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Research Papers

Determining Prestrains in an Elastomer Through Elliptical Indentation

[+] Author and Article Information
Yue Zheng

Department of Mechanical and Aerospace Engineering,
University of California,
La Jolla, San Diego, CA 92093
e-mail: yuz485@ucsd.edu

Shengqiang Cai

Department of Mechanical and Aerospace Engineering,
University of California,
La Jolla, San Diego, CA 92093
e-mail: shqcai@ucsd.edu

1Corresponding author.

Contributed by the Applied Mechanics Division of ASME for publication in the journal of applied mechanics. Manuscript received June 5, 2019; final manuscript revised July 15, 2019; published online August 5, 2019. Assoc. Editor: Shaoxing Qu.

J. Appl. Mech 86(10), (Aug 05, 2019) (8 pages) Paper No: JAM-19-1290; doi: 10.1115/1.4044306 History: Received June 05, 2019; Accepted July 16, 2019

Residual stresses or prestrains could strongly affect the properties and functionalities of soft materials and tissues. However, non-destructive measurements of residual stresses or prestrains in materials are generally challenging. Previous studies have shown that residual stresses or prestrains can affect indentation tests of different materials, including metals, glassy polymers, soft elastomers, and gels. Nevertheless, an indentation method for determining the state of residual stresses or prestrains of large magnitude in an elastomer is not yet available. In this article, we propose to use elliptical indentation to measure large prestretches in a Neo-Hookean elastomer. We have analytically derived the relationship between the indentation force and indentation depth for both a flat-ended elliptical indenter and an ellipsoidal indenter. We have further shown that such a relationship greatly depends on the rotational angle of the indenter with respect to the direction of principle stretches in the elastomer. Based on the derived analytical results, we construct two diagrams for the flat-ended elliptical indenter and ellipsoidal indenter, respectively, which can be directly used to determine quantitatively large prestrains in an elastomer.

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Copyright © 2019 by ASME
Topics: Elastomers
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Figures

Grahic Jump Location
Fig. 1

Schematics of indentation of a biaxially prestretched elastomer with (a) a flat-ended elliptical indenter with semi-axes R1 and R2 (R1 > R2) and (b) an ellipsoidal indenter with principle semi-axes R1, R2, and R3 (R1 > R2). λ1 and λ2 are two lateral principle stretches applied onto the elastomer before indentation. ϕ denotes the angle between axis R1 and direction x1. The shaded elliptical area denotes the contact area between the indenter and the elastomer. For a flat-ended elliptical indenter, the shape of the contact area is identical to the shape of the cross section of the indenter. For an ellipsoidal indenter, the shape of the contact area can be very different from the that of the cross section of the indenter. In particular, the rotational angles of the ellipsoidal indenter (ϕ) and the contact area (β) are generally different.

Grahic Jump Location
Fig. 2

Indentation force versus indentation depth at different rotational angles for (a) a flat-ended elliptical indenter with semi-axes R1 and R2 and (b) an ellipsoidal indenter with principle semi-axes R1, R2, and R3. Solid curves are analytical predictions and circular dots are finite element simulations.

Grahic Jump Location
Fig. 3

Normalized indentation force varies with the rotational angle of (a) a flat-ended elliptical indenter with semi-axes R1 and R2 and (b) an ellipsoidal indenter with principle semi-axes R1, R2, and R3 for several different principle prestretches. Analytical predictions are plotted by solid curves and finite element computations are shown as circular dots.

Grahic Jump Location
Fig. 4

Contact pressure distribution between a flat-ended elliptical indenter (R1 = 2 and R2 = 1) and a biaxially prestretched elastomer (λ1 = 1 and λ2 = 2) for three different rotational angles. The left column shows the result of our analytical predictions, while the right column is the results from finite element simulations. In the plot, the indentation depth D/R1 = 0.077.

Grahic Jump Location
Fig. 5

Contact pressure distribution between an ellipsoidal indenter (R1′=R12/R3=2 and R2′=R22/R3=1) and a biaxially prestretched elastomer (λ1=1andλ2=2) for three different rotational angles. The left column shows the result of our analytical predictions, while the right column is the results from finite element simulations. ϕ denotes the angle between the major axis R1 and x1 direction (or the direction of λ1), and β denotes the angle between semi-major axis a of the contact area and the x1 direction. In the plot, the indentation depth D/R1 = 0.096.

Grahic Jump Location
Fig. 6

Contour of normalized maximal and minimal indentation forces for (a) the flat-ended elliptical indenter with semi-axes R1 and R2 and (b) the ellipsoidal indenter with principle semi-axes R1, R2, and R3. With given elastic modulus of the elastomer, as shown in both the figures, the magnitude of the biaxial prestretches in the elastomer can be determined based on the measurements of maximal and minimal indentation forces.

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