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Research Papers

Constructing Continuous Strain and Stress Fields From Spatially Discrete Displacement Data in Soft Materials

[+] Author and Article Information
Wanru Liu

Department of Mechanical Engineering,
University of Alberta,
Edmonton, AB T6G 2G8, Canada

Rong Long

Department of Mechanical Engineering,
University of Colorado Boulder,
Boulder, CO 80309
e-mail: rong.long@colorado.edu

1Corresponding author.

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received July 16, 2015; final manuscript received October 7, 2015; published online November 5, 2015. Assoc. Editor: Kyung-Suk Kim.

J. Appl. Mech 83(1), 011006 (Nov 05, 2015) (15 pages) Paper No: JAM-15-1370; doi: 10.1115/1.4031763 History: Received July 16, 2015; Revised October 07, 2015

A recent study demonstrated that three-dimensional (3D) continuous displacement fields in transparent soft gels can be constructed from discrete displacement data obtained by optically tracking fluorescent particles embedded in the gels. Strain and stress fields were subsequently determined from gradients of the displacement field. This process was achieved through the moving least-square (MLS) interpolation method. The goal of this study is to evaluate the numerical accuracy of MLS in determining the displacement, strain, and stress fields in soft materials subjected to large deformation. Using an indentation model as the benchmark, we extract displacement at a set of randomly distributed data points from the results of a finite-element model, utilize these data points as the input for MLS, and compare resulting displacement, strain, and stress fields with the corresponding finite-element results. The calculation of strain and stress is based on finite strain kinematics and hyperelasticity theory. We also perform a parametric study in order to understand how parameters of the MLS method affect the accuracy of the interpolated displacement, strain, and stress fields. We further apply the MLS method to two additional cases with highly nonuniform deformation: a plate with a circular cavity subjected to large uniaxial stretch and a plane stress crack under large mode I loading. The results demonstrate the feasibility of using optical particle tracking together with MLS interpolation to map local strain and stress field in highly deformed soft materials.

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Figures

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Fig. 2

Illustration of the MLS interpolation method. In the zone of interest Ω (the square), A is a point where the interpolation is performed. The data points (n = 40) in Ω are shown by star symbols. The circle Ωb, centered at A and with radius rc, is the region within which the data points are assigned nonzero weight.

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Fig. 1

Schematic of the indentation experiment where a rigid sphere is indented on a soft gel substrate. The deformed shape of the gel substrate is illustrated by the dashed lines. The inset shows locations of the particles (star symbols) and displacements (arrows) in the highlighted region of the gel substrate.

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Fig. 3

An example of the zone of interest containing grid points and data points inside. The star symbols represent data points for the MLS method. The black circles are gird points where we perform interpolation and calculate relative errors between FEA and MLS results.

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Fig. 6

Cauchy stress field in the zone of interest for components σ11 (a)–(c), σ12 (d)–(f), and σ22 (g)–(i): (a), (d), and (g) reference fields given by FEA results; (b), (e), and (h) fields reproduced by the MLS method; and (c), (f), and (i) histogram of relative errors between FEA and MLS results. For the MLS result, the cutoff radius rc is 0.4h, the cubic basis in Eq. (12) is used, and the number of data points is 1200. The contour plots of the corresponding FEA and MLS results share the same color map, and all contour plots are shown using the undeformed zone of interest.

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Fig. 9

Median relative error η̃ (in percentage and logarithmic scale) versus the normalized cutoff radius rc/h for the three stress components σ11, σ12, and σ22. The cubic basis is used and the number of data points is 1200 (γ = 0.01641).

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Fig. 4

Displacement field in the zone of interest for components u1 (a)–(c) and u2 (d)–(f): (a) and (d) reference fields given by FEA results; (b) and (e) fields reproduced by the MLS method; and (c) and (f) histogram of relative errors between FEA and MLS results. For the MLS result, the cutoff radius rc is 0.4h, the cubic basis is used, and the number of data points is 800. The contour plots of the corresponding FEA and MLS results share the same color map, and all contour plots are shown using the undeformed zone of interest.

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Fig. 5

Hencky strain field in the zone of interest for components E11 (a)–(c), E12 (d)(f), and E22 (g)–(i): (a), (d), and (g) reference fields given by FEA results; (b), (e), and (h) fields reproduced by the MLS method; and (c), (f), and (i) histogram of relative errors between FEA and MLS results. For the MLS result, the cutoff radius rc is 0.4h, the cubic basis is used, and the number of data points is 800. The contour plots of the corresponding FEA and MLS results share the same color map, and all contour plots are shown using the undeformed zone of interest.

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Fig. 7

Median relative error η̃ (in percentage and logarithmic scale) versus the normalized cutoff radius rc/h (h is the substrate thickness): (a) displacement component u2 and (b) Hencky strain component E22. Results with linear, quadratic, and cubic bases are shown together. The number of data points is 800 (γ = 0.01979).

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Fig. 8

Median relative error η̃ (in percentage and logarithmic scale) versus normalized average nearest-neighbor distances γ between data points: (a) displacement component u2 and (b) Hencky strain component E22. The cubic basis is used.

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Fig. 10

Effect of noise in displacement data: median relative error η̃ (in percentage and logarithmic scale) versus the noise level d for the indentation example. The cubic basis is used, the number of data points is 1200 (γ = 0.01641), and the cutoff radius rc/h = 0.4.

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Fig. 11

Thin plate with a circular cavity under uniaxial tension. A horizontal displacement Δ1 is applied to both the right and left edges of the plate. The deformed shape of the plate is illustrated by the dashed lines. The inset shows the zone of interest with data points (star symbols).

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Fig. 12

Distributions of the displacement component u1 (a)(c) and strain component E11 (d)–(f) in the zone of interest: (a) and (d) reference fields given by FEA results; (b) and (e) fields reproduced by the MLS method; and (c) and (f) histogram of relative errors between FEA and MLS results. For the MLS result, the cutoff radius rc is 0.7r1, the cubic basis is used, and the number of data points is 800. The contour plots of corresponding FEA and MLS results share the same color map, and all contour plots are shown using the undeformed zone of interest.

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Fig. 13

Distribution of the Cauchy stress component σ11 in the zone of interest: (a) reference fields given by FEA result; (b) fields reproduced using method A; (c) histogram of relative error with method A; (d) field reproduced using method B; and (e) histogram of relative errors with method B. The contour plots of the corresponding FEA and MLS results share the same color map, and all contour plots are shown using the undeformed zone of interest.

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Fig. 14

Plane stress crack under mode I loading. Vertical displacement Δ2 is applied to both the top and bottom edges of the specimen. The deformed shape of the plate is illustrated by the dashed lines. The inset shows the zone of interest with data points (star symbols).

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Fig. 15

Distributions of the displacement component u2 (a)–(c) and strain component E22 (d)–(f) in the zone of interest: (a) and (d) reference fields given by FEA results; (b) and (e) fields reproduced by the MLS method; and (c) and (f) histogram of relative errors between FEA and MLS results. For the MLS result, the cutoff radius rc is 0.00025h2, the cubic basis is used, and the number of data points is 800. The contour plots of the corresponding FEA and MLS results share the same color map, and all contour plots are shown using the undeformed zone of interest.

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Fig. 16

Distribution of the Cauchy stress component σ22 in the zone of interest: (a) reference fields given by FEA result; (b) fields reproduced using method A; (c) histogram of relative error with method A; (d) field reproduced using method B; and (e) histogram of relative errors with method B. The contour plots of the corresponding FEA and MLS results share the same color map, and all contour plots are shown using the undeformed zone of interest.

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Fig. 17

(a) Illustration of two different integration paths from the reference X0 to the target point X when determining the hydrostatic term p using method A. (b) The distribution of the Cauchy stress σ22 calculated using method A to determine p and path 2. The result obtained by using path 1 has been shown in Fig. 16(b).

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