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Indentation of two-dimensional solids: The signatures of geometrical and material nonlinearity

The material characterization of ultra-thin solid sheets, including two-dimensional materials like graphene, is often performed through indentation tests. While indentation is a convenient means of measuring properties such as the stretching (two-dimensional) modulus of such materials, experiments on ostensibly similar systems have reported very different material properties. In this paper, we present a modelling study of this indentation process assuming elastic behaviour. In particular, we investigate the possibility that the reported differences may arise from different geometrical parameters and/or non-Hookean deformations, which lead to the system exploring nonlinearities with geometrical or material origins.


Introduction
Just as it is natural to test the inflation of a tyre and ripeness of fruit by poking with a finger, a common means of testing the mechanical properties of solids is via indentation tests (Harding and Sneddon, 1945;Perriot and Barthel, 2004;Butt et al., 2005;McKee et al., 2011) This is particularly the case for very thin materials, which are difficult to manipulate in a tensile testing machine because of their extreme flexibility. Models of the response of floating elastic plates and thin membranes to point indentation date back to the work of Hertz (1884) and Schwerin (1929), respectively, but have remained a fertile area of research. A key topic of interest has been determining the effects of, for example, an underlying substrate (Perriot and Barthel, 2004), pretension (Norouzi et al., 2006), bending stiffness (Wan et al., 2003) and indenter geometry (Begley and Mackin, 2004;Komaragiri et al., 2005).
Recently, there has been particular interest in indentation tests of ultra-thin, or two-dimensional, solids such as graphene, molybdenum disulphide, and mica (Castellanos-Gomez et al., 2015). Such solids have sparked intense interest because of their high material strength, novel electronic properties, and the interaction between imposed elastic strain and electronic properties (Vozmediano et al., 2010;Akinwande et al., 2017). In this regard then, it is the stretching stiffness, or two-dimensional Young's modulus, E 2D , that is of particular interest, since it governs the force required to induce strain, though the measurement of any pre-stress is also of some interest. The first measurement of E 2D in graphene was made by Lee et al. (2008), who reported a value E 2D ≈ 340 N/m as determined by indentation with an Atomic Force Microscope (AFM). While this value is in good agreement with that predicted from first principles (Kudin and Scuseria, 2001), values as low as E 2D ≈ 20 N/m at room temperature (Nicholl et al., 2015) and as high as E 2D ≈ 680 N/m with imposed strain (López-Polín et al., 2017) have been reported. It has also been reported that an optimal number of defects may increase the value of E 2D (López-Polín et al., 2015).
There are many complicating factors involved in the indentation of truly two-dimensional materials like graphene including the importance of thermal crumpling (flexural phonons) that give rise to 'hidden area' (Nicholl et al., 2017), the anisotropy induced by the underlying hexagonal lattice (Kumar and Parks, 2015) as well as the possibility of slip at the boundary. Nevertheless, it has also been pointed out that some inconsistencies may exist in the way that the predictions of classical elasticity theory are used to interpret experimental data (Vella and Davidovitch, 2017;Jia and Ben Amar, 2020). Here, our aim is to set out clearly the predictions of the standard models of mechanics and to highlight the potential pitfalls that experimental attempts to characterize material properties by indentation may fall into. By doing so, we hope that future experiments will be more readily reconciled, and more clearly highlight the truly unique properties of these unusual solids.
The paper is organized as follows. In §2 we present our general modelling approach, together with a simple scaling analysis that highlights the variety of possible behaviours; the regimes in which these different behaviours are expected is summarized in Fig. 2. We then move on to consider in more detail the effect of indenter geometry with a linear stress-strain relation by considering a cylindrical-tipped indenter ( §3) and a spherical-tipped indenter ( §4). In §5 we consider the effect of material nonlinearity (i.e. non-Hookean behaviour), before discussing the significance of our results for the experimental determination of elastic constants in §6 and then summarizing our results and concluding in §7.

Physical model of indentation
Typical experimental measurements of the mechanical properties of two-dimensional solids by indentation involve a sheet being suspended over a circular hole, of radius R out , on an otherwise planar substrate. While the boundary conditions at the edge of the hole are not generally wellcontrolled, it is usually assumed that the sheet is perfectly clamped at this boundary (i.e. there is no additional radial displacement at the edge as the sheet is deformed by indentation) by the film-substrate adhesion. We shall also make this assumption. When the sheet is deposited it is typically subject to a pre-existing tension, or pre-stress, T pre , which may be caused by the processing or fabrication of the sheet. We shall assume that the pre-tension is uniform and isotropic for simplicity.
A quantity of considerable practical interest is the two-dimensional Young's modulus of the sheet, E 2D . For a thin, Hookean sheet of thickness t, and Young's modulus E, E 2D = Et, but we use E 2D throughout so that our theory describes equally thin elastic sheets and two-dimensional solids. We assume the sheet has a (two-dimensional) Poisson's ratio ν.
Indentation is typically performed by an Atomic Force Microscope (AFM) tip, which applies the force F required to impose a vertical displacement δ of the central region (measured relative to the clamped edges). We consider two tip shapes in detail: a cylindrical indenter (of radius R in < R out ), which facilitates our analysis, and a spherical tip (of radius of curvature R s < R out ), which is more representative of the tips used experimentally (see López-Polín et al., 2017, for example). We shall see that the shape of the indenting tip can be important for the final interpretation of the force-displacement relationship. A sketch of the typical experimental setup is shown in Fig. 1. Figure 1: Cross-sectional sketch of the indentation of a clamped sheet by: (a) a cylindrical punch of known radius Rin and (b) a spherical-capped indenter of known radius of curvature, Rs, but unknown contact radius, Rin(δ).

Scalings and physical arguments
The relationship between the indentation force and depth, F (δ), holds important information for understanding the elastic properties of thin sheets. To get a first sense of the possible different behaviours of the force-indentation relation, we begin by considering at a scaling level the various energies in the problem.
The work done by the indenter, which scales as U work ∼ F δ, must be stored predominantly in either the stretching or bending energies of the sheet. To estimate the stretching energy stored within the sheet, we note that the sheet tension has two components: one caused by the pre-tension, T pre , and another caused by the stretching in response to the imposed strain ε ∼ (δ/R out ) 2 , which is ∆T ∼ E 2D · ε. (Note that, for small displacements δ R out , the strain estimate ε ∼ (δ/R out ) 2 can be derived from elementary geometrical considerations.) We therefore write T ∼ T pre + E 2D ε and note that the stretching energy of the sheet must therefore scale according to U s ∼ R 2 out (T pre + E 2D ε)ε. Finally, we note that the bending energy of the sheet scales as U b ∼ BR 2 out K 2 , where K ∼ δ/R 2 out is the typical sheet curvature and B is the bending modulus (for an isotropic, thin, Hookean solid, B = Et 3 /12(1 − ν 2 ), though we use a general bending modulus to account for truly two-dimensional solids, such as graphene, for which the effective value of B may differ significantly from this value 1 ).
Before considering the different limits, we note that the work done, U work , must equal the sum of these different energies, i.e. U work ∼ U b + U s , and hence Written in this way, there are three possible dominant balances, depending on which term dominates the RHS of (1): (i) pre-tension dominant so that F ∼ T pre δ, corresponding to linear membrane theory with constant tension (see Begley and Mackin, 2004;Komaragiri et al., 2005; Davidovitch, 2017, for example); (ii) stretching dominant so that F ∼ E 2D δ 3 /R 2 out , recovering the scaling of the classical Schwerin solution (see Schwerin, 1929;Begley and Mackin, 2004;Komaragiri et al., 2005;Vella and Davidovitch, 2017); (iii) bending stiffness dominant so that F ∼ Bδ/R 2 out , and the sheet responds as a classical plate (see Timoshenko and Woinowsky-Krieger, 1959;Komaragiri et al., 2005, for example).
Taking the pre-tension dominated case as a reference state, natural choices of dimensionless indentation depth, force, and bending stiffness are respectively. Alternatively, one could have taken the bending or stretching dominated cases as the reference state (Komaragiri et al., 2005, for example). As we shall show shortly, in many applications of current interest (especially for two-dimensional materials) the bending dominated region is of limited interest: the pre-tension dominates the bending stiffness, so that B 1. Similarly, as suggested by Vella and Davidovitch (2017), many recent experiments do not always reach the Schwerin regime. As such, we shall use the non-dimensionalization in (2) and shall consider the transition between the dominant balances (i) and (ii).
In dimensionless variables the above dominant balances become: These behaviours/regions were the main discussion of Komaragiri et al. (2005) who considered indentation by an idealized point-indenter (with our regions (i)-(iii) respectively corresponding to regions 3-1 in Komaragiri et al. 2005). Regions (i)-(iii) are shown in the regime diagrams Fig. 2 along with subregions (associated with the onset of bending and stretching) and asymptotic results -which are both established in Sections 3 & 4. We emphasize that region (i) corresponds to a constant compliance, d/F , while region (ii) corresponds to a constant 'cubic compliance', d/F 1/3 . We shall be focussed in this paper on understanding the dependence of each of these compliances on the material properties of the system, and the behaviour of the system in-between the asymptotic regimes that correspond to regions (i) and (ii).
From these simple energy arguments, one might assume that the indenter's geometry has little effect on the response d(F). This assumption has been made implicitly across a range of experimental work (Lee et al., 2008;Castellanos-Gomez et al., 2015;López-Polín et al., 2015; 3.4 × 10 −18 J. As a result, graphene is 10 times more bendable than would be expected for a corresponding isotropic solid; this reflects the fact that, since it is only a single molecule thick, the usual mechanism for generating a bending stiffness (i.e. differential strain through the sheet thickness) is not relevant for graphene.

Stretching dominant
Bending dominant Pre-tension dominant Non-linear elasticity

Stretching dominant
Bending dominant Pre-tension dominant et al., 2017, to name a few) -in a complex experimental setup, applying the point-indenter 'solutions' to experimental data allows progress to be made. In this paper, we investigate the circumstances in which the radius of contact, R in , and the shape of the indenter matter. For a cylinder, we therefore introduce the dimensionless radius For a sphere, it is not immediately clear what length scale should be used to measure the sphere size; here we use the tensile length R out (E 2D /T pre ) 1/2 , which is used to rescale vertical deflections; we thus introduce the dimensionless sphere radius:

Moderate strains and rotations
The energy arguments described in the last section assumed that the stress and strain were linearly related, i.e. that the material remains Hookean throughout. This assumption even fails before indentation occurs if the pre-strain ∼ T pre /E 2D = O(1) or during indentation if the stretching induced strain ∼ F/rE 2D = O(1) for R in ≤ r ≤ R out . In dimensionless variables these two conditions are equivalent to requiring: for a Hookean response, where we have introduced the new dimensionless variable as a measure of the pre-strain in the sheet caused by the pre-tension T pre . (The variable E is only required for the nonlinear elastic model in §5.) In the case of a spherical indenter (for which the radius of contact R is unknown) an analogous bound to (5b), can be formulated by noting that our detailed analysis in §4 shows that R ∼ (R 3 s F) 1/4 , eq. (25b), so that the linear analysis holds provided that F R s /E 2 . When the strains remain small, it is possible to make reasonable amounts of analytical progress (see Vella and Davidovitch, 2017, for example). However, much of the recent interest in graphene has focussed on whether its material properties change measurably with strain (see e.g. Nicholl et al. 2015;López-Polín et al. 2017). We shall, therefore, be interested here in presenting models of indentation in which the material behaviour becomes nonlinear at some point during indentation. In §5 we consider the effects of large strains/slopes and nonlinear constitutive relationships using the theory of finite elasticity. For now, however, we focus on describing the linearly elastic behaviour more fully.

Cylindrical indentation of a linear elastic sheet
We begin by considering the case of a cylindrical indenter, for which the specification of the problem (especially the boundary conditions) is relatively simple.

Föppl-von Kármán formulation
We initially confine our attention to linear elasticity with the Kirchhoff assumptions (i.e. small strains and plate rotations), so that the Föppl-von Kármán (FvK) equations hold (Mansfield, 2005). The FvK equations link the out-of-plane displacement of the sheet, z(r), to the stress via a stress potential ψ(r), which is defined such that the principal stresses are σ rr = ψ(r)/r and σ θθ = ψ (r), thereby ensuring that the in-plane equation holds automatically (Mansfield, 2005). The out-of-plane force balance and the compatibility of strains condition may both be integrated once to give: respectively. Note that, since we are assuming linear elasticity, the deformed and undeformed configurations are interchangeable; the radial coordinate in the sheet is thus denoted r ∈ [R in , R out ]. At the outer-rim, r = R out , we assume a perfectly clamped boundary: Here (8a,b) are geometric conditions of zero vertical displacement and slope, while (8c) ensures the radial displacement is fixed to be that caused by the initial isotropic tension T pre . At the inner-rim, r = R in , we assume a perfect-slip boundary (in reality there may be a small amount of adhesion/friction between the indenter and sheet): Here (9a,b) are geometric conditions of continuous vertical displacement and slope, while (9c) comes from a force balance with the inner (known-geometry) solution [ψ(r) ∝ r]. Throughout this paper we shall assume that the sheet and indenter remain in contact for r ≤ R in .

Non-dimensionalization
We use the dimensionless variables introduced in the discussion of energy and scalings in §2.2; in particular, we define Substitution of (10) into (7)-(9) gives the dimensionless system for R ≤ ρ ≤ 1, subject to the boundary conditions (12d-f) (Recall that F, B, and R are defined in equations (2a), (2c), and (3), respectively.) For given parameters ν, B, F, and R, (11) subject to (12) may be solved by a standard numerical integrator (in our work we used bvp4c in Matlab). To make analytical progress we consider separately two asymptotic limits that allow for simplifications: (i) small deflections from the pre-stretched state and (ii) negligible bending stiffness. In §3.2 we consider the first of these, by considering perturbations to the isotropic pre-tensed state (ψ ∼ T pre r, Ψ ∼ ρ), thereby extending the work of e.g. Jennings et al. (1995); Wan et al. (2003), while in §3.3 we consider the second case by considering the zero bending stiffness limit (B = 0), similarly to e.g. Bhatia and Nachbar (1968); Vella and Davidovitch (2017). This analysis allows us to reproduce some previously known results in a systematic way, whilst also uncovering new results in some regimes; we discuss the broader context of these results as they are derived.
3.2. Small indentation forces (F max{B 1/2 , R}) For sufficiently small indentations, the stretching of the sheet is negligible compared to the isotropic pre-tension and the (also small) bending stiffness, B 1. To investigate how bending and pre-tension interact, we follow the approach of e.g. Vella et al. (2012); Box et al. (2017), and linearize the governing equations (11) about the initial pre-tensed state: we let Ψ(ρ) ∼ ρ +Ψ(ρ) and Z(ρ) ∼ 0 +Z(ρ) forΨ,Z 1. The two equations (11) thus reduce to a single third-order differential equation forZ that can be solved by a linear combination of logarithms and modified Bessel functions (Appendix A). Applying the boundary conditions (12) we obtain an explicit relationship between the force and indentation depth that is linear (i.e. F ∝ d). We, therefore, express this through the compliance d/F: (14c,d) and I j (x) and K j (x) are the jth-order modified Bessel functions of the first and second kind, respectively. Equation (13) corresponds to a constant stiffness regime: the compliance d/F is a function of the indenter radius, R, and bending stiffness, B, only. One could use this solution as an explicit formula to describe the small indentation compliance of a tense clamped plate. As a tool for inferring the bending stiffness from an experimental measure of the compliance, however, the complexity of this equation is daunting since the various dimensionless quantities are coupled inside the Bessel functions. Instead, recall that we are specifically interested in the limit of small bending stiffnesses B 1 and note, from (14b,d), that B/R 2 is a key parameter. We, therefore, consider separately the cases B R 2 and R 2 B 1. We find the leading-order results where γ ≈ 0.577 is the Euler-Mascheroni constant. The evolution of the compliance with indentation force is shown in Fig. 3 for a fixed R and four values of B; we see that, for small F, the numerical results agree with our asymptotic results. Note, in particular, that the results for B = 10 −8 and B = 10 −9 are essentially indistinguishable at the scale of the plot (Fig. 3a): in both cases, R 2 = 10 −6 B and so the small-indentation compliance is controlled by the indenter radius, rather than bending stiffness, as predicted by (15). In these cases, the compliance is essentially indistinguishable from that of an ideal membrane with B = 0.
Overall, for small displacements by a cylindrical indenter (F max{B 1/2 , R}), the indentation compliance d/F is a constant determined by the relative size of the indenter radius R and the bending stiffness B 1/2 . We note also that the fact that the form of the stiffness in each of the cases in (15) is functionally similar -both are logarithmic -is not a coincidence: in the bendingdominated case, the sheet is approximately flat over a region of horizontal scale B 1/2 , and it is as if the sheet were deformed by a virtual cylindrical indenter of radius R ∼ 2e −γ B 1/2 .
While the indenter-dominated solution (15)[B R 2 < 1] has been previously derived (see Jennings et al., 1995, for example), we believe that the bending-dominated solution (15)[R 2 B 1] is novel; this describes scenarios when the bending stiffness is small, but cannot be neglected because the indenter size is smaller than the virtual bending-induced indenter radius, which is therefore the relevant length scale.

Small bending stiffnesses
In many experimental setups the dimensionless bending stiffness B is small enough that the sheet can be modelled as a thin membrane. In this case, we can simplify the model by taking B → 0 in (11a) and dropping the highest-order boundary conditions (12b,e) -ignoring the effects of the edge boundary layers, which occur over a typical length scale ρ ∼ B 1/2 . Vella and Davidovitch (2017, App. B) solved the membrane problem for a point indenter (R = 0); we extend their work by solving the finite cylinder case (the full derivation can be found in Appendix B). Ultimately, we obtain the parametric force-displacement relation: which is parametrized by the boundary stresses Φ 1 := AΨ(1) and Φ 0 := ARΨ(R), these 'stresses' must further satisfy the equation For a given indentation force F, equations (16b) and (16d) form a pair of equations for the two unknowns Φ 0 and Φ 1 . Equation (16a) therefore yields an implicit force-displacement relation d(F; R, ν). In the limit of the inner stress (or equivalently the radius) taken to zero,  (14)).
To make further analytical progress with a finite indenter radius, we concentrate on the two asymptotic limits F → 0 ( §3.3.1) and F → ∞ ( §3.3.2). This allows us to determine explicit forms that are formally valid in these limits, but that might be expected to apply more broadly.

Moderate indentation forces
where Φ 0 is given by The parameter Φ 0 can be eliminated from (17) to give an explicit equation for d in terms of F; however, the result is complicated and does not readily reveal the limiting behaviours of the indenter. Instead we consider the sub-cases F R and F R, which correspond to taking Φ 0 → ∞ and Φ 0 → 0, respectively.
At leading-order we find The very small displacement solution in (18) (found when F R) matches precisely with our earlier solution from (15) with B R 2 . Moreover, the moderate displacement solution in (18) (found when R F 1) is that presented by Vella and Davidovitch (2017) for a point indenter, and is shown as the red-dashed curve in Fig. 3. In summary, for a cylinder, as the indentation force increases, the indentation compliance d/F evolves from being a constant (controlled by the indenter radius) to a logarithmic behaviour (controlled by an F-dependent stretching-induced radius).

Large indentation forces
Inserting these into (16) gives the force-displacement relation: where Φ 1 and Φ 0 are the solutions to The expression in (19a) corresponds to a constant cubic compliance: d/F 1/3 is a function of the indenter size R and Poisson's ratio ν only. Experimentally, the indenter is often orders of magnitude smaller than the sheet clamping radius, so that R 1 (Lee et al., 2008); this limit corresponds to Φ 0 = O(R 4/3 ) → 0, and so the cylindrical result (19) simplifies to where Φ 1 is the solution to This is the point-indenter result from Vella and Davidovitch (2017) with an additional (2R) 2/3 term to account for the small (but finite) indenter size. We find that the cubic compliance where q c [ν, R] and q p [ν] are given by the leading-order equations in (19) and (20) respectively. Thus, for large indentations by a cylindrical indenter (B 1/2 1 F), the cubic compliance d/F 1/3 is a constant controlled by the sheet's Poisson's ratio ν and indenter radius R. An analogous result was found by Vella and Davidovitch (2017, eq. (48)), for the case of a no-slip indenter: Fig. 4 shows the comparison between numerical simulations, the prediction for a point indenter R = 0, and the expression (21) for R = 0.1. We see that the effect of finite indenter size is non-negligible in the large force limit.
The asymptotic predictions (15), (18), and (21) are presented in the regime diagram Fig. 2a. We now move on from the case of a cylindrical indenter to a spherical one.

Spherical indentation of a linearly elastic sheet
Having considered in some detail the simplest case of a finite, cylindrical indenter, we now move on to a case of more experimental relevance: an indenter with a hemispherical cap (e.g. Bhatia and Nachbar 1968;Jennings et al. 1995;Begley and Mackin 2004;Lee et al. 2008;López-Polín et al. 2017). The important difference with the cylindrical indenter already considered is that the radial position of the edge of contact, R in , is initially unknown and evolves with the indentation depth, δ, as the sheet wraps more of the indenter.  21)[full] in green. Observe the difference made by geometry: results for a spherical indenter never reach the constant cubic compliance regime d/F 1/3 = cst, while even for cylindrical indenters the finite indenter size may play a significant role.

Föppl-von Kármán formulation
The Föppl-von Kármán formulation for a spherical-capped indenter is the same as the cylindrical indenter formulation in §3.1 but with a modified inner boundary condition, to account for the new indenter geometry. These take the dimensionless form Here (22a-c) express the geometric conditions of continuous vertical displacement, slope, and curvature, while (22d) comes from an in-plane force balance, since within the contact region it may easily be shown that Ψ(r) + ρ 3 /16R 2 s ∝ ρ. We again emphasize that the point of contact R is unknown here and must be determined as part of the solution, explaining why we require an extra boundary condition (22c) compared to the cylindrical case.
The system of equations (11) subject to (12a-c) and (22) can be solved for given parameters ν, B, F, and R s by using a standard numerical integrator with unknown R. To facilitate this computation, it is more convenient to fix the value of R and solve for an unknown F instead. By doing so, one avoids the problems associated with an unknown domain size.
To make analytical progress we apply the same asymptotic simplifications as in the cylindrical case: (i) linearize around the pre-stretched base state [i.e. take Ψ(ρ) ∼ ρ +Ψ(ρ) and Z(ρ) ∼ 0 +Z forΨ,Z 1]; or (ii) consider the membrane theory limit [i.e. take B → 0 and drop the boundary conditions (12b) and (22c)]. The analysis of these asymptotic limits are analogous to the cylindrical indenter case presented in §3.2 & §3.3, but with the added detail that R is unknown; we shall only present the concluding results below (the full analysis can be found in Appendix A, small indentation forces, and Appendix B, small bending stiffnesses).
4.2. Small indentation forces (F max{B 1/2 , R s }) Linearizing around the pre-tensed state, we find that with where γ ≈ 0.577 is again the Euler-Mascheroni constant. Thus, for small displacements by a spherical-capped indenter (F max{B 1/2 , R s }), the indentation compliance d/F evolves from being a constant (controlled by the bending-induced radius) to a logarithmic behaviour (controlled by the radius of contact, which in turn depends on the force). This evolution is shown in Fig. 3 and confirms that the numerical results reproduce the expected asymptotic results in the relevant limits.
The asymptotic compliance (23a) is directly equivalent to that for a cylindrical indenter, i.e. (15), and can be recovered by accounting for the extra indentation depth due to the sphere geometry [d → d + R 2 /2R s , i.e. compare (12d) and (22a)] and inserting the contact radius expression (23b). It should be noted that the indenter-dominated solution (23)[B R s F] has also been derived by Bhatia and Nachbar (1968, eq (43b)), from analytical solutions to small-rotation Reissner theory (analogous to FvK), and Norouzi et al. (2006, eq. (14)), by minimizing the energy of the constant-tension problem.
4.3. Moderate indentation forces (min{B 1/2 , B/R s } F 1) Taking the membrane theory limit and assuming small indentation forces (F 1) we find that with Thus, for moderate indentations by a spherical-capped indenter (min{B 1/2 , B/R s } F 1), the indentation stiffness F/d evolves from one logarithmic behaviour (controlled by the radius of contact) to another (controlled by a stretching-induced radius); these asymptotic results are also shown in Fig. 3.
Solution (24) is directly equivalent to the cylindrical result (18), and can again be obtained by accounting for the extra indentation depth due to the sphere geometry [d → d + R 2 /2R s , i.e. compare (12d) and (22a)] and inserting the contact radius expression (24b).

Large indentation forces (F 1)
Taking the membrane theory limit and assuming large indentation forces (F 1), we find that with Note that this result suggests a different effect of the geometry of the indenter, and the associated change in contact radius, than that proposed by Jia and Ben Amar (2020) based on a linear fitting procedure. In particular, our result predicts a weakly force-dependent cubic compliance, O(R 1/2 s F 1/6 ). Above, we have implicitly assumed that R 3 s F 1, which must be true for the assumption of small slopes to hold (R R s 1). Thus, for large indentations by a spherical indenter (F 1), the cubic compliance d/F 1/3 is controlled by the sheet's Poisson's ratio ν and contact radius R ∼ (R 3 s F) 1/4 . This is shown in Fig. 4, and demonstrates that in the spherical case the cubic compliance never saturates at a constant value (while it does for a cylindrical indenter). Nevertheless, the numerically-observed behaviour of the cubic compliance is in good agreement with the asymptotic prediction (25).
To our knowledge, the solution (25) is new. Moreover, while it is functionally similar to the corresponding cylindrical result (21), it cannot be obtained by accounting for the extra indentation depth [d → d + R 2 /2R s ] and inserting the contact radius expression (25b), as we did for small indentation depths. This difference is because the stress in the membrane that is in contact with the indenter is different between the two models -compare (12f) and (22d). The asymptotic results (23a), (24a), and (25a) are presented in the regime diagram Fig. 2b. Figure 5 shows numerical results for the force-indentation relationship and the contact radius obtained from our FvK model with a spherical indenter. The results of Fig. 5a are plotted to show how the instantaneous estimate of q(ν)E 2D from the point indenter model, defined as q(ν)E 2D := F R 2 out /δ 3 with q(ν) := 2πq p [ν] −3 in our notation, evolves with increasing force. Results are shown for a range of assumed pre-tensions and sphere radii, with typical maximal applied loads used in AFM indentation indicated (López-Polín et al., 2017).

Numerical results for a spherical indenter
These numerical results show three key features. Firstly, increasing the pre-tension may have the (undesired) effect of pushing measurements further from the desired horizontal asymptote (since in practice, only a finite indentation force can be applied). Secondly, results with the larger radii of curvature, R s /R out , may ultimately lead to the geometric nonlinearity that pushes results further from the ideal point indenter solution of Schwerin (1929). Thirdly, it seems that the fraction of the indenter that is wrapped by the membrane is relatively large, with R in /R s lying in the interval 0.3 R in /R s 0.7 in the regime of experimental interest.
Overall, and more qualitatively, our key result is that the cubic compliance is not constant for spherical indenters with realistic material parameters. It seems plausible that such non-constant behaviour might be interpreted experimentally as a nonlinear material effect, either softening or stiffening. Before discussing this possibility further, we turn now to consider such material nonlinearities.

Nonlinear elastic materials
Thus far, through the application of the Föppl-von Kármán equations, we have retained the leading-order geometrical nonlinearities associated with deformation, but have assumed that the material response remains Hookean: we have neglected any effect of nonlinear constitutive response. A simple energetic scaling ( §2.3) shows that this assumption is valid provided that both E 1 and F RE −3/2 , where E is the dimensionless pre-tension defined in (6). Many recent experiments have shown behaviour different to that expected on the basis of the FvK model of point indentation, and concluded that they are probing the nonlinear mechanical response of graphene (e.g. Lee et al. 2008;López-Polín et al. 2017). However, we have also seen that the geometry of a spherical indenter can give behaviour that differs from the usual cubic response expected from the FvK equations. The question, therefore, is what happens when the indentation advances beyond the small-strain limit and how should one distinguish this regime from the geometrically nonlinear effects associated with the indenter shape? We introduce a model that allows for the possibility of large slopes and strains. Specifically, we use the work of Green and Adkins (1960), who derived a generalized model for the large deformations of an elastic membrane. The formulation presented below follows similar work by Yang and Feng (1970) We will concentrate on the specific case of the indentation by a perfect-slip, spherical-capped wedge (with radius of curvature R s and wedge angle 2 1). This type of indenter is commonly used in experiments (e.g. Lee et al. 2008;López-Polín et al. 2017) and reduces to the sphericalcapped indenter used in §4 provided the indentation depth is sufficiently small. (Hence the results of this section should deviate from those of §4 only at large indentation depths.) Here, we also ignore the effects of the bending stiffness (B = 0), since we are ultimately interested in the stretching dominant limit.

Governing ODEs
To allow for large rotations of the sheet, it is useful to introduce intrinsic coordinates (the radial arc-length ξ and angle of rotation α, which is measured with respect to the radial-r-axis). We then have the geometrical conditions dz dξ = sin α and dr dξ = cos α, (26a,b) where we recall that 0 ≤ r ≤ R out and z are the radial and vertical coordinates of the sheet, respectively. These variables are sketched in Fig. 6. The membrane is then split into two regions: the region in which the membrane contacts the tip and a non-contacting region. In the contacting region (which occupies 0 ≤ r ≤ R in with R in as yet unknown) we require for wedge angle 2 . In the out-of-contact region (i.e. R in ≤ r ≤ R out ) we impose the (integrated) vertical force balance where T ξ and T φ are the thickness-averaged, in-plane, radial and azimuthal stresses, which must satisfy the in-plane force balance d dr Note that the out-of-plane stress is forced to be zero for thin sheets (t 1) [Green and Adkins 1960]. In modelling finite deformations of a thin membrane, one must distinguish between the reference and deformed configurations. Here, we take the reference configuration to be the planar sheet, subjected to an isotropic tension T pre ; this configuration is parametrized by ( , 0) for 0 ≤ ≤ R out (where clamping is imposed at = r = R out ). The variables describing the deformed configuration are expressed as functions of and hence we define the principal stretches as Here λ ξ is the longitudinal stretch (along a cross-sectional curve in the r-z plane), λ φ is latitudinal stretch (along the direction normal to r-z plane), and λ z is the out-of-plane stretch (a measure of membrane thickness t compared to its reference value t 0 ). We also introduce the planar and out-of-plane pre-stretches Λ := Λ ξ = Λ φ and Λ z which measure the initial isotropic stretching of the sheet; we will relate these pre-stretches to the pre-tension T pre shortly. System (26) is closed by imposing a constitutive relation that links the stresses with the principal stretches (27). In particular, by assuming a hyperelastic isotropic medium, we can introduce a constitutive strain energy density (per unit-volume) function W (λ 1 , λ 2 , λ 3 ) [see e.g. Holzapfel 2002] which directly links the stretches and stresses.
Using this formalism, we first calculate the pre-stretch Λ by solving the initial stress state, witht 0 being the undeformed sheet thickness and Λ z chosen to satisfy the zero out-of-plane stress condition ∂W ∂λ 3 (Λ, Λ, Λ z ) = 0.
The stresses in the deformed configuration are then computed using where λ z satisfies the zero out-of-plane stress condition Together, (26)-(29) form a system of three ordinary differential equations for the three unknowns -the in-plane stretches λ ξ (r) and λ φ (r) and the vertical displacement z(r): where α(r) is given by (26c), for 0 ≤ r ≤ R in , and (26d), for R in ≤ r ≤ R out ; and T ξ (λ ξ , λ φ ), T φ (λ ξ , λ φ ), and T ξ, j (λ ξ , λ φ ) := dT ξ / dλ j are given by (28) and (29). To proceed further requires a particular choice of strain energy function W , and so we turn to discuss this now.

Choice of strain energy density function
The choice of strain energy density function W (λ 1 , λ 2 , λ 3 ) is dependent on the material of interest. In this paper, we will present results for two hyperelastic models to show the influence of this choice. In particular, we present results for a neo-Hookean material (the natural extension of the Hookean response implicit in the FvK equations) and for a Gent hyperelastic material (which is a model developed for polymeric materials with finite extensibility, but is chosen here as a qualitative way to account for the finite bond-lengths in graphene). The Gent model contains a parameter b that captures the finite chain length; we note that the neo-Hookean strain energy function is recovered as this maximal chain length diverges, i.e. as b → 0. The formulation of the compressible strain energy density functions is a lengthy process -the details are presented in Appendix C. The final form of the Gent energy density function used is where β := ν/(1 − 2ν) is some known constant, ν = 1/2 is the Poisson ratio, I i are the usual tensor invariants (defined in Appendix C), and b is some empirical parameter based on the finite extensibility of the material, defined such that 1/b ≡ max{I 1 − 3}.

Boundary conditions
Having split the domain into contacting (0 ≤ r ≤ R in ) and non-contacting regions (R in ≤ r ≤ R out ), the problem is a multi-point boundary value problem and so we require boundary conditions at three positions r = 0, R in , and R out .
At the outer edge, r = R out : the sheet is perfectly clamped, At the origin, r = 0: we require an isotropic stretch (due to the symmetry of the problem), At the interface between the contacting and non-contacting regions, r = R in : a local force balance reveals that we require continuity in radial stress T ξ (λ ξ , λ φ ) and membrane slope α. Coupling this with the physical requirement of continuity of deformed variables z and r = λ φ , we require continuity in all our variables α, z, λ ξ , and λ φ . [This would not have been the case if there was a corner in the imposed indenter geometry (e.g. for a cylindrical punch).] Note that continuity in α gives an extra equation for the unknown point of contact R in (F ).

Non-dimensionalization
To non-dimensionalize the problem we use the same choice of dimensionless radius and vertical coordinate (ρ and Z) as the Föppl-von Kármán formulation (10a,c), and define also were we use E 2D :=t 0 E to be the two-dimensional Young's Modulus of the undeformed sheetthis is equivalent to the FvK choice under small strains. Substitution of (10a,c) and (33) into equations (26c,d), (30), and (32) gives a dimensionless system for λ ξ (ρ), λ φ (ρ), Z(ρ), and α(ρ), with Cauchy stresses given bŷ where the principal stretches Λ and Λ z solve For a given strain energy density functionŴ (λ 1 , λ 2 , λ 3 ) [we use (31)] the above system can be solved by a standard numerical integrator (in our work we use bvp4c in Matlab) for given parameters ν, F, E, and R s , with unknown R. The associated indentation depth δ(F) can then be calculated from (32d). In practice, however, it is move convenient to impose R instead of F and use the first derivative of (26d) to form a differential equation for the out-of-contact α: for R ≤ ρ ≤ 1, with a continuity boundary condition at ρ = R; F can then be extracted post computation, along with δ. By doing so, one avoids the issue of unknown domain size.

Results
The aim of introducing a nonlinear elastic model was to quantitatively understand when the effects of material (as opposed to geometric) nonlinearity are felt through their impact on the key force-displacement curve. Although the quantitative results are highly dependent on the chosen constitutive strain energy density function (34), the transition from the linearized-material asymptotics of §4 occurs at a similar indentation force, independent of the choice of strain energy function. Thus, we can analyse the numerical effect of varying the sphere radius R s = (R s /R out )E 1/2 and pre-tension E. In Fig. 7 we present a table of force-displacement curves obtained as E and R s vary; observe that the dimensionless force at which the results deviate from the Föppl-von Kármán solutions increases with R s /R out and decreases with E = T pre /E 2D -reminiscent of the prediction from the energetic analyses, F ∼ R s /E 2 , presented in §2.2. It is also interesting to note that, close to the place at which the non-Hookean results deviate from the FvK results, the effect of material nonlinearity is to soften the material response. We discuss these results, and their significance for indentation probes of the elastic constants of thin materials, now.  Table of plots showing how the onset of nonlinear elasticity depends on the spherical cap curvature (rows) and the applied pre-tension (columns) of the membrane (ν = 1/3, B = 0) -here we have used an indenter wedge angle of 2 = π/9 (matching the value in e.g. López-Polín et al. 2017). Numerical solutions are shown as solid curves: Föppl-von Kármán (linear elastic) in black, Gent (b = 1) in blue, Gent (b = 0.5) in green, and neo-Hookean (b = 0) in yellow. The asymptotic results from the FvK analysis for a point and spherical indenter, (25a) with Rs = 0 and Rs = (Rs/Rout)(Tpre/E2D) 1/2 , are shown as dotted and dashed lines respectively. The vertical dashed lines show where the relative difference between the Neo-Hookean and FvK models reaches 10%.

Discussion: Application to fitting protocols
The controlled indentation of thin circular sheets is a common, but delicate, experimental technique used to extract mechanical properties of materials. Our detailed analysis of the cylindrical and spherical indentation has led to a number of asymptotic results that highlight the complexity of this problem (these results are summarized in §7). Consequently, there are a number of potential pitfalls in this fitting procedure that must be appreciated if they are to be avoided and a reliable measurement of the quantity of interest to be obtained. Below, we shall discuss these pitfalls in the context of measurements of three properties: the sheet pre-tension T pre , the two-dimensional Young's modulus E 2D , and the non-Hookean material behaviour (i.e. the behaviour not governed by linear elasticity). We shall concentrate on spherical-capped indenters, since these are among the most commonly used in practice.

Measuring sheet pre-tension
Our results showed the existence of a small-indentation regime in which the pre-tension dominates and F ∝ T pre δ (Fig. 2). Based on our analysis of this region, there are two potential pitfalls that may cause errors when attempting to infer T pre from experimental measurements of F : • Bending stiffness. Although the bending stiffness is small at a macroscopic level, i.e. B T pre R 2 out , it is not necessarily negligible in the early stages of indentation with a sphericalcapped indenter (see Fig. 2b): at small indentations, the spherical cap is barely wrapped by the sheet, the effective contact radius is small and it is the bending stiffness of the sheet that dominates. In this case, the indentation is constrained by the bending-induced radius, (B/T pre ) 1/2 , instead of the contact radius. Avoiding this requires that the applied force be sufficiently large; in particular, that F B/R s for small bending stiffnesses (B 1/2 R s T pre /E Combining these requirements, we find that indentation tests aimed at measuring the pretension in a sheet should focus on indentation forces F T such that

Measuring 2D Young's modulus
The measurement of the two-dimensional Young's modulus of a sheet requires experiments to be performed in the stretching dominated regime, where F ∝ E 2D δ 3 /R 2 out (Fig. 2). Based on our analysis of this regime, there are two immediate potential pitfalls that may cause fitting errors: • Sheet pre-tension. If the sheet is not sufficiently indented, the effect of the pre-tension may still be significant, leading to a non-cubic response (i.e. the sheet is still transitioning between the linear F ∼ d and cubic F ∼ d 3 behaviours, see Fig. 7). To avoid this, sufficiently large forces should be applied; in particular F R out T 3/2 pre /E 1/2 2D . • Mechanical nonlinearities. If the sheet is indented too much, the stress in the sheet exceeds that for which the linear elastic (Hookean) constitutive response is valid, and nonlinearities become important. In this case, the expected cubic response may not be observed (see the solution divergence in Fig. 7) causing errors in the fitted value of E 2D . To ensure this possibility is avoided requires F E 2D R s .

21
Combining these requirements, we find that indentation tests aimed at measuring the stretching modulus in a sheet should focus on indentation forces F E 2D such that A third potential pitfall is less obvious and independent of the indentation force applied: • Geometrical nonlinearities. When fitting experimental data to obtain a value for the 2D Young's modulus, it is common practice to use the Schwerin (1929) point indenter solution, which in our notation reads F = Q(ν)d 3 for Q(ν) := q p [ν] −3 . For example, a point-wise estimate, E 2D = F R 2 out /2πQ(ν)δ 3 is often used (e.g. in López-Polín et al. 2017). However, the analysis presented here shows that the indenter geometry is important and may lead to large errors in the fitted value of the 2D Young's modulus if not accounted for. In particular, for spherical indenters the point-wise estimate of E 2D may not converge as the force increases, so the constant value (as predicted by Schwerin) is never observed (see Fig. 7). To avoid a significant effect of the spherical geometry requires R s /R out 1; in practice, R s /R out 10 −2 appears to be sufficient (see Fig. 7); in many experimental setups R s /R out ∼ 0.1 (see López-Polín et al., 2017, for example) so the indenter geometry may play a role.

Measuring non-Hookean material behaviour
If interested in examining the non-Hookean material behaviour of the sheet (i.e. the behaviour beyond the Hookean linear stress-strain relation), there are two potential pitfalls to avoid: • Remaining in the linear elastic regime. The main concern when measuring non-Hookean behaviour is whether the strains induced by indentation are large enough to be controlled by a nonlinear constitutive law. For a spherical-capped indenter, this requires • Geometrical nonlinearities. A less obvious trap, is that deviating from Schwerin's solution [F = Q(ν)d 3 ] might be interpreted as nonlinear mechanical effects, but actually result from nonlinear geometry. For example, for sufficiently large spherical indenters, the Föppl-von Kármán response will never achieve the expected Schwerin result (Fig. 7). This failure to reach the Schwerin regime might be interpreted as a material nonlinearity, rather than a universal geometric property. To avoid this possibility requires geometric effects to be negligible throughout the linear elastic regime, that is R s R out .
Together, (36)-(38) provide limits on the validity of fitting force-indentation curves for mechanical properties. (Of course, their validity can only be checked with some initial parameter estimate or a posteriori.) It should be noted that these are only based on order arguments and are not concrete cut off points. Instead, when fitting, one should always ensure the wanted asymptotic behaviour is distinguishable (e.g. ensure that there is indeed a cubic plateau, F/d 3 ∼ constant, when fitting for the Young's modulus). These solutions can be generalised to other indenter geometries by examining the parameter bounds in the cylindrical regime diagram, Fig. 2a, and taking the contact point as an observable quantity, R in (F ).

Summary
We have presented a series of asymptotic solutions that may be used in combination with experimental force-indentation data to fit mechanical properties of thin materials, subject to an appreciation of the pitfalls described in §6. These results apply in various asymptotic regimes determined by the relative sizes of the dimensionless force F, bending stiffness B, indenter radius R (or R s ), and pre-tension E. Here, we summarize these results for a cylindrical and sphericalcapped indenter; these are most easily expressed in terms of the vertical indentation depth d achieved for a fixed force F.
Cylindrical indenter. For indentation by a cylindrical-capped indenter, the asymptotic response is dependent on the relative size of the bending stiffness B and indenter area R 2 . We have that Here, γ ≈ 0.577 is the Euler-Mascheroni constant and q c [ν, R] ∼ q p [ν] − (2R) 2/3 is defined in (21). These solutions are valid provided E 1 and F RE −3/2 ; if either of these conditions fail then the material instead behaves according to a nonlinear constitutive law (non-Hookean behaviour). The regions of the regime diagram Fig. 2a are delineated by the expressions in (39).
Spherical indenter. For indentation by a spherical-capped indenter, we find the asymptotic response: with the (a priori unknown) contact radius given by Here, γ ≈ 0.577 is the Euler-Mascheroni constant and q s [ν, R 3 s F] ∼ q p [ν] − 2(2 − √ 2)(R 3 s F) 1/6 is defined in (25). These solutions are valid provided E 1 and F R s E −2 ; if either of these conditions fail then the material instead behaves according to a nonlinear constitutive law (non-Hookean behaviour). The regions of the regime diagram Fig. 2b are delineated by the expressions in (40).

Conclusions
Altogether, our work provides a comprehensive description of the Hookean response of a sheet accounting for geometrical nonlinearities, while additionally providing information about when a non-Hookean response can be expected. It is common in experiments to use dimensional versions of similar asymptotic solutions to extract information about the material of interest (e.g. its 2D Young's modulus E 2D or pre-tension T pre ). The asymptotic results presented in (39) and (40)  max ) errors in the fitted Young's modulus -which can be significant in such a sensitive process. We therefore emphasize the importance of using the correct response when fitting parameters.
Finally, we note that the asymptotic regimes considered here have been motivated by recent indentation experiments on ultra-thin materials, including few layer graphene. For such materials, B T pre R 2 out , corresponding to B 1, and so the different regimes described in eqns (39) and (40) are only valid when B 1. As the dimensionless bending stiffness becomes larger, B = O(1), the balances that lead to these results are expected to change. In particular, we expect that for dimensionless forces F 1 and B ∼ 1 new results would be required (indicated by the greyed-out regions in Fig. 2).
Most importantly (A.1a) no longer couples the out-of-plane deflection with the in-plane stress; the third order differential equation for the sheet profile Z(ρ) ∼Z(ρ) can immediately be integrated to give with the constants c 1 , c 2 , and c 3 yet to be determined. (The stress profile may be determined by substituting (A.2) into (A.1b); we omit this here since it affects the force-indentation response only at higher order in d.) The sheet profile (A.2) is to be solved subject to the boundary conditions for a cylindrical indenter (R known), or for a spherical indenter (R unknown). Implementation of (A.3)-(A.5) gives explicit expressions for the coefficients and for a cylindrical indenter. For a spherical indenter, we have and with R implicitly given by (A.7d) (In each case,Î j ,Î R j ,K j , andK R j are defined in eq. (14).) The small force-displacement relations are then found by imposing the boundary conditions (A.4a) and (A.5a), to give: for a cylindrical indenter; and with R implicitly given by for a spherical indenter.
In the limit of small bending stiffnesses which is relevant here, B 1, (A.8) and (A.9) can be simplified using the asymptotic behaviour of modified Bessel functions to give Finally, considering the limits B R 2 < 1 and R 2 B 1 leads to the leading-order solutions (15) for a cylindrical-capped indenter and (23) for a spherical-capped indenter.

Appendix B. Asymptotic solutions for small bending stiffnesses
In this Appendix, we expand the arguments of §3.3, §4.3, and §4.4 to derive the asymptotic solutions for large indentation depths, and negligible bending stiffnesses (18), (21), (24), and (25). We follow the analysis of a point-indenter by Vella and Davidovitch (2017, App. B) and present the case of cylindrical and spherical-capped indenter simultaneously.
Applying the boundary conditions (B.2)-(B.4) leads to a system for the unknown constants A, B, and C. To simplify these equations, we define Φ 0 := AΦ(R 2 ) and Φ 1 := AΦ(1); the system then takes the form Φ 1/2 for a spherical indenter. Eliminating B and C from system (B.7) with (B.8) or (B.9), leaves a force-displacement equation given implicitly by parameters Φ 0 , Φ 1 , and A (with R as an extra unknown in the spherical case); the resulting system for a cylindrical indenter is given in (16). We now consider the two limits F → 0 and F → ∞ to derive explicit asymptotic forcedisplacement relations; in particular, equation (B.7c) determines how A and Φ 1 relate to the applied force, F. Appendix B.1. Moderate indentation forces (F → 0) In the small indentation limit, F → 0, eq. (B.7c) gives A ∼ Φ 1 = 4/F 2 + O(1), which leads to the parametric force-displacement relationship for a cylindrical indenter: For a spherical indenter, we find where Φ 0 and R are given by Taking the limits F R < 1 and R F 1 leaves the leading-order solutions (18) for a cylindrical-capped indenter and (24) for a spherical-capped indenter.