Hertzian pressure

Hertzian pressure between two curved bodies

The Hertzian pressure (after the German physicist Heinrich Hertz ) is understood to mean the greatest tension that prevails in the middle of the contact area between two elastic bodies.

If two rigid bodies with a curved surface ( cylinders or balls ) are pressed against each other, then in this idealized case they only touch each other in a linear or point-like manner. In the elastic body, however, the elasticity at the point of contact creates a flattening and a contact surface and a characteristic stress distribution ( surface pressure ) on the contact surface in both bodies .

According to Hertz, the size and shape of the contact area as well as the level and distribution of the mechanical stresses under the contact area can be calculated. The level of Hertzian pressure depends on the force with which the two bodies are pressed against one another, on their radii of curvature , their modulus of elasticity and the Poisson's ratio of their materials.

Shape of the contact surfaces:

If two spheres , a sphere and a plane or two crossed cylinders touch each other , a contact ellipse is created .
When two parallel cylinders or a cylinder touch a plane, a rectangular, elongated contact surface is created; one also speaks here of roller pressing.

requirements

Requirements for the calculation of the surface pressure according to the Hertz equations are

linear-elastic , homogeneous and isotropic materials
Contact surface flat and small (compared to the dimensions of the body)
Friction-free , no shear stresses in the contact area
the body can be used as elastic half spaces are considered

calculation

General

The Hertzian pressure on contact with curved surfaces is calculated accordingly

{\ displaystyle p _ {\ mathrm {max}} = {\ frac {1} {\ xi \ cdot \ eta}} \ cdot {\ frac {1} {\ pi}} \ cdot {\ sqrt [{3}] {{\ frac {3F \ cdot (\ sum k) ^ {2}} {8}} \ cdot \ left ({\ frac {E} {1- \ nu ^ {2}}} \ right) ^ {2 }}}}

where:

${\ displaystyle \ xi, \ eta}$ - Hertz coefficients for touching curved surfaces
${\ displaystyle F}$ - strength between the bodies
${\ displaystyle k}$ - Curvature = reciprocal of the radius
${\ displaystyle {\ frac {1 - {\ nu} ^ {2}} {E}} = {\ frac {1} {2}} \ cdot \ left ({\ frac {1 - {\ nu} _ { 1} ^ {2}} {E_ {1}}} + {\ frac {1 - {\ nu} _ {2} ^ {2}} {E_ {2}}} \ right)}$

${\ displaystyle {\ nu} _ {1,2}}$ - Poisson's number (also: Poisson's ratio) body 1, body 2

{\ displaystyle E_ {1,2}}

- E-modulus of the materials body 1, body 2.

Point contact ball - ball

For the simple case of contact between sphere and sphere:

{\ displaystyle p _ {\ mathrm {max}} = {\ frac {1} {\ pi}} \ cdot {\ sqrt [{3}] {{\ frac {1,5F} {r ^ {2}}} \ cdot \ left ({\ frac {E} {1- \ nu ^ {2}}} \ right) ^ {2}}}}

With

${\ displaystyle r = {\ frac {r_ {1} \ cdot r_ {2}} {r_ {1} + r_ {2}}} = {\ frac {1} {\ sum k}}}$ with - sphere radii sphere 1, sphere 2 ${\ displaystyle r_ {1,2}}$
${\ displaystyle E = 2 {\ frac {E_ {1} \ cdot E_ {2}} {E_ {1} + E_ {2}}}}$

special cases

Sphere - hollow sphere: If the larger sphere encloses the smaller one, a negative is used. ${\ displaystyle r_ {2}}$
Sphere - plane: and with it ${\ displaystyle r_ {2} \ rightarrow \ infty}$ ${\ displaystyle r = r_ {1}}$

Line contact cylinder - cylinder

For the simple case of contact between cylinder and cylinder (or plane), the following applies:

{\ displaystyle p _ {\ mathrm {max}} = {\ sqrt {{\ frac {F} {2 \ pi rl}} \ cdot {\ frac {E} {1- \ nu ^ {2}}}}} }