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TECHNICAL PAPERS: Cell

# Use of Rigid Spherical Inclusions in Young’s Moduli Determination: Application to DNA-Crosslinked Gels

[+] Author and Article Information
David C. Lin

Department of Mechanical and Aerospace Engineering, Rutgers,  The State University of New Jersey, 98 Brett Road, Piscataway, NJ 08854

Bernard Yurke

Bell Laboratories, Lucent Technologies Inc., 600 Mountain Avenue, Murray Hill, NJ 07974

Noshir A. Langrana1

Department of Mechanical and Aerospace Engineering, Rutgers,  The State University of New Jersey, 98 Brett Road, Piscataway, NJ 08854langrana@rutgers.edu

1

Corresponding author.

J Biomech Eng 127(4), 571-579 (Feb 23, 2005) (9 pages) doi:10.1115/1.1933981 History: Received March 18, 2004; Revised February 23, 2005

## Abstract

Current techniques for measuring the bulk shear or elastic $(E)$ modulus of small samples of soft materials are usually limited by materials handling issues. This paper describes a nondestructive testing method based on embedded spherical inclusions. The technique simplifies materials preparation and handling requirements and is capable of continuously monitoring changes in stiffness. Exact closed form derivations of $E$ as functions of the inclusion force-displacement relationship are presented. Analytical and numerical analyses showed that size effects are significant for medium dimensions up to several times those of the inclusion. Application of the method to DNA-crosslinked gels showed good agreement with direct compression tests.

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## Figures

Figure 1

A schematic diagram of two polymer chains crosslinked by a system of three DNA strands (SA1, SA2, and the crosslinking strand L2). When a crosslink-dissociating strand (not shown) with a complementary base sequence to the crosslinking strand is added, branch migration will initiate at the toehold region, eventually forming a double-stranded waste product.

Figure 2

The elasticity test fixture. The schematic on the right shows how the size of the medium relative to the inclusion, q, is measured.

Figure 3

A rigid spherical inclusion in three different media: (a) infinite, (b) spherical, and (c) cylindrical

Figure 4

Half view of meshed models with hollow inclusion: (a) spherical (R1=2mm) 10,610 elements in complete model and (b) cylindrical (H=r1=2mm), 13,193 elements in the complete model.

Figure 7

Plots of the vertical displacement fields as predicted by theory and by FEA. Values are expressed as a fraction of the inclusion displacement (uz∕δ, where δ=0.001mm). In both cases, a medium radius equal to six times the inclusion radius was used. Similar color maps were used.

Figure 8

The force-displacement plot for the sample with 27.78% of full crosslinking. The electromagnet was used to apply the force to the inclusion, whose displacement was then recorded digitally.

Figure 9

Figure 5

The ratio of Young’s modulus as calculated from using the infinite medium assumption (Ei) to the actual modulus (Ea), as a function of the relative medium size. The four cases shown are: linear elasticity theory for a spherical medium (a); FEA of a spherical medium (b); FEA of a square cylindrical medium (c); and FEA of the cylindrical medium with relaxed displacement restraints on the top surface (d).

Figure 6

Individual effects of cylinder radius and height on the ratio of Young’s modulus values, for cylindrical media with an unrestrained top surface. (Left) Effect of height for constant radius. (Right) Effect of radius for constant height.

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