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Research Papers

Effect of Properties and Turgor Pressure on the Indentation Response of Plant Cells

[+] Author and Article Information
Viggo Tvergaard

Professor
Department of Mechanical
Engineering—Solid Mechanics,
Technical University of Denmark,
Lyngby DK-2800 Kgs., Denmark

Alan Needleman

Professor
Fellow ASME
Department of Materials Science
and Engineering,
Texas A&M University,
College Station, TX 77843

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received March 5, 2018; final manuscript received March 8, 2018; published online March 30, 2018. Editor: Yonggang Huang.

J. Appl. Mech 85(6), 061007 (Mar 30, 2018) (8 pages) Paper No: JAM-18-1129; doi: 10.1115/1.4039574 History: Received March 05, 2018; Revised March 08, 2018

The indentation of plant cells by a conical indenter is modeled. The cell wall is represented as a spherical shell consisting of a relatively stiff thin outer layer and a softer thicker inner layer. The state of the interior of the cell is idealized as a specified turgor pressure. Attention is restricted to axisymmetric deformations, and the wall material is characterized as a viscoelastic solid with different properties for the inner and outer layers. Finite deformation, quasi-static calculations are carried out. The effects of outer layer stiffness, outer layer thickness, turgor pressure, indenter sharpness, cell wall thickness, and loading rate on the indentation hardness are considered. The calculations indicate that the small indenter depth response is dominated by the cell wall material properties, whereas for a sufficiently large indenter depth, the value of the turgor pressure plays a major role. The indentation hardness is found to increase approximately linearly with a measure of indenter sharpness over the range considered. The value of the indentation hardness is affected by the rate of indentation, with a much more rapid decay of the hardness for slow loading, because there is more time for viscous relaxation during indentation.

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Figures

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Fig. 1

Sketch of the configuration analyzed

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Fig. 2

Finite element mesh used for the calculations with ΔR/Ro = 0.05 and ΔRoR = 0.2. The mesh consists of 22×210 quadrilateral elements with uniform radial spacing for 1≤r/Ro≤0.99 and uniform θ spacing for 90 deg≤θ≤80 deg: (a) full mesh and (b) mesh near r = 0.

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Fig. 3

Curves of normalized hardness, H/Ei, versus normalized indenter depth, h/Ro, for a shell with Eo/Ei=1, pinit/Ei=0.0154, ΔR/Ro = 0.05, β=19 deg, and ΔRoR = 0.2

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Fig. 4

For a shell with pinit/Ei = 0.0154, ΔR/Ro = 0.05, β=19 deg, and ΔRoR = 0.2: (a) normalized hardness, H/Ei, versus normalized indenter depth, h/Ro and (b) normalized contact area, Acont/πRo2, versus normalized indenter depth, h/Ro

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Fig. 5

Distribution of normalized Mises effective stress in thevicinity of the indenter for a shell with pinit/Ei = 0.0154, ΔR/Ro = 0.05, β=19 deg, and Eo/Ei = 1 at h/Ro = 0.139

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Fig. 6

Distribution of normalized Mises effective stress in thevicinity of the indenter for a shell with pinit/Ei = 0.0154, ΔR/Ro = 0.05, β=19 deg, ΔRoR = 0.2, and Eo/Ei = 4 at h/Ro = 0.154

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Fig. 7

For a shell with pinit/Ei = 0.0077, ΔR/Ro = 0.05, β=19 deg, and ΔRoR = 0.2: (a) normalized hardness, H/Ei, versus normalized indenter depth, h/Ro and (b) normalized contact area, Acont/πRo2, versus normalized indenter depth, h/Ro

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Fig. 8

For a shell with pinit/Ei = 0.0154, ΔR/Ro = 0.05, β=19 deg, and ΔRoR = 0.1: (a) normalized hardness, H/Ei, versus normalized indenter depth, h/Ro and (b) normalized contact area, Acont/πRo2, versus normalized indenter depth, h/Ro

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Fig. 9

For shells with Eo/Ei = 2, pinit/Ei = 0.0154, ΔR/Ro = 0.05, ΔRoR = 0.2, and various values of β: (a) normalized hardness, H/Ei, versus normalized indenter depth, h/Ro and (b) normalized contact area, Acont/πRo2, versus normalized indenter depth, h/Ro

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Fig. 10

Distribution of normalized Mises effective stress in thevicinity of the indenter for a shell with β=29 deg and pinit/Ei = 0.0154, ΔR/Ro = 0.05, ΔRoR = 0.2, and Eo/Ei = 2: (a) at h/Ro = 0.065 and (b) at h/Ro = 0.127

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Fig. 11

Dependence of H/Ei at h/Ro = 0.01 on  tan2β

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Fig. 12

Dependence of H/Ei on cell wall thickness with Eo/Ei = 2, ΔRoR = 0.2, and pinit/Ei = 0.0154

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Fig. 13

Dependence of H/Ei on normalized indenter speed V,where tiViref/Ro=0.333 and with Eo/Ei=2, ΔRo/ΔR=0.2, pinit/Ei=0.0154, and ΔR/Ro=0.05

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