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Update models.rst
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DavidMercier authored Jul 20, 2021
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Expand Up @@ -281,7 +281,7 @@ on an elastic-plastic (ductile) substrate, pop-in were observed experimentally a
*Schematic of indentation load-displacement curve with a pop-in event and the plot of the load carried by the elastic-plastic substrate.*

Some authors explained that a circumferential crack appears at the location of the elastic–plastic boundary in the substrate [#Bahr_2003]_ and [#Morash_2007]_.
The radius :math:`c` of this plastic zone in the ductile substrate is defined by the following equation :
The radius :math:`c` of this plastic zone in the ductile substrate is defined by the following equation:

.. math:: c = \sqrt{\frac{3F_\text{crit,s}}{2\pi\sigma_\text{e}}}
:label: plastic_zone_radius
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.. [#Stauffer_2012] `Stauffer D.D. et al., "Plastic response of the native oxide on Cr and Al thin films from in situ conductive nanoindentation" (2012). <https://doi.org/10.1557/jmr.2011.432>`_
.. [#Wang_2004] `Wang M.G. and Ngan A.H.W., "Indentation strain burst phenomenon induced by grain boundaries in niobium" (2004). <https://doi.org/10.1557/JMR.2004.0316>`_
.. [#Wang_2012] `Wang L., "Influences of sample preparation on the indentation size effect and nanoindentation pop-in on nickel" PhD Thesis (2012). <http://trace.tennessee.edu/utk_graddiss/1371>`_
.. [#Wang_2021] `Wang D., "Understanding the hydrogen effect on pop-in behavior of an equiatomic high-entropy alloy during in-situ nanoindentation", Journal of Materials Science & Technology (2021). <https://doi.org/10.1016/j.jmst.2021.04.060>`_
.. [#Wang_2021] `Wang D. et al., "Understanding the hydrogen effect on pop-in behavior of an equiatomic high-entropy alloy during in-situ nanoindentation", Journal of Materials Science & Technology (2021). <https://doi.org/10.1016/j.jmst.2021.04.060>`_
.. [#Weibull_1951] `Weibull W., "A statistical distribution function of wide applicability", J. Appl. Mech.-Trans. ASME (1951), 18(3). <http://www.barringer1.com/wa_files/Weibull-ASME-Paper-1951.pdf>`_
.. [#Wo_2006] `Wo P.C. et al., "Time-dependent incipient plasticity in Ni3Al as observed in nanoindentation" (2006). <https://doi.org/10.1557/JMR.2005.0056>`_
.. [#Wu_2014_1] `Wu D. et al., "Effect of tip radius on the incipient plasticity of chromium studied by nanoindentation" (2014). <https://doi.org/10.1016/j.scriptamat.2014.09.017>`_
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