Spherical indentation of a membrane on an elastic half-space
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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.
I. INTRODUCTION
Indentation is a remarkably flexible mechanical test due to its relative experimental simplicity. Coupled with advances in instrument development, ease of implementation has placed indentation as a viable research tool for a number of different systems across size scales (nano to macro) and scientific or engineering disciplines. Recent attention has focused on contact probing of biomaterials, both soft (e.g., tissue) and hard (e.g., bone). Indentation is attractive for this, largely because of the lack of necessary specimen preparation, allowing quick in vitro or in vivo mechanical tests, for physiologic study or possibly diagnosis (note that palpation could be considered a type of indentation). Indentation of “soft” biomaterials raises some issues to be considered for analysis and/or property extraction, including viscoelasticity, adhesion due to moist surfaces, and structural heterogeneities in tissue samples. In the context of contact mechanics, this presents a number of problems to be solved. In this paper, we address one problem of spherical indentation of an elastic half-space, covered with a tensed membrane. This is equivalent to indentation of an elastic thin plate/film on an elastic solid, where the thickness and stiffness of film are sufficiently small to neglect bending rigidity.1 The problem arises in a few instances, for example, the macroscopic contact of the lung, which can be treated as a quasielastic solid surrounded by a thin (pleural) membrane, or other viscera, or skin over muscle or fat. Spherical indentation of these systems would be useful in the extrac-
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Address all correspondence to this author. e-mail: [email protected] DOI: 10.1557/JMR.2008.0278 2212
http://journals.cambridge.org
J. Mater. Res., Vol. 23, No. 8, Aug 2008 Downloaded: 27 Mar 2015
tion of material properties for modeling purposes, or quantitative changes, for physiology or diagnosis. Beyond measurement, this specific form of loading could take place in life, e.g., contact between heart and lung. For other biologically relevant problems where bending rigidity and membrane tension are comparable, the reader is referred to a numerical study by Zamir and Taber.2 Briefly (see Fig. 1), the membrane imposes a size effect during contact; that is, a relatively large contact dimension “feels” the underlying solid (relationship between indentation l
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