Adhesion.ReferenceSolutions.sinewave package

Submodules

Adhesion.ReferenceSolutions.sinewave.JKR module

References:

Original paper

Johnson, K. L. The adhesion of two elastic bodies with slightly wavy surfaces. International Journal of Solids and Structures 32, 423–430 (1995) DOI: 10.1016/0020-7683(94)00111-9

Asymmetric case and stress intensity factor:

Carbone, G. et al. Journal of the Mechanics and Physics of Solids 52, 1267–1287 (2004) DOI: 10.1016/j.jmps.2003.12.001

Physical quantities:

The geometry of the sinusoidal indenter is

\[2\ h\ sin^2(\pi\ a\ / \lambda )\]

  • \(\lambda\): wavelength of the sinusoidal indenter

  • \(h\): half peak to tale distance

  • \(E^*\): contact modulus

Nondimensional units

  • lateral lengths are in unit of \(\lambda\)

  • vertical lengths and displacements are in unit of \(h\) (or :math:`2 h `)

  • Pressures are in units of \(p_{wfc} = \pi E^* h/\lambda\) the amplitude

of the sinusoidal pressures in the full contact.

  • Johnson Parameter \(\alpha\).

\[\alpha = \frac{\sqrt{2}}{\pi} \frac{\lambda}{h} \sqrt{\frac{w}{E^* \lambda}}\]

\(\alpha^2\) can be interpreted as a ratio of surface energy and the elastic energy necessary for full contact.

\[\alpha^2 = \frac{w}{h p_{wfc}} \frac{2}{\pi}\]

\(\alpha\) is also the stress intensity factor in nondimensional form

\[\alpha = \frac{K}{p_{wfc} \sqrt{\lambda}} = \frac{K}{\pi E^* h / \sqrt{\lambda}}\]
Adhesion.ReferenceSolutions.sinewave.JKR.contact_radius(mean_pressure, alpha)[source]
Parameters:

  • mean_pressure – mean pressure in units of \(\pi Es h/\lambda\)

  • alpha (float) – johnson parameter

Return type:

half the contact width

Adhesion.ReferenceSolutions.sinewave.JKR.elastic_energy(a, mean_pressure)[source]

\(\frac{U - U_{flat}}{h p_{wfc} A}\)

Parameters:

  • mean_pressure (in units of \(p_{wfc}\)) –

  • a (in units of the period \(\lambda\)) –

Return type:

energy per unit area in units of \(h p_{wfc}\)

Adhesion.ReferenceSolutions.sinewave.JKR.flatpunch_pressure(x, a)[source]

solution by koiter

Flat punch solution (uniform deformation on periodic strides)

Parameters:

  • x (float, np.array) – in units of lambda

  • a (float) – half width of the flat punch

Return type:

Pressure distribution with mean -1

References

Johnson, K. L. The adhesion of two elastic bodies with slightly wavy surfaces. International Journal of Solids and Structures 32, 423–430 (1995).

Koiter, W. T. An infinite row of collinear cracks in an infinite elastic sheet. Ing. arch 28, 168–172 (1959).

Zilberman, S. & Persson, B. N. J. Adhesion between elastic bodies with rough surfaces. Solid State Communications 123, 173–177 (2002).

Adhesion.ReferenceSolutions.sinewave.JKR.mean_gap(a, alpha)[source]

from carbon mangialardi equation (39)

Parameters:

  • a (float) – half contact width in units of lambda

  • alpha (float) – Johnson Parameter

Returns:

  • mean gap in units of \(h\)

  • h is half the peak to valley distance

Examples

>>> import matplotlib.pyplot as plt
>>> fig, ax = plt.subplots()
>>> a = np.linspace(0, 0.5 )
>>> ls = [ax.plot(a, mean_gap(a, alpha), label="{}".format(alpha) ) for alpha in (0., 0.1,0.2,0.5)]
>>> leg = ax.legend()
>>> _ = ax.set_xlabel(r"contact radius $a$ ($\lambda$)");
>>> _ = ax.set_ylabel(r"mean gap ($h$)");
>>> plt.show()
Adhesion.ReferenceSolutions.sinewave.JKR.mean_pressure(a, alpha, der='0')[source]

mean pressure in units of \(\pi E^* h/ \lambda\)

and it’s derivatives

Parameters:

  • a (half contact width in units of lambda) –

  • alpha (float or array) – johnson parameter

  • der ({"0", "1", "2"}) – order of the derivative with respect to (a/lambda)

Returns:

  • der=”0” – mean pressure in units of pi Es h/lambda

  • der=”1” – d (mean pressure) / d (a) in units of pi Es h / lambda^2

  • der=”2” – d (mean pressure)^2 / d^2 (a) in units of pi Es h / lambda^3

Adhesion.ReferenceSolutions.sinewave.JKR.pressure(x, a, mean_pressure)[source]
Parameters:

  • x – position in units of the period \(\lambda\)

  • a – half contact width in units of the period \(\lambda\)

  • mean_pressure – externally applied mean pressure in units of the westergaard full contact pressure pi E^* h / lambda

Adhesion.ReferenceSolutions.sinewave.JKR.stress_intensity_factor_asymmetric(a_s, a_o, P, der='0')[source]

Stress intensity factor at the crack front at a_s

partial derivatives are taken at constant mean pressure P

units for a_s and a_o: wavelength of the sinewave units of pressure: westergaard full contact pressure \(\pi E^* h / \lambda\)

returns the stress intensity factor in units of \(\pi E^* h / \sqrt{\lambda}\) or it’s partial derivative.

Notation for the derivative flag: for example “1_a_s” returns the partial derivative according a_s, holding a_o and P constant

Parameters:

  • a_s (float between 0 and 0.5) – position of the (positive x) crack front at which the SIF is computed

  • a_o (float between 0 and 0.5) – position of the (negative x) crack front opposite to where the SIF is computed

  • P – mean pressure

  • der (string {"0", "1_a_s", "1_a_o"}) – choose the partial derivative of K to be returned

References

Carbone, G. et al. Journal of the Mechanics and Physics of Solids 52, 1267–1287 (2004) DOI: 10.1016/j.jmps.2003.12.001

Adhesion.ReferenceSolutions.sinewave.JKR.stress_intensity_factor_symmetric(a, P, der='0')[source]

Module contents

Analytical or semi-analytical reference solutions

for the periodic 2-dimensional contact of a sinusoidal indenter