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Research Papers

# Effect of Properties and Turgor Pressure on the Indentation Response of Plant CellsOPEN ACCESS

[+] 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

## Abstract

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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## Introduction

There is significant interest in understanding the mechanical aspects of the growth of plants [1,2]. In systems biology attempts are made to integrate the biological sciences with the quantitative approaches of applied mathematics, physics, and engineering in order to model biological processes computationally. The mechanical behavior of individual plant cells plays an important role in the growth process, including in cell division. Geitmann [3] discussed the mechanical behavior of the three most important elements determining structure in plant and hyphal cells, the cytoskeleton, the cell wall, and turgor pressure. The cytoskeleton is the elastic scaffold, while the individual cell behavior is determined by the cell wall properties and the turgor pressure.

Much work has focused on the interaction of the turgor pressure and the cell wall mechanics. The reviews [3,4] mention a number of experimental methods including microcapillary, that can be used to directly measure the turgor pressure, and applications of indentation of the cell. Routier-Kierzkowska and Smith [5] showed how micro-indentation results are sensitive to turgor pressure, while nano-indentation applies so little force that it only deforms the cell wall locally. Several experimental results on cell wall thickness, turgor pressure, and the elastic modulus are presented by Hayot et al. [6] for Arabidopsis ecotypes, both for young leaves, intermediate leaves, and old leaves. Beauzamy et al. [7] used atomic force microscopy together with osmotic treatments to carry out indentation studies aimed at quantifying turgor pressure. Variations in Young's modulus between different cell wall layers have been studied for oak wood fibers by Clair et al. [8], using an atomic force microscope in force modulation mode, and for spruce wood by Wimmer and Lucas [9], using nano-indentation. Also, Gindl et al. [10] carried out nano-indentation experiments to study the effects of structural variability in spruce wood on the mechanical response. Mosca et al. [11] found that variations in the indentation response of plant cells in tissue did not necessarily mean a difference in cell properties, but could be due to differences in the surrounding environment. Nano-indentation was used by Branco et al. [12] to stimulate a structural defense response in a plant cell as a model for exploring the response of plants to potentially pathogenic microbes.

In this study of indentation of a plant cell, the cell is represented as a spherical shell loaded by internal pressure prior to the indentation. Among recent investigations of indentation in shells under internal pressure, both polymeric capsules and yeast cells were studied in Ref. [13], and in Ref. [14], the cell wall in living shoot apical meristems were studied, which are highly organized structures that contain the plant stem cells. Vella et al. [13] also analyzed the relation between indentation depth, indenter force, and internal pressure, using linear shallow shell theory, or nonlinear shell theory in cases where the indentation depth is much larger than the shell thickness. Kol et al. [15] used indentation to investigate the mechanical properties of a leukemia virus particle, and Zelenskaya et al. [16] used atomic force microscopy together with indentation analyses to understand the mechanical behavior of cochlear hair cells. Ogbonna and Needleman [17] carried out a numerical study of indentation of thick-walled elastic spherical shells using a finite deformation formulation, and Tvergaard and Needleman [18] analyzed the indentation of pressurized polymer spherical shells.

For the plant cells analyzed here, the cell wall material is modeled as hyperelastic, using a finite strain formulation, and also viscous effects are accounted for, with a consistent representation of the hydrostatic turgor pressure inside the cell. In a parametric study of the effect of different combinations of cell wall stiffness, cell wall thickness, and turgor pressure, the possibility is included that the stiffness varies through the thickness. It is shown that for very small depth of indentation, the hardness is nearly independent of the turgor pressure, whereas for larger depth indentation, where shell wall bending starts to play a role, the hardness depends on the level of the turgor pressure.

## Governing Equations

The cell is modeled as a bimaterial spherical shell with a thin outer stiff layer and an inner softer layer. Attention restricted to axisymmetric deformations and the undeformed inner radius is denoted by Ri, the undeformed outer radius by Ro, and the undeformed inner radius of the outer stiff layer by Rw, as sketched in Fig. 1. The shell thickness, denoted by ΔR, is given by $ΔR=Ro−Ri$, and the thickness of the outer stiff layer is denoted by ΔRo and given by $ΔRo=Ro−Rw$.

As in previous studies, e.g., see Refs. [17] and [18], a finite deformation convected coordinate Lagrangian formulation is used. Here, the quasi-static calculations are based on the principle of virtual work written in incremental form as Display Formula

(1)$∫Ω(τ˙ijδEij+τiju˙k,iδu,jk) dV+p∫Spα˙irnrδui dS+p˙∫Spαirnrδui dS=0$

with Display Formula

(2)$Eij=12(ui,j+uj,i+u,ikuk,j)$

and, see Sewell [19] Display Formula

(3)$αir=12εijkεlmr(gjl+uj,l)(gkm+uk,m)$

Here, τij are the contravariant components of Kirchhoff stress on the deformed convected coordinate net ($τij=Jσij$, with σij being the contravariant components of the Cauchy stress and J the ratio of current to reference volume), nr and uj are the covariant components of the reference surface normal and displacement vectors, respectively, ρ is the mass density, p is the pressure, gij are the covariant components of the reference metric tensor, $( ),i$ denotes covariant differentiation in the reference frame, and Ω and S are the volume and surface of the body in the reference configuration. A superposed $(˙)$ denotes $∂( )/∂t$ at a fixed material point.

All field quantities are considered to be functions of convected coordinates, yi, and time, t. A polar coordinate system is used in the reference configuration with $y1=r, y2=θ$, and $y3=z$.

The spherical shell is indented by a rigid conical indenter, with the cone angle β defined as shown in Fig. 1. The quasi-static loading is applied in two phases: (i) the internal pressure p is increased monotonically until a specified pressure, pinit, is reached; (ii) the sphere is then indented along the axis of symmetry by a conical indenter. During the indentation phase of loading, symmetry about the central plane (z = 0) is assumed, so that the results correspond to simultaneous indentation at two opposite poles of the shell. Also, perfect sticking is presumed when the indenter comes into contact with the sphere so that the rate boundary conditions are Display Formula

(4)$u˙3=−Vi on Scont$

and h, the indenter depth, is given by Display Formula

(5)$h=∫0tVidt$

In Eq. (4), Scont denotes the portion of the surface of the sphere in contact with the indenter. We note that h defined in Eq. (5) is the displacement of the rigid indenter and is, in general, not the depth of indentation into the material due to the overall deformation of the shell.

The contact area, Acont is specified as Display Formula

(6)$Acont=πrcont2$

where $rcont$ is defined as the largest distance in the deformed configuration from the symmetry axis, where the indenter and outer surface of the shell are in contact, and the hardness is defined by Display Formula

(7)$H=PAcont$

where P is the indentation force.

## Constitutive Relation

The logarithmic strain rate $ε˙$ is written as the sum of a nonlinear elastic part, $ε˙e$, and a linear viscous part, $ε˙v$ so that Display Formula

(8)$ε˙=ε˙e+ε˙v$

A principal axis formulation of nonlinear elasticity due to Hill [20,21] is used. The principal axes of elastic strain are taken to coincide with the principal axes of Kirchhoff stress. The principal strain values $εie$ and principal stress values τi are related via the isotropic relation Display Formula

(9)$εie=1+νEτi−νE(τ1+τ2+τ3)$

In Eq. (9), E is the Young's modulus, and ν is the Poisson's ratio.

The rate constitutive relation is expressed on the current principal axes, and the rate elastic relation can be written in the form Display Formula

(10)$ε˙e=R−1:τ̂$

where $τ̂$ is the Jaumann derivative of Kirchhoff stress and $A:a=Aijklalk$.

On the principal axes, the components of the moduli Rijkl for i = j, k = l have the form Display Formula

(11)$Rijkl=E1+ν[δikδjl+ν1−2νδijδkl]$

where, for axisymmetric deformations, the $r−z−$ plane shear moduli are given by [20,21] Display Formula

(12)$R1313=R3131=R3113=R1331=qE2(1+ν)$

with Display Formula

(13)$q=(ε1e−ε3e)coth(ε1e−ε3e)$

On the principal axes Display Formula

(14)$τ̂ij=Rijkl(ε˙kl−1μmτkl′)$

The linear viscous relation is Display Formula

(15)$ε˙v=1μmτ′$

Here, $τ′=τ−tr(τ)/3$.

Hence, we can write Display Formula

(16)$εe=∫(ε˙−1μmτ′)dt$

In the limit $μm→∞$, the relation reduces to a hyperelastic relation.

To update the stress, we first express Eq. (14) in general tensor notation on the deformed base vectors as $τ˙cij=Lijkl(E˙kl−qkl)$ with the convected rate given by Display Formula

(17)$τ˙cij=τ̂ij−τikg¯jlE˙kl−τjkg¯ilE˙kl$

where $E˙kl$ are the Lagrangian strain rate components, and update the stress via Display Formula

(18)$τij(t+dt)=τij(t)+τ˙cijdt=τij(t)+Lijkl(E˙kl−qkl)dt$
with qkl being a consequence of the second term on the right hand side of Eq. (14).

## Numerical Results

The cell wall is modeled as consisting of two layers: an outer stiff layer and an inner more compliant layer. Quantities associated with the outer layer are denoted with a subscript ()o, and those associated with the inner layer are denoted with a subscript ()i.

There is no characteristic length in the quasi-static analyses carried out here, so, when appropriately normalized, the computed results only depend on the ratios of geometric quantities, not on their absolute values; specifically on the ratio of the shell thickness to its outer radius, ΔR/Ro and on the ratio of the outer stiff layer thickness to total thickness, ΔRoR. In this regard, we note that representative values of cell wall thicknesses mentioned for various plant cells are in a range of about 700 nm, see, for example, Ref. [6]. Hayot et al. [6] also mentioned representative values of the Young's modulus in the range 30–90 MPa, representative values of the turgor pressure in the range 0.10–0.30 MPa, and a representative cell radius in the range of 7 μm. All stress quantities are normalized by the value of Young's modulus of the inner layer, Ei, and all time quantities are normalized by the value of the material characteristic time, $ti=μmi/Ei$.

Calculations are carried out for two values of $ΔRo/ΔR, ΔRo/ΔR=0.2$, and $ΔRo/ΔR=0.1$ and for two values of the ratio of outer layer Young's modulus, Eo, to inner layer Young's modulus, namely Eo/Ei = 2 and Eo/Ei = 4. For comparison purposes, calculations are also carried out with Eo/Ei = 1. The value of Poisson's ratio ν is taken to have the uniform value 0.42 throughout the shell.

The finite element mesh for the calculations with ΔR/Ro = 0.05 and ΔRoR = 0.2 and is composed of 22 × 210 quadrilateral elements, with each quadrilateral element consisting of four “crossed” linear displacement triangular subelements. There are eight uniformly spaced, in the radial direction, elements between r/Ro = 1 and r/Ro = 0.99, and then, the element size is gradually increased in the radial direction. In the θ direction, there are 100 uniformly spaced elements between $90 deg$ and $80 deg$. This mesh is shown in Fig. 2. A 22 × 210 quadrilateral mesh is also used for the calculations with $ΔRo/ΔR=0.1$ and with the same distribution in the θ direction. However, in these calculations, there are four uniformly spaced elements in $1≤r/Ro≤0.995$ in the r direction, and then, the element size is gradually increased.

In the calculations here, the characteristic time, to, for the outer layer is ti, 2ti, and 4ti, for Eo/Ei = 4, 2, and 1, respectively. The value of to increases linearly with $μmo$, and a longer characteristic time gives a more nearly elastic response, see Eq. (16).

During pressure loading, the prescribed loading rate is $tip˙/Ei=0.051$ and, unless specified otherwise, Vi in Eq. (4) is taken to have the value $Viref$ given by $tiViref/Ro=0.333$.

In the calculations, acont in Eq. (6) varies in discrete steps when a new nodal point comes in contact with the indenter. This gives a highly oscillating numerically induced variation of hardness H with indenter depth that masks the underlying material governed response, particularly for small indenter depths. In order to provide a clearer picture of the material-governed indentation response, the “raw” output curves from the computer code are smoothed by plotting the points midway between the peaks and the troughs, as shown in Fig. 3. Even with this smoothing, the values of H for small values of h/Ro, with the precise value depending on the material properties and shell geometry, are not reliable since the oscillations are so large. In the following, the results corresponding to less than about 0.003 are not presented. We note that with ΔR/Ro = 0.05 as in Fig. 3, h/Ro = 0.003 corresponds to hR = 0.06.

Figure 4 shows the indentation response of a shell with pinit/Ei = 0.0154 and ΔRoR = 0.2 for three values of Eo/Ei. At a small value of indenter depth, the hardness essentially scales with Eo/Ei. For example, in Fig. 4(a), at h/Ro = 0.005, H/Ei = 0.24, 0.45, and 0.91 for Eo/Ei = 1, 2, 4, respectively. For larger values of h/Ro, the response is shell bending dominated and is independent of the ratio Eo/Ei. On the other hand, in Fig. 4(b), the contact area is nearly independent of Eo/Ei for small indenter depths and dependent on Eo/Ei for larger indenter depths. At h/Ro = 0.01, the values of H/Ei are 0.25, 0.41, and 0.77 for Eo/Ei = 1, 2, 4, respectively. An indenter depth of h/Ro = 0.01 corresponds to hRo = 1.0, so that at least for these parameter values an indenter depth of the outer shell thickness is still largely governed by the outer shell stiffness. For an indenter depth of h/Ro = 0.1, the corresponding values of H/Ei are 0.075, 0.090, and 0.10, so that for large indenter depths, where bending dominates, the value of H is much less dependent on the outer layer stiffness.

Figures 5 and 6 show contours of normalized Mises effective stress, σe/Ei in the vicinity of the indenter for cases with pinit/Ei = 0.0154, ΔR/Ro = 0.05, and ΔRoR = 0.2. In Fig. 5, Eo/Ei = 1, while in Fig. 6Eo/Ei = 4. Here, the Mises effective stress is based on the Kirchhoff stress deviator $τ′$ and given by Display Formula

(19)$σe2=32τ′:τ′, τ′=τ−13tr(τ)I$

where tr () is the trace, and I is the identity tensor.

In Fig. 5, where Eo/Ei = 1 (so that the material properties are uniform through the shell thickness), there is a high stress concentration, σe/Ei ≈ 0.65, under the indenter, a location similar to what is expected for indentation into a half space, but there is also a stress concentration of σe/Ei ≈ 0.4 near the inner radius due to shell bending. With Eo/Ei = 4 in Fig. 6, the area of high stress, i.e., $σe/Ei≥0.6$, under the indenter is much larger and extends further away from the central axis. The maximum value of H/Ei in this region is about 1.25. However, since the stress concentration is confined to the stiffer outer shell, the ratio of σe to the local Young's modulus, Eo is ≈ 0.3. The bending stress concentration is reduced compared with that for the shell with Eo/Ei = 1.

In Fig. 7, all parameters are fixed except for the value of the internal pressure. In these calculations, the value of p is reduced by half to pint/Ei = 0.0077. The values of H/Ei at h/Ro ≈ 0.01 in Fig. 7(a) for Eo/Ei = 1, 2, 4 are 0.23, 0.40, and 0.77, respectively. These differ little from the corresponding values with p/Ei = 0.0154 in Fig. 4(a) indicating that the value of the hardness H for small values of h/Ro is set by the material properties and is largely independent of the internal pressure. For $h/Ro≈0.1,H/Ei=0.070$, 0.079, and 0.086, for $Eo/Ei=1,2,4$. As in Fig. 4(a), the dependence of H on the outer layer stiffness is not great, and the values of H are reduced by up to ≈15% with a reduction of a factor of 2 in the value of pint. In Fig. 7(b), the contact area is essentially independent of the ratio Eo/Ei, whereas in Fig. 4(b) there is a significant dependence on Eo/Ei.

Calculations were also carried out with $pinit/Ei=0.00385$. In these calculations, the values H/Ei = 0.23, 0.41, and 0.80 at h/Ro ≈ 0.01 were obtained for Eo/Ei = 1, 2, 4, which are essentially the same values as with the two greater values of internal pressure. At h/Ro ≈ 0.1, the corresponding values of H/Ei are 0.076, 0.084, and 0.090.

Figure 8 shows the evolution of the normalized indentation hardness, H/Ei, and contact area, $Acont/πRo2$, with indenter depth, h/Ro, where the stiff outer layer thickness is $ΔRo/ΔR=0.1$. The values of H/Ei in Fig. 8(a) at h/Ro ≈ 0.01 are 0.25, 0.34, and 0.51 for Eo/Ei = 1, 2, 4, respectively. Thus, there is a substantial reduction in indentation hardness with reduced outer layer thickness that increases with increasing outer layer stiffness. There is a much smaller effect for larger indenter depths, with H/Ei = 0.082, 0.090 for Eo/Ei = 2, 4 at h/Ro ≈ 0.10. The contact area in Fig. 8(b) shows a qualitatively similar variation with h/Ro as seen in Fig. 4(b).

The effect of the sharpness of the conical indenter on the evolution of the indentation hardness and on the contact area is shown in Fig. 9 for various values of the indenter angle β. The case $β=19 deg$ corresponds to the calculation with Eo/Ei = 2 in Fig. 4. The value of H at a given value of h/Ro in Fig. 9(a) increases monotonically with β (increasing β corresponds to increasing cone sharpness), while the corresponding value of the contact area, Acont, in Fig. 9(b) decreases monotonically with β.

The evolution of the effective stress distribution is shown in Fig. 10. At the earlier stage of deformation, h/Ro = 0.065, as shown in Fig. 10(a), the curvature-induced stress concentration is evident. Also, the increased stress level in the inner layer associated with the increased value of β can be seen. The qualitative features of the stress distribution are consistent with those in Figs. 5 and 6. In Fig. 10(b), where h/Ro = 0.127, the stress concentration associated with the curvature has relaxed, and a more nearly uniform region of high effective stress occurs throughout the softer inner layer under the indenter in Fig. 10(b). This corresponds to large stretching in that region. Thus, for the parameter values considered and at least over the range computed, smaller values of β give rise to a bending-dominated deformation mode under the indenter, whereas larger values of β induce large stretching under the indenter at relatively small indenter depths.

The shell effect can be quantified by considering the dependence of the indentation hardness on the cone angle β. For indentation into a half-space, the nominal contact area is proportional to $1/ tan2β$. Hence, the indentation hardness H is proportional to $tan2β$. The actual contact area can involve sink-in or pile-up depending on material properties, but for fixed material properties, the proportionality is expected to remain at least a good approximation. Figure 11 shows the variation of H/Ei at h/Ro = 0.01, denoted by H0.01/Ei, with $tan2β$ for the four values of β in Fig. 9. Although there is near linearity, the ratio, $H0.01/(Ei tan2β)$ does decrease with increasing β, being 4.35, 3.46, 2.88, and 2.41 for $β=14 deg, 19 deg, 24 deg$, and $29 deg$, respectively. Thus, the effect of the shell geometry comes into play at h/Ro = 0.01, which corresponds to hR = 0.20 with ΔR/Ro = 0.05 as in Figs. 9 and 11.

Figure 12 shows the effect of shell thickness for a case with $Eo/Ei=2, ΔRo/ΔR=0.2$, and pinit/Ei = 0.0154. In the early stages of indentation, the amplitude of the oscillations is too large to accurately assess the effect of shell thickness on the indentation hardness. However, at h/Ro ≈ 0.01, there seems to be little variation as would be expected for a sufficiently small indenter depth. At h/Ro = 0.02, the values of normalized indentation hardness are 0.34, 0.40, and 0.45 for ΔR/Ro = 0.05, 0.10, and 0.20, respectively. The corresponding values of H/Ei at h/Ro = 0.05 are 0.20, 0.29, and 0.37. Consistent with the steeper drop seen in Fig. 12, the value of H/Ei decreases more rapidly with indenter depth for thinner shells.

The effect of indentation rate is shown in Fig. 13. The normalized indentation rate used in all previous calculations is $Viref=tih˙/Ro=0.333$. Figure 13 shows results for indentation rates between 1/10 and 10 times the reference value. In these calculations, $Eo/Ei=2, ΔRo/ΔR=0.2, pinit/Ei=0.0154$, and $ΔR/Ro=0.05$. For sufficiently small values of h/Ro, the indentation response is relatively independent of the indentation rate. In this regime, the stiff outer layer and the relatively short time for stress relaxation play key roles in determining this. Subsequently, if the indentation rate is sufficiently rapid, there is little time for viscous stress relaxation so that the material response is essentially elastic. As can be seen in Fig. 13, there is little difference in the response for indentation rates between $Viref/2$ and $Viref$. Furthermore, increasing the indentation rate by a factor of 10 has only a small effect on the evolution of the indentation hardness. On the other hand, for an indentation rate of $Viref/10$ stress relaxation leads to a significantly lower value of indentation hardness, H/Ei, over a range of values of h/Ro. For example, at $h/Ro=0.03$, the value of normalized indentation hardness, H/Ei, is 0.10 for $Vi=Viref/10$ and varies between 0.23 and 0.30 for Vi in the range $Viref/3$ to $10Viref$. For sufficiently, large values of indenter depth, the turgor pressure plays a dominant role, and the effects of differences in indentation rate are reduced.

## Conclusions

The indentation analyses show that for very small indenter depths, smaller than the thickness of the outer stiff layer, the hardness depends mainly on the elastic modulus of that layer, not on the turgor pressure. As the indenter depth increases slightly, the deformation field under the tip grows into the softer inner layer of the shell, which results in a rapid decay of the hardness. This rapid decay is not seen for the shell with no outer stiff layer. At larger indenter depth, the hardness is much affected by shell bending effects and by the level of the turgor pressure. This agrees with the experimental observation that micro-indentation results are sensitive to the turgor pressure, while nano-indentation only deforms the cell wall locally.

The difference in Young's modulus between two layers of the cell wall has been studied to mimic observations of such variations. It is clear that the thickness of the hard outer layer plays a role; in particular, a thinner layer gives an earlier transition to the situation dominated by bending and turgor pressure. If there were a stiff layer both on the inside and on the outside of the cell wall, this would increase the bending stiffness and thus add to the indentation hardness at larger indenter depth.

The general continuum mechanics formulation used for the present analyses has the advantage that large rotations are fully accounted for. Near the indenter tip, the material will certainly rotate by the rather large angle β, and away from the tip, the material in contact will have rotated more. It is also ensured that the internal pressure always acts normal to the deformed inner surface of the cell wall. The formulation allows for arbitrarily large strains, but the calculations here are limited to moderate elastic strains, because the elastic constitutive relation, Eq. (9), gives rise to stress levels of a significant fraction of the Young's modulus at moderate values of elastic strain. However, near the tip where very large stresses develop relative to the Young's modulus, there are significant geometry changes. In reality how large stresses the plant material can carry before failure will be limited, but the effect of damage is not incorporated in the present analyses.

The cell wall thickness has a noticeable effect on the development of hardness with indenter depth, in that the hardness drops more rapidly for a thinner shell and less rapidly for a thicker shell. However, if the curves in Fig. 12 were plotted against hR rather than h/Ro the difference would apparently be much less, which is reasonable, since the change to the turgor pressure-dependent range depends on the indenter depth relative to the wall thickness.

Variation of the indenter angle β gives rise to a continuous increase of the hardness with increasing β. Also the rate of indentation plays a role, with a much more rapid decay of the hardness in the case of slow loading, because there is more time for viscous relaxation during indentation.

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Kol, A. , Gladnikoff, M. , Barlam, D. , Shneck, R. Z. , Rein, A. , and Rousso, I. , 2006, “ Mechanical Properties of Murine Leukemia Virus Particles: Effect of Maturation,” Biophys. J., 91(2), pp. 767–774. [PubMed]
Zelenskaya, A. , de Monvel, J. B. , Pesen, D. , Radmacher, M. , Hoh, J. H. , and Ulfendahl, M. , 2005, “ Evidence for a Highly Elastic Shell-Core Organization of Cochlear Outer Hair Cells by Local Membrane Indentation,” Biophys. J., 88(4), pp. 2982–2993. [PubMed]
Ogbonna, N. , and Needleman, A. , 2011, “ Conical Indentation of Thick Elastic Spherical Shells,” J. Mech. Mater. Struct., 6(1–4), pp. 443–452.
Tvergaard, V. , and Needleman, A. , 2016, “ Indentation of Pressurized Viscoplastic Polymer Spherical Shells,” J. Mech. Phys. Solids, 93, pp. 16–33.
Sewell, M. J. , 1967, “ On Configuration-Dependent Loading,” Arch. Ration. Mech. Anal., 23(5), pp. 327–351.
Hill, R. , 1970, “ Constitutive Inequalities for Isotropic Elastic Solids Under Finite Strain,” Proc. R. Soc. London Ser. A, 314(1519), pp. 457–472.
Hill, R. , 1978, “ Aspects of Invariance in Solid Mechanics,” Advances in Applied Mechanics, Vol. 18, C.-S. Yih , ed., Academic Press, New York, pp. 1–75.
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Mosca, G. , Sapala, A. , Strauss, S. , Routier-Kierzkowska, A. L. , and Smith, R. S. , 2017, “ On the Micro-Indentation of Plant Cells in a Tissue Context,” Phys. Biol., 14(1), p. 015003. [PubMed]
Branco, R. , Pearsall, E.-J. , Rundle, C. A. , White, R. G. , Bradby, J. E. , and Hardham, A. R. , 2017, “ Quantifying the Plant Actin Cytoskeleton Response to Applied Pressure Using Nanoindentation,” Protoplasma, 254(2), pp. 1127–1137. [PubMed]
Vella, D. , Ajdari, A. , Vaziri, A. , and Boudaoud, A. , 2012, “ The Indentation of Pressurized Elastic Shells: From Polymeric Capsules to Yeast Cells,” J. R. Soc. Interface, 9(68), pp. 448–455. [PubMed]
Milani, P. , Gholamirad, M. , Traas, J. , Arnodo, A. , Boudaoud, A. , Argoul, F. , and Hamant, O. , 2011, “ In Vivo Analysis of Local Wall Stiffness at the Shoot Apical Meristem in Arabidopsis Using Atomic Force Microscopy,” Plant J., 67(6), pp. 1116–1123. [PubMed]
Kol, A. , Gladnikoff, M. , Barlam, D. , Shneck, R. Z. , Rein, A. , and Rousso, I. , 2006, “ Mechanical Properties of Murine Leukemia Virus Particles: Effect of Maturation,” Biophys. J., 91(2), pp. 767–774. [PubMed]
Zelenskaya, A. , de Monvel, J. B. , Pesen, D. , Radmacher, M. , Hoh, J. H. , and Ulfendahl, M. , 2005, “ Evidence for a Highly Elastic Shell-Core Organization of Cochlear Outer Hair Cells by Local Membrane Indentation,” Biophys. J., 88(4), pp. 2982–2993. [PubMed]
Ogbonna, N. , and Needleman, A. , 2011, “ Conical Indentation of Thick Elastic Spherical Shells,” J. Mech. Mater. Struct., 6(1–4), pp. 443–452.
Tvergaard, V. , and Needleman, A. , 2016, “ Indentation of Pressurized Viscoplastic Polymer Spherical Shells,” J. Mech. Phys. Solids, 93, pp. 16–33.
Sewell, M. J. , 1967, “ On Configuration-Dependent Loading,” Arch. Ration. Mech. Anal., 23(5), pp. 327–351.
Hill, R. , 1970, “ Constitutive Inequalities for Isotropic Elastic Solids Under Finite Strain,” Proc. R. Soc. London Ser. A, 314(1519), pp. 457–472.
Hill, R. , 1978, “ Aspects of Invariance in Solid Mechanics,” Advances in Applied Mechanics, Vol. 18, C.-S. Yih , ed., Academic Press, New York, pp. 1–75.

## Figures

Fig. 1

Sketch of the configuration analyzed

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.

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

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

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

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

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

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

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

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

Fig. 11

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

Fig. 12

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

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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