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Research Papers

# Strain Uniformity in Biaxial Specimens is Highly Sensitive to Attachment Details

[+] Author and Article Information
Armin Eilaghi

Department of Mechanical and Industrial Engineering, and Institute for Biomaterials and Biomedical Engineering, University of Toronto, Toronto, Ontario, M5S 1A1

John G. Flanagan

Department of Ophthalmology and Vision Sciences, University of Toronto; School of Optometry School, University of Waterloo, Waterloo, Ontario, N2L 3G1

G. Wayne Brodland

Department of Civil and Environmental Engineering, and Department of Biology,University of Waterloo

C. Ross Ethier1

Department of Mechanical and Industrial Engineering, Institute for Biomaterials and Biomedical Engineering, and Department of Ophthalmology and Vision Sciences, University of Toronto; Department of Bioengineering, Imperial College, London SW7 2AZ, UKr.ethier@imperial.ac.uk

1

Corresponding author.

J Biomech Eng 131(9), 091003 (Aug 04, 2009) (7 pages) doi:10.1115/1.3148467 History: Received September 29, 2008; Revised December 19, 2008; Published August 04, 2009

## Abstract

Biaxial testing has been used widely to characterize the mechanical properties of soft tissues and other flexible materials, but fundamental issues related to specimen design and attachment have remained. Finite element models and experiments were used to investigate how specimen geometry and attachment details affect uniformity of the strain field inside the attachment points. The computational studies confirm that increasing the number of attachment points increases the size of the area that experiences sensibly uniform strain (defined here as the central sample region where the ratio of principal strains $E11/E22<1.10$), and that the strains experienced in this region are less than nominal strains based on attachment point movement. Uniformity of the strain field improves substantially when the attachment points span a wide zone along each edge. Subtle irregularities in attachment point positioning can significantly degrade strain field uniformity. In contrast, details of the apron, the region outside of the attachment points, have little effect on the interior strain field. When nonlinear properties consistent with those found in human sclera are used, similar results are found. Experiments were conducted on $6×6 mm$ talc-sprinkled rubber specimens loaded using wire “rakes.” Points on a grid having $12×12 bays$ were tracked, and a detailed strain map was constructed. A finite element model based on the actual geometry of an experiment having an off-pattern rake tine gave strain patterns that matched to within 4.4%. Finally, simulations using nonequibiaxial strains indicated that the strain field uniformity was more sensitive to sample attachment details for the nonequibiaxial case as compared to the equibiaxial case. Specimen design and attachment were found to significantly affect the uniformity of the strain field produced in biaxial tests. Practical guidelines were offered for design and mounting of biaxial test specimens. The issues addressed here are particularly relevant as specimens become smaller in size.

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## Figures

Figure 1

A typical finite element mesh. The 20 attachment points are indicated by semitransparent circles. The mesh contains 4989 elements and 15084 nodes.

Figure 2

Number of attachment points. Changing the number of attachment points n along each edge from three to six significantly changes the strain field inside the region they define, as indicated by the color contour plots of the principal strain ratio E11/E22. The black dots indicate the attachment points and are drawn to scale. The percentages on the top of each panel indicate the fraction of the interior area where the principal strain ratio is sensibly constant (E11/E22≤1.1). The color legend also applies to Figs.  345678.

Figure 3

Attachment point spacing changes in the spacing of the attachment points along each edge affect the strain field significantly. The dimensions shown indicate the spacing between the tines along each side. The best field is obtained when the attachment points from adjacent edges are as closely spaced as practical (Fig. 3 is identical to Fig. 2 and is repeated for reference purposes). For color legend and further description, see Fig. 2 caption.

Figure 4

Irregularities in attachment point locations in: (a) the attachment points on only the left edge were moved slightly closed to each other (0.9 mm spacing rather than 1.0 mm, as in Fig. 2), (b) points on only the left edge are spaced 0.8 mm apart, (c) the center point on the left edge was moved outward by 0.2 mm, and (d) the center point on the left edge was moved toward the bottom of the figure by 0.2 mm.

Figure 5

Relative movement between attachment point and specimen: (a) when nonadhering contact elements are used between the 300 μm-diameter attachment points and the matching holes in the specimen, relative motion occurs; (b) uniformity of the strain field did not change significantly; however, the magnitude of the central strain was less than in the otherwise similar case as shown in Fig. 2

Figure 6

(a) Apron geometry increasing the apron width and (b) changing its shape had little effect on the interior strain field. In (a) the apron width was doubled compared with the baseline geometry, producing a specimen that is 7×7 mm, and in (b) the top right corner of the specimen was moved upward by 1 mm.

Figure 7

Nonlinear materials: (a) stress-strain curves for sclera (see text for details) and (b) a contour plot of the ratio of first to second principal strains for multilinear elastic properties. The resulting strain field is similar to that for a linear material (Fig. 2).

Figure 8

Representative experiment with matching FE model: (a) a 6×6 mm sheet of rubber was mounted, such that the second attachment point from the right along its upper edge was intentionally off-pattern, and then stretched equibiaxially to 8% strain. Points on the surface were tracked (the grid), and the principal strain ratios were calculated for each area; (b) regional principal strain ratios based on image tracking; (c) strain ratios produced by a finite element model based on the initial geometry of the experimental specimen and actual attachment point positions; (d) to facilitate comparison with the experimental results (b), the FE data were pixilated. The normalized rms difference between the experimental (b) and pixilated FE (d) principal strain ratios was 4.4%. The narrowed pixels along the perimeters of figure parts (b) and (d) are an artifact of the graphics program used.

Figure 9

Nonequibiaxial strain field contour plots of normalized principal strain ratios for: (a) the baseline geometry, with the attachments displaced twice as much in the X-direction as in the Y-direction; (b) the baseline geometry, with the attachments displaced twice as much in the Y-direction as in the X-direction; (c) the sample geometry shown in Fig. 8, with the attachments displaced twice as much in the X-direction as in the Y-direction; and (d) the sample geometry shown in Fig. 8, with the attachments displaced twice as much in the Y-direction as in the X-direction. Note that all principal strain ratios were normalized by the ratio of the principal strains at the center of the idealized sample geometry, and that the contour legend is different than that used in previous figures. With the indicated normalization, deviations of the principal strain ratios away from one are due to the influence of attachment point geometry on the strain field.

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