

Results

F_c		N	Force normal to the contact zone
a		μm	X-radius of the elliptical contact
b		μm	Y-radius of the elliptical contact
e		-	Eccentricity of the elliptical contact, i.e., a >> b
P_max		MPa	Maximum Hertzian contact pressure
δ_c		μm	Deflection of the contact zone
δ_z		μm	Displacement of the top body
k_c		10⁶N/m	Contact zone stiffness


Body 1 Stress
σ_tmax,1a		MPa	Max tensile stress at radius a, Body 1
σ_tmax,1b		MPa	Max tensile stress at radius b, Body 1
τ_max,1		MPa	Max shear stress in Body 1
z₁		μm	Depth of max shear stress in Body 1


Body 2 Stress
σ_tmax,2a		MPa	Max tensile stress at radius a, Body 2
σ_tmax,2b		MPa	Max tensile stress at radius b, Body 2
τ_max,2		MPa	Max shear stress in Body 2
z₂		μm	Depth of max shear stress in Body 2

Limits: Based on Max Allowable Hertzian Stress / Pressure

F_max		N	Max allowable contact force

Directions

These calculators are designed to resemble spreadsheet calculators that can be printed as reports for project recordkeeping. The term project, with regards to these calculators, may refer to an engineered design, homework problem, tutorial problem, hobby design or self-study. The workflow for each calculator is from top to bottom.

Enter the diameter of the sphere or ball.
Enter the applied load or preload.
Select or enter the material properties for Body I, i.e., the sphere.
Select or enter the material properties for Body II, i.e., the flat plate.
Press the calculate button.
Print a report for your project recordkeeping by pressing the print button.

Assumptions

Surfaces in contact are perfectly smooth.
Material limits are not exceeced.
Materials are homogeneous.
No frictional forces within the contact area.
The primary plane is the XZ-plane of the body's coordiante system.
The secondary plane is the YZ-plane of the body's coordinate system.
Radii are positive if the center of curvature lies within the given body, i.e., the surface is convex and negative otherwise.
As the bodies approach sphericity, the approximate and exact solutions converge.
When the bodies are far from being spherical, the calculations yield convervative results, by up to 30%.

Equations

Equivalent Modulus of the Two Bodies
$$E_e = {1\over {{{1-\nu_1^2}\over E_1} + {{1-\nu_2^2}\over E_2}}}$$
Equivalent Diameter of the Two Bodies
$$D_e = \frac{3}{2} \frac{1}{\frac{1}{R_{1x}}+\frac{1}{R_{1y}}+\frac{1}{R_{2x}}+\frac{1}{R_{2y}}}$$
Radius of Elliptical Contact along X
$$a = \alpha\left(\frac{F_c D_e}{E_e}\right)^{1\over 3}$$
Radius of Elliptical Contact along Y
$$b = \beta\left(\frac{F_c D_e}{E_e}\right)^{1\over 3}$$
Eccentricity of the Elliptical Contact
$$e = \left(1-\frac{b^2}{a^2}\right)^{1\over2}\quad for \quad a \gg b$$
Maximum Hertzian Pressure
$$P_{max} = \frac{3F_c}{2\pi ab}$$
Deflection of the Contact Zone
$$\delta_c = \lambda\left(\frac{F_c^2}{D_eE_e^2}\right)^{1\over 3}$$
Stiffness
$$k_c = \frac{3}{2\lambda}\left({F_cD_eE_e^2}\right)^{1\over 3}$$
Max Tensile Stress at Radius a, Body 1
$$\sigma_{tmax, 1a} = P_{max}\left(1-2\nu_1\right)\frac{b}{ae^2}\left[\frac{1}{e} tanh^{-1}e-1\right]$$
Max Tensile Stress at Radius b, Body 1
$$\sigma_{tmax,1b} = P_{max}\left(1-2\nu_1\right)\frac{b}{ae^2}\left[1-\frac{b}{ae}tan^{-1}\left(\frac{ea}{b}\right)\right]$$
Max Tensile Stress at Radius a, Body 2
$$\sigma_{tmax,2a} = P_{max}\left(1-2\nu_2\right)\frac{b}{ae^2}\left[\frac{1}{e} tanh^{-1}e-1\right]$$
Max Tensile Stress at Radius b, Body 2
$$\sigma_{tmax,2b} = P_{max}\left(1-2\nu_2\right)\frac{b}{ae^2}\left[1-\frac{b}{ae}tan^{-1}\left(\frac{ea}{b}\right)\right]$$
Max Shear Stress in both Bodies
$$\tau_{max} = \left[0.3906\left(\frac{b}{a}\right)^5-1.1198\left(\frac{b}{a}\right)^4+1.2448\left(\frac{b}{a}\right)^3-0.7177\left(\frac{b}{a}\right)^2+0.2121\left(\frac{b}{a}\right)+.3\right]P_{max}$$
Depth of Max Shear Stress in both Bodies
$$z = \left[-0.651\left(\frac{b}{a}\right)^4+1.6262\left(\frac{b}{a}\right)^3-1.2726\left(\frac{b}{a}\right)^2-0.0077\left(\frac{b}{a}\right)+.7851\right]*b;$$
Max Allowable Contact Force
$$F_{c,~max} = \left(\frac{2\pi \alpha\beta P_{f}}{3}\right)^3\left(\frac{D_e}{E_e}\right)^2$$
$$cos\theta = \frac{2D_e}{3}\sqrt{\left(\frac{1}{R_{1x}}-\frac{1}{R_{1y}}\right)^2+\left(\frac{1}{R_{2x}}-\frac{1}{R_{2y}}\right)^2+2\left(\frac{1}{R_{1x}}-\frac{1}{R_{1y}}\right)\left(\frac{1}{R_{2x}}-\frac{1}{R_{2y}}\right)\cos2\phi}$$
Transcendental functions of the auxilliary angles, expressed in terms of elliptic integrals
$$for\quad cos\theta\le 0.9 $$

$$\alpha = 23.43 cos^5\theta-41.83 cos^4\theta+27.38 cos^3\theta-6.72 cos^2\theta+1.29 cos\theta+0.998$$

$$\beta = -0.999 cos^5\theta+1.77 cos^4\theta-1.33 cos^3\theta+0.59 cos^2\theta-0.68865 cos\theta+1$$

$$\lambda = -1.55 cos^5\theta2.68 cos^4\theta-1.76 cos^3\theta0.3 cos^2\theta-0.04 cos\theta+0.75$$

$$for\quad cos\theta>0.9 $$

$$\alpha = 6269014.5cos^5\theta-29371139.2cos^4\theta+55036391.2cos^3\theta-51557740.6cos^2\theta+24146276.0cos\theta-4522790.6;$$

$$\beta = -70684.5cos^5\theta+331436.0cos^4\theta-621625cos^3\theta+582926cos^2\theta-273306.3cos\theta+51254.0;$$

$$\lambda = -98958.3cos^5\theta+462760.4cos^4\theta-865562.5cos^3\theta+809436.1cos^2\theta-378446.2cos\theta+70770.7;$$

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Hertzian Contact: General Calculator

Directions

Assumptions

Equations

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