Spherical indentation of a membrane on an elastic half-space


A number of physiological systems involve contact or indentation of solids with tensed surface layers. In this paper the contact problem of spherical indentation of a linear elastic solid, covered with a tensed membrane is addressed. Semianalytical solutions are obtained relating indentation force to contact radius, as well as contact radius to depth. Good agreement is found between derived equations and results from finite element method (FEM) simulations. In addition, effect of membrane on subsurface stresses is shown quantitatively and compared favorably to FEM results. This work is applicable to mechanical property assessment of a number of biological systems.

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JHK and AG gratefully acknowledge the NSF Faculty Early Career Award CMS 0449268 for supporting this work.

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Corresponding author

Correspondence to Andrew Gouldstone.


Appendix A

Modification constant b in Eq. (6)

$$b = {A_1} - {{{A_2}} \over {{{\left( {1 + {A_3}\,\cdot\,{T_{\rm{o}}}\left( {1 - v} \right)/{\mu }} \right)}^{1/{A_4}}}}}$$

Appendix B

II (ro), I2(ro), L1(ro), L2(ro), M in Eqs. (22) to (26)

$${I_1}\left( {{r^{\rm{o}}}} \right) = \int_{\rm{o}}^\infty {{{\left[ {\int_{\rm{o}}^1 {{{\left( {1 - {{{r^{{\rm{o}}2}}} \over {{b^2}}}} \right)}^{1/2}}{r^{\rm{o}}}{J_{\rm{o}}}\left( {k{r^{\rm{o}}}} \right)d{r^{\rm{o}}}} } \right]} \over {\left[ {1 + epsilonk} \right]}}{J_{\rm{o}}}\left( {k{r^{\rm{o}}}} \right)dk} $$

(b is modification factor in Appendix A)

$${ {{I_2}\left( {{r^{\rm{o}}}} \right) = \int_{\rm{0}}^\infty {{{\left[ {\int_{\rm{o}}^1 {\left( {1 - {{{r^{{\rm{o}}2}}} \over {{b^2}}}} \right){r^{\rm{o}}}{J_{\rm{o}}}\left( {k{r^{\rm{o}}}} \right)d{r^{\rm{o}}}} } \right]} \over {\left[ {1 + \epsilon{k}} \right]}}k{J_{\rm{o}}}\left( {k{r^{\rm{o}}}} \right)dk\,\,,} }{{L_1}\left( {{r^{\rm{o}}}} \right) = \int_{\rm{0}}^\infty {{{{J_1}\left( k \right)} \over {k\left[ {1 + \epsilon{k}} \right]}}{J_{\rm{o}}}\left( {k{r^{\rm{o}}}} \right)dk,\,\,} }{{L_2}\left( {{r^{\rm{o}}}} \right) = \int_{\rm{0}}^\infty {{{{J_1}\left( k \right)} \over {k\left[ {1 + \epsilon{k}} \right]}}k{J_{\rm{o}}}\left( {k{r^{\rm{o}}}} \right)dk} \,\,\,,}{M = \int_{\rm{0}}^1 {{{\left( {1 - {{{r^{{\rm{o}}2}}} \over {{b^2}}}} \right)}^{1/2}}2{\pi}}{r^{\rm{o}}}d{r^{\rm{o}}},} }{\epsilon = {{{T_{\rm{o}}}} \over {{\mu }a}\left( {1 - v} \right),}{{p_0} = {{4a{\mu}} \over {{\pi }R\left( {1 - v} \right)}}}}$$

Appendix C

Displacement and stress distribution underneath indenter


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Kim, J.H., Gouldstone, A. Spherical indentation of a membrane on an elastic half-space. Journal of Materials Research 23, 2212–2220 (2008). https://doi.org/10.1557/JMR.2008.0278

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