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ORIGINAL PAPER

Nonlinear contact mechanics for the indentation of hyperelastic cylindrical bodies Amy M. Dagro1

· K. T. Ramesh2

Received: 21 January 2019 / Accepted: 10 March 2019 © Springer Nature Switzerland AG 2019

Abstract The mechanical properties of biological materials are commonly found through the application of Hertzian theory to forcedisplacement data obtained through micro-indentation techniques. Due to their soft nature, biological specimens are often subjected to large indentations, resulting in a nonlinear deformation behavior that can no longer be accurately described by Hertzian contact. Useful models for studying the large deformation response of cylindrical specimens under indentation are not readily available, and the morphologies of biological materials are often closer to cylinders than spheres (e.g., cellular processes, fibrin, collagen fibrils, etc.). In this study, a computational model is used to analyze the large deformation indentation of an incompressible hyperelastic cylinder in order to provide a generalized formulation that can be used to extract mechanical properties from indentation into soft cylindrical bodies. The effects of specimen size and indentation depth are examined in order to quantify the deformation at which the proposed force-displacement relationship remains accurate. Keywords Indentation · Hyperelastic · Nonlinear contact

1 Introduction Measuring the mechanical response of biological specimens is an important key to developing new biomedical materials [1], understanding injury or disease progression [2–4], and potentially improving clinical diagnostic technologies [5, 6]. However, obtaining useful mechanical properties of most biological specimens (e.g., individual cells) is a difficult task due to the nonlinearity, anisotropy, and heterogeneity [7–9] that they exhibit. Within the last century, a common approach to measuring the mechanical properties in cells is through indentation testing—an experimental technique that gained popularity due to the relative convenience of probing adherent cells grown on flat, conventional substrates [10]. By indenting the cell surface with a simple geometry probe (e.g., sphere, cone, etc.), it is possible to use the relationship between the applied force and the concurrently measured indentation depth, to estimate an effective local modulus of the cell. The prevalence of indentation testing, such as atomic force microscopy (AFM), is also attributable to the fact that previously used techniques, such as micropipette aspiration, might disturb the attachment of the cell membrane to the underlying cytoskeleton [10]. AFM was formally introduced in 1986 [11], although similar indentation-like devices appeared before this, such as the “cell-poker” device of the Elson group in the 1970s [12]. By the 1990s, Radmacher and others began using AFM on individual cells to create “elasticity maps,” or images that depict elastic properties across the various regions of the cell with high spatial resolution [13–15].

 Amy M. Dagro

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