Point Contacts and Contact Stiffness

  • Diethelm JohannsmannEmail author
Part of the Soft and Biological Matter book series (SOBIMA)


If an external object (such as a sphere) touches the resonator surface across a contact with a diameter much below the wavelength of sound and, also, much below the size of the particle, one can infer the stiffness of the contact from the frequency shift. The situation is particularly transparent for particles, which are heavy enough to be clamped in space by inertia, so that they do not follow MHz motion of the resonator. In this case, the frequency shift is positive and proportional to the contact stiffness. Smaller particles give rise to coupled resonances. Coupled resonances can be viewed as absorption lines in shear-wave spectroscopy.


Tangential Stress Contact Mechanic Rotational Mode Contact Stiffness Hertzian Contact 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.




Definition (Comments)


As in index: undamped (Exception: f 0)


Contact radius


(Effective) area of the resonator plate (See Sect.  7.4)


Scattering length (Sect. 11.7)

bshear, bbend

Numerical coefficients

(Eq. 11.5.4)


Diffusivity (Sect. 11.6)


Young’s modulus


Effective Young’s modulus (Eq. 11.2.2)




Oscillator strength (a number, not a frequency)


Frequency of zero-crossing (Eq. 11.4.7)


Resonance frequency at the fundamental (f 0 = Z q /(2m q ) = Z q /(2ρ q d q ))


Normal force


Tangential force


Standard acceleration (g = 9.81 m/s2)


Shear modulus


Effective shear modulus (Eq. 11.2.12)


As an index: heterogeneously broadened (Sect. 11.4)


As an index: interface (Sect. 11.8)


Moment of area (Eq. 11.2.12)




Thermal energy


As an index: liquid




Overtone order


Number of particles per unit area


Normal stress


As an index: Particle


As an index: Polar Diagram


Distance from the center of a contact, distance from a scattering center


As in index: rocking mode


As in index: rotational mode


Particle radius


Radius of circle in polar diagram (Eq. 11.5.7)


Reflectivity evaluated at the resonator surface (Sect. 11.7)


Location at the resonator surface


In-plane distance from the scattering center (Sect. 11.7)


As an index: Surface

Contact stiffness (Sect. 11.8)


As an index: scattered (Sect. 11.7)


Tangential displacement



Load impedance


Acoustic wave impedance of AT-cut quartz (Z q  = 8.8 × 106 kg m−2 s−1)


Damping factor of a coupled resonance (Sect. 11.4)


Surface energy (Sect. 11.2)


Imaginary part of a resonance frequency


Normal compression (Fig. 11.2)


Scattering phase (Sect. 11.7)


As a prefix: A shift induced by the presence of the sample


Spring constant


Friction coefficient (µ = F x /F N )


Poisson number


Angle of rotation




Tangential stress


Momentum relaxation time (also: slip time Sect. 11.6)


A constant tangential stress in the sliding zone of a contact experiencing partial slip (Sect. 11.2)


Drag coefficient


Angular frequency


Particle resonance frequency


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Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Institute of Physical ChemistryClausthal University of TechnologyClausthal-ZellerfeldGermany

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