Last updated Aug. 21th, 2019
- F | the force, |
- x | the chain extension, |
- p | the persistence length, |
- L | the contour length, |
- kB | the Boltzmann constant, |
- T | the absolute temperature. |
- F | the force, |
- x | the chain extension, |
- a | the Kuhn length, |
- L | the contour length, |
- kB | the Boltzmann constant, |
- T | the absolute temperature. |
- F | the force, |
- x | the chain extension, |
- r | the rotating unit length, |
- L | the contour length, |
- kB | the Boltzmann constant, |
- T | the absolute temperature. |
\[F_{ Sphere } = \frac{ 4 }{ 3 } \frac{ E }{ 1 - \nu^2 } \sqrt{R} \; \delta^{ 3 / 2 }\] | \[F_{ Cone } = \frac{ 2 }{ \pi } \frac{ E }{ 1 - \nu^2 } \; \operatorname{ tan } \alpha \; \delta^2\] | |
\[\operatorname{ log } ( F_{ Sphere } ) = \frac{ 3 }{ 2 } \; \operatorname{ log } \; ( \delta ) + \operatorname{ log } \left( \frac{ 4 }{ 3 } \frac{ E }{ 1 - \nu^2 } \sqrt{R} \right)\] | \[\operatorname{ log } ( F_{ Cone } ) = 2 \; \operatorname{ log } \; ( \delta ) + \operatorname{ log } \left( \frac{ 2 }{ \pi } \frac{ E }{ 1 - \nu^2 } \; \operatorname{ tan } \alpha \right)\] | |
which is under the form of | ||
\[F = a \; D + E\] |
- F | the force, |
- d | the sample deformation, |
- E | the Youngs modulus, |
- n | the Poisson ratio, |
- R | the radius of the indenting sphere, |
- a | the opening angle of the indenting cone. |
\[F_{ Sphere } = \frac{ 4 }{ 3 } \frac{ E }{ 1 - \nu^2 } \sqrt{R} \; \delta^{ 3 / 2 }\] |
- F | the force, |
- d | the sample deformation, |
- E | the Youngs modulus, |
- n | the Poisson ratio, |
- R | the radius of the indenting sphere. |
\[F_{ Cone } = \frac{ 2 }{ \pi } \frac{ E }{ 1 - \nu^2 } \; \operatorname{ tan } \alpha \; \delta^2\] |
- F | the force, |
- d | the sample deformation, |
- E | the Youngs modulus, |
- n | the Poisson ratio, |
- a | the opening angle of the indenting cone. |
\[F_{ Pyramid } = \frac{ 1 }{ \sqrt{ 2 } } \frac{ E }{ 1 - \nu^2 } \; \operatorname{ tan } \alpha \; \delta^2\] |
- F | the force, |
- d | the sample deformation, |
- E | the Youngs modulus, |
- n | the Poisson ratio, |
- a | the opening angle of the indenting cone. |
\[F_{ Sphere } = \frac{ 4 }{ 3 } \frac{ E }{ 1 - \nu^2 } \sqrt{R} \; \delta^{ 3 / 2 }\] | \[F_{ Cone } = \frac{ 2 }{ \pi } \frac{ E }{ 1 - \nu^2 } \; \operatorname{ tan } \alpha \; \delta^2\] | \[F_{ Pyramid } = \frac{ 1 }{ \sqrt{ 2 } } \frac{ E }{ 1 - \nu^2 } \; \operatorname{ tan } \alpha \; \delta^2\] |
- F | the force, |
- d | the sample deformation, |
- E | the Youngs modulus, |
- n | the Poisson ratio, |
- R | the radius of the indenting sphere, |
- a | the opening angle of the indenting cone or pyramid. |
\[F_{ Sphere } = \frac{ 4 }{ 3 } \frac{ E }{ 1 - \nu^2 } \sqrt{R} \; \delta^{ 3 / 2 } \; - \; F_0\] | \[F_{ Cone } = \frac{ 2 }{ \pi } \frac{ E }{ 1 - \nu^2 } \; \operatorname{ tan } \alpha \; \delta^2 \; - \; F_0\] | \[F_{ Pyramid } = \frac{ 1 }{ \sqrt{ 2 } } \frac{ E }{ 1 - \nu^2 } \; \operatorname{ tan } \alpha \; \delta^2 \; - \; F_0\] |
- F | the force, |
- d | the sample deformation, |
- E | the Youngs modulus, |
- n | the Poisson ratio, |
- R | the radius of the indenting sphere, |
- a | the opening angle of the indenting cone or pyramid, |
- F0 | the adhesion force. |
\[F_{ Sphere } = \frac{ 4 }{ 3 } \frac{ E }{ 1 - \nu^2 } \sqrt{R} \; \delta^{ 3 / 2 }\] | ||
\[F_{ Sphere }^{2/3} = \left( \frac{ 4 }{ 3 } \frac{ E }{ 1 - \nu^2 } \sqrt{R} \right)^{2/3} \; \delta\] | ||
\[\left( \frac{ 4 }{ 3 } \frac{ E }{ 1 - \nu^2 } \sqrt{R} \right)^{2/3} \; = \frac{ \Delta F_{ Sphere }^{2/3} }{ \Delta \delta } = slope\] | ||
\[E = \frac{ 3 }{ 4 } \left( \frac{ \Delta F_{ Sphere }^{2/3} }{ \Delta \delta } \right)^{3/2} \frac{ 1 - \nu^2 }{ \sqrt{ R } } = \frac{ 3 }{ 4 } slope^{ 3/2 } \frac{ 1 - \nu^2 }{ \sqrt{ R } }\] |
- F | the force, |
- d | the sample deformation, |
- E | the Youngs modulus, |
- n | the Poisson ratio, |
- R | the radius of the indenting sphere. |
\[F_{ Cone } = \frac{ 2 }{ \pi } \frac{ E }{ 1 - \nu^2 } \; \operatorname{ tan } \alpha \; \delta^2\] | ||
\[F_{ Cone }^{ 1 / 2 } = \left( \frac{ 2 }{ \pi } \frac{ E }{ 1 - \nu^2 } \; \operatorname{ tan } \alpha \right)^{ 1 / 2 } \; \delta\] | ||
\[\left( \frac{ 2 }{ \pi } \frac{ E }{ 1 - \nu^2 } \; \operatorname{ tan } \alpha \right)^{ 1 / 2 } = \frac{ \Delta F_{ Cone }^{ 1 / 2 } }{ \Delta \delta } = slope\] | ||
\[E = \frac{ \pi }{ 2 } \left( \frac{ \Delta F_{ Cone }^{ 1 / 2 } }{ \Delta \delta } \right)^2 \frac{ 1 - \nu^2 }{ \operatorname{ tan } \alpha } = \frac{ \pi }{ 2 } slope^2 \frac{ 1 - \nu^2 }{ \operatorname{ tan } \alpha }\] |
- F | the force, |
- d | the sample deformation, |
- E | the Youngs modulus, |
- n | the Poisson ratio, |
- a | the opening angle of the indenting cone. |
- F | the force, |
- d | the sample deformation, |
- E | the Youngs modulus, |
- n | the Poisson ratio, |
- R | the radius of the indenting sphere. |
- E | the Youngs modulus, |
- H | the Hardness, |
- n | the Poisson ratio, |
- S | the contact stiffness, |
- A | the area function, |
- Fmax | the maximum force. |
- F | the force, |
- d | the sample deformation, |
- R | the radius of the indenting sphere, |
- a | the opening angle of the indenting cone, |
- hC | the contact depth. |
- F | the force, |
- d | the sample deformation, |
- a | the contact radius, |
- R | the tip radius, |
- W | the adhesion energy, |
- K | the combined elastic modulus given by the following relation, |
- E | the Youngs modulus, |
- n | the Poisson ratio. |
- F | the force, |
- d | the sample deformation, |
- E | the Youngs modulus, |
- n | the Poisson ratio, |
- R | the radius of the indenting sphere, |
- F0 | the adhesion force, |
- W0 | the adhesion work. |
- E | the Youngs or elastic modulus modulus of the cantilever's material, |
- w | the cantilever's width, |
- t | the cantilever's thickness, |
- L | the cantilever's Length. |
- fn | resonnance frequencey of a mode number n of a cantilever, |
- an | the solution of a set of flexural vibration modes of mode number n, |
- t | the cantilever's thickness, |
- L | the cantilever's Length. |
- E | the Youngs or elastic modulus modulus of the cantilever's material, |
- r | the mass density. |
for more details, see equation 6 of: K Babaei Gavan, E W J M van der Drift, W J Venstra, M R Zuiddam and H S J van der Zant Effect of undercut on the resonant behaviour of silicon nitride cantilevers J. Micromechanics Microengineering 2009; 19 035003 |
|
||
|
||
|
|
|
|
\[F_{ Sphere } = \frac{ 4 }{ 3 } \frac{ E }{ 1 - \nu^2 } \sqrt{R} \; \delta^{ 3 / 2 }\] |
\[F_{ Cone } = \frac{ 2 }{ \pi } \frac{ E }{ 1 - \nu^2 } \; \operatorname{ tan } \alpha \; \delta^2\] |
\[F_{ Sphere } = \frac{ 4 }{ 3 } \frac{ E }{ 1 - \nu^2 } \sqrt{R} \; \delta^{ 3 / 2 }\] |
\[F_{ Cone } = \frac{ 2 }{ \pi } \frac{ E }{ 1 - \nu^2 } \; \operatorname{ tan } \alpha \; \delta^2\] |