Nonlinear contact mechanics for the indentation of hyperelastic cylindrical bodies


The mechanical properties of biological materials are commonly found through the application of Hertzian theory to force-displacement data obtained through micro-indentation techniques. Due to their soft nature, biological specimens are often subjected to large indentations, resulting in a nonlinear deformation behavior that can no longer be accurately described by Hertzian contact. Useful models for studying the large deformation response of cylindrical specimens under indentation are not readily available, and the morphologies of biological materials are often closer to cylinders than spheres (e.g., cellular processes, fibrin, collagen fibrils, etc.). In this study, a computational model is used to analyze the large deformation indentation of an incompressible hyperelastic cylinder in order to provide a generalized formulation that can be used to extract mechanical properties from indentation into soft cylindrical bodies. The effects of specimen size and indentation depth are examined in order to quantify the deformation at which the proposed force-displacement relationship remains accurate.


Measuring the mechanical response of biological specimens is an important key to developing new biomedical materials [1], understanding injury or disease progression [2,3,4], and potentially improving clinical diagnostic technologies [5, 6]. However, obtaining useful mechanical properties of most biological specimens (e.g., individual cells) is a difficult task due to the nonlinearity, anisotropy, and heterogeneity [7,8,9] that they exhibit.

Within the last century, a common approach to measuring the mechanical properties in cells is through indentation testing—an experimental technique that gained popularity due to the relative convenience of probing adherent cells grown on flat, conventional substrates [10]. By indenting the cell surface with a simple geometry probe (e.g., sphere, cone, etc.), it is possible to use the relationship between the applied force and the concurrently measured indentation depth, to estimate an effective local modulus of the cell. The prevalence of indentation testing, such as atomic force microscopy (AFM), is also attributable to the fact that previously used techniques, such as micropipette aspiration, might disturb the attachment of the cell membrane to the underlying cytoskeleton [10]. AFM was formally introduced in 1986 [11], although similar indentation-like devices appeared before this, such as the “cell-poker” device of the Elson group in the 1970s [12]. By the 1990s, Radmacher and others began using AFM on individual cells to create “elasticity maps,” or images that depict elastic properties across the various regions of the cell with high spatial resolution [13,14,15].

Since AFM now offers the convenience of being commercially available and compatible with traditional cell culturing techniques, it has become a commonly employed tool in biology and biomechanics. However, the results from indentation testing are only useful when considerations are taken with regards to subcellular heterogeneities [9, 10, 16]. Furthermore, it is unfortunately common to find reports that incorrectly apply certain contact mechanics formulations to cellular indentation tests. The well-known classical Hertz solution assumes linear elasticity, which should only be used to estimate the elastic moduli when describing small indentations. However, cells can be subjected to large deformations due to their soft nature, often surpassing the linear elastic regime associated with classic Hertz theory [17,18,19]. Therefore, the usefulness of the elastic modulus is limited, and it is desirable to obtain hyperelastic parameters that can account for the finite deformations [20, 21] and strain stiffening behavior [22] of biological materials.

In classical Hertz theory [23], aside from assuming an isotropic linear elastic material, the solution is limited to small deformations since the shape of the spherical indenter is given by a parabolic function approximated as r2/2Rs, where r is the radial coordinate and and Rs is the indenter radius [24, 25]. It is possible to model larger indentation depths (up to 0.8 times the indenter radius) by using a higher order function to describe the spherical indenter [26]. However, such a solution will still be limited by providing only the linear elastic modulus from the experimentally found forces and displacements.

In recent decades, several approaches have been taken to modify the Hertz solution, in order to obtain parameters that describe the material of the specimen as hyperelastic. The first approach involves the derivation of an approximate analytical expression, and validation of the expression with a finite element calculation [27, 28]. A second approach is to use a finite element model to investigate the effects of hyperelasticity on indentation and then create correction factors to use in conjunction with the equations of Hertzian theory [29, 30]. Finally, if the sole purpose of a study is to determine the material properties in a specific experiment, it is possible to disregard the classical Hertzian solution, and parametrically run a series of finite element simulations, varying the material properties for a given constitutive model, until the experimentally found force-displacement relationships are obtained [31, 32].

However, the aforementioned solutions are for the case of a rigid sphere indenting a half-space or another spherical body. There are no force-displacement relationships that exist for finding hyperelastic parameters in structures with a cylindrical morphology. Quantifying these relationships is essential for finding properties of cylindrical cellular structures—neuronal axons, cell processes, filopodia, and certain types of bacteria, to name a few—that undergo large deformations during indentation testing. Here, we only consider a spherical indenter, as previous works have found that pyramidal indenters produce stresses and indent depths that are too large for cells [33, 34].

The ratio of length scales between the indentation depth, indenter radius, and specimen radius is important for selecting the appropriate indentation formulation. The indentation depth is given by δ, Rs is the radius of a spherical indenter tip, and Rc is the radius of the cell. For indenter tips with a radius much smaller than the radius of the cell (Rs << Rc), one can approximate that the cell is a half-space, and it is no longer necessary to consider the cylindrical morphology. Although AFM indenter tips with small radii (<100 nm) are readily available, a larger indenter tip might be desirable in applications where it is necessary to probe mechanical properties over a larger area [35,36,37], to avoid nonlinear effects [38, 39], or create loads over a larger area when studying cellular injury [40] and mechanotransduction [41].

In this paper, we use a computational model to investigate the force-displacement relationships for spherical indentation into hyperelastic cylindrical bodies (Fig. 1). We provide analytical expressions, with the calculated corrective functions, that can be used to obtain hyperelastic material properties from indentation testing on flat and cylindrical bodies (using both flat and spherical indenters). The force-displacement relationships were found to be dependent on the ratio of indenter to specimen radius (Rs/Rc), as well as the ratio of indentation to indenter size (δ/Rs). We observe the strains at which the indentation is large enough that the solution deviates from linear elastic theory. Finally, we evaluate the proposed hyperelastic corrective functions for large indentations in order to quantify the extent of error due to interactions with the substrate.

Fig. 1

a Example of a cylindrical body (e.g., axon) subjected to indentation by a spherical indenter. b Experiment is simplified as a sphere indenting a cylindrical body c A finite element model of spherical indentation. Note that due to symmetry, only one-fourth of the model needed to be simulated. The indenter was modeled as a rigid body and displaced along the negative x-direction by δ


Elliptical contact onto linear elastic bodies

In the case of a rigid spherical indenter in contact with a flat half-space made of an isotropic linear elastic material, the force-displacement relation is given by the classical Hertz solution:

$$ F_{H}=\frac{4}{3}R_{s}^{1/2}\frac{E}{1-\nu^{2}}\delta^{3/2} $$

where FH is the magnitude of the force applied to the indented body (e.g., a cell), Rs is the effective radius of the spherical indenter, ν is the Poisson’s ratio of the cell, and E is the Young’s modulus of the cell. This relationship is for a spherical contact area and can only be used when probing the somal (spherical) compartment of a cell, or a large, flat cell with a radius of curvature much larger than the indenter.

In the case of the silica bead indenting an axon or cell process, this can be idealized as a rigid sphere (with radius Rs) indenting a cylindrical body (with radius Rc), and thus an elliptical contact area is formed. A schematic of two generalized ellipsoidal shaped bodies and their respective geometries is shown in Fig. 2. To account for the eccentricity in the contact area, correction factors are used to modify (1). We will also introduce the parameters Rz and Ry which are the effective radii of curvature in the z-plane and y-plane, respectively. The effective curvature sum R, a quantity of importance in the analysis of contact mechanics, is defined by [42, 43]:

$$ \frac{1}{R}=\frac{1}{R_{z}}+\frac{1}{R_{y}} $$


$$ \frac{1}{R_{z}}=\frac{1}{R_{c,z}}+\frac{1}{R_{s,z}} $$
$$ \frac{1}{R_{y}}=\frac{1}{R_{c,y}}+\frac{1}{R_{s,y}} $$

For a cylinder where Rc, y goes to infinity, the last expression simplifies to Ry = Rs, y. For a spherical indenter, Rs, z = Rs, y. Now, the relation for force-displacement during an elliptical contact for this linear elastic case can be written as:

$$ F_{ellip}^{LE}=\frac{2 \pi}{3}(2R)^{1/2}\frac{E}{1-\nu^{2}}\delta^{3/2}\chi_{ellip} \\ $$

where χellip is a product which is solely dependent on the ratio between the indenter radius Rs and the cylinder radius Rc and can be written as:

$$ \chi_{ellip}=k\mathcal{F}^{(-3/2)}\mathcal{E}^{1/2} $$

where k is the eccentricity parameter of the contact area (given as the ratio of semimajor to semiminor axis), \(\mathcal {F}\) is the complete elliptic integral of the first kind, and \(\mathcal {E}\) is the complete elliptic integral of the second kind. The eccentricity parameter is related to the elliptic integrals of the first and second kind [24, 43]:

$$ k=\sqrt{\frac{2\mathcal{F}-\mathcal{E}(1+\gamma)}{\mathcal{E}(1-\gamma)}} $$


$$ \gamma=R(\frac{1}{R_{z}}-\frac{1}{R_{y}}) $$
$$ \mathcal{F}={\int}_{0}^{\pi/2}[1-(1-\frac{1}{k^{2}})\sin^{2}\phi]^{-1/2}d\phi $$
$$ \mathcal{E}={\int}_{0}^{\pi/2}[1-(1-\frac{1}{k^{2}})\sin^{2}\phi]^{1/2}d\phi $$
Fig. 2

Generalized case of two ellipsoidal volumes (“Body S” and “Body C”) and their respective geometries. The equivalent radii of curvature are given in Eqs. 3 and 4. In the case of a spherical indenter (Body S), Rs, y = Rs, z. For a cylindrical specimen (Body C) with the longitudinal axis aligned in the y-direction, \(R_{c,y}\rightarrow \infty \)

One can numerically evaluate the elliptical equations using iterative procedures or utilizing lookup tables [44]. Here, in order to provide the most useful result to the reader, we will utilize the algebraic approximations from Brewe and Hamrock [42] to replace the elliptical integrals of \(\mathcal {F}\) and \(\mathcal {E}\) with numerically found approximations of \(\overline {\mathcal {F}}\) and \(\overline {\mathcal {E}}\). These are given by:

$$ \overline{\mathcal{F}}= 1.5277+.6023\ln{\frac{R_{y}}{R_{z}}} $$
$$ \overline{\mathcal{E}}= 1.0003+\frac{0.5968R_{z}}{R_{y}} $$

The approximate eccentricity parameter is now approximated:

$$ k \approx \sqrt{\frac{2\overline{\mathcal{F}}-\overline{\mathcal{E}}(1+\gamma)}{\overline{\mathcal{E}}(1-\gamma)}} $$

It follows that the approximate expression for χellip is now given by:

$$ \chi_{ellip} \approx k \overline{\mathcal{F}}^{(-3/2)} \overline{\mathcal{E}}^{1/2} $$

Thus, we can rewrite the relation for force-displacement during an elliptical contact as a product of the classic Hertzian force relationship (FH) and a correction factor that depends on the ratio of Rs to Rc, which we will call \({\Omega }_{ellip}^{LE}\):

$$ \begin{array}{@{}rcl@{}} F_{ellip}^{LE} &=&\frac{2 \pi}{3}(2R)^{1/2}\frac{E}{1-\nu^{2}}\delta^{3/2}k \overline{\mathcal{F}}^{-3/2}\overline{\mathcal{E}}^{1/2} \end{array} $$
$$ \begin{array}{@{}rcl@{}} &=&F_{H} {\Omega}_{ellip}^{LE} \end{array} $$


$$ {\Omega}_{ellip}^{LE}=\frac{\pi \chi_{ellip}}{\sqrt{2}}\frac{\sqrt{\frac{R_{s}R_{c}}{2R_{c}+R_{s}}}}{\sqrt{R_{S}}}. $$

The superscript LE denotes that these expressions are for linear elastic bodies. Note that the force-displacement Fδ relationship retains the 3/2 power, arising out of linear elasticity.

A comparison between the force-displacement relationships of Eqs. 1 and 16 is shown in Fig. 3a. The ratio between the indenter tip size versus specimen size will effect the ellipticity of the contact area, determining the usefulness of Eq. 16. Figure 3a shows that for sufficiently small indenter tips (Rs/Rc ≤ 0.4), the elliptical contact solution differs from the circular contact solution (1) by < 6% for extremely large indentations (δ/Rc = 0.9). In other words, if the radius of the cylindrical specimen is at least 2.5 times larger than the indenter tip, the elliptical contact solution approaches the classic Hertzian solution (analogous to a sphere indenting a half-space). For large values of the size ratio Rs/Rc, the eccentricity will become large enough that the behavior will approach a line contact formulation (k ≈ 10).

Fig. 3

a Nondimensional plot of the force-displacement relationship for different ratios of Rs/Rc. Solid lines show Fδ curves using the elliptical contact formulation of Eq. 16, whereas dashed lines show the Fδ relationships with a circular contact area (classical Hertz theory). b Plot of the elliptical contact correction factor \({\Omega }_{ellip}^{LE}\) (orange) and eccentricity (\(\overline {k}\), blue) as a function of varying Rs/Rc (and hence, Ry/Rz)

Figure 3b plots the eccentricity parameter k and the elliptical contact correction factor \({\Omega }_{ellip}^{LE}\) as a function of Rs/Rc. Note that if Ry/Rz= 1, then \(\mathcal {F}=\mathcal {E}=\pi /2\), k = 1, and there is no need to use \({\Omega }_{ellip}^{LE}\) (\({\Omega }_{ellip}^{LE}= 1\)) since it is a circular contact area. As Rs/Rc becomes large, k also becomes large, and will eventually approach the limit of a rectangular line contact. For all values of Rs/Rc, \({\Omega }_{ellip}^{LE}\) must be ≤ 1 (and hence \(F_{ellip}^{LE} \underline {<} F_{H}\)).

Elliptical contact on hyperelastic neo-Hookean bodies

The preceding section considered a linearly elastic material subjected to small deformation. In the case of biological materials subjected to large deformation, a hyperelastic material model is more appropriate. We begin by considering a neo-Hookean material (perhaps the simplest constitutive model within the class of hyperelastic models). The neo-Hookean model for an incompressible material uses a strain energy density of the form:

$$ W_{NH}=C_{10}(I_{1}-3); $$

where C10 is a material property and I1 is the first invariant of C (the right Cauchy-Green deformation tensor). In the small strain regime, the material constant C10 can be related to the linear elastic shear modulus μ and linear elastic Young’s modulus through \(C_{10}=\mu /2=\frac {E}{4(1+\nu )}\), where ν is the Poisson’s ratio. For an incompressible material, ν = 0.5.

For the case of a neo-Hoookean body subjected to spherical indentation, we propose a solution which takes the form:

$$ \begin{array}{@{}rcl@{}} F_{NH} &=&F_{H} {\Omega}_{ellip}^{LE} {\Omega}_{NH} \end{array} $$
$$ \begin{array}{@{}rcl@{}} &=&\frac{2 \pi}{3}(2R)^{1/2}\frac{E}{1-\nu^{2}}\delta^{3/2} \chi_{ellip} {\Omega}_{NH} \end{array} $$

where ΩNH is a new correction function that must be solved for. Although \({\Omega }_{ellip}^{LE}\) is solely dependent on the ratio between the indenter radius and specimen radius (Rs/Rc), the correction factor for the neo-Hookean material will have an additional dependence on the extent of deformation since hyperelastic materials exhibit stiffening at larger strains. Here, the extent of deformation is characterized by the ratio between indentation depth and the indenter radius (δ/Rs). Thus, ΩNH will be dependent on (δ/Rs) and (Rs/Rc). For simplicity in calculating the material property C10 (explained in later sections), the following sections will obtain expressions for the corrective function ΓNH (the inverse of ΩNH):

$$ {\Gamma}_{NH}(\delta /R_{s}, R_{s}/R_{c})=\frac{1}{{\Omega}_{NH}}. $$

Elliptical contact on hyperelastic Mooney-Rivlin bodies

A Mooney-Rivlin material model is another hyperelastic constitutive model that is considered an extension of the neo-Hookean model since it incorporates a second invariant of the left Cauchy-Green tensor. For an incompressible material, its strain energy takes the form:

$$ W_{MR}=C_{10}(I_{1}-3)+C_{01}(I_{2}-3) $$

where C10 and C01 are material constants. Generally speaking, expressions which include the dependence on the second invariant can model the stress response of rubber-like materials and soft biomaterials more accurately [45]. For small deformations, the constants C10 and C01 are related to the elastic modulus by E = 4(1 + ν)(C10 + C01). For C01 = 0, Eq. 22 reduces to the neo-Hookean strain energy. Similar to the correction factor for the neo-Hookean body ΩNH, an expression for a Mooney-Rivlin body subjected to spherical indentation will have a correction factor ΩMR:

$$ F_{MR} =F_{H} {\Omega}_{ellip}^{LE}{\Omega}_{MR} $$

Again, we will be finding solutions to the inverse corrective function: \({\Gamma }_{MR}=\frac {1}{{\Omega }_{MR}}\). In this case, the corrective function will have an additional dependence on the ratio between the parameters C10 and C01. Therefore, ΓMR will be a function of (δ/Rs), (Rs/Rc), and κ, where κ is a dimensionless parameter defined by:

$$ \kappa=\frac{C_{10}}{C_{10}+C_{01}} $$

Note that κ = 1 corresponds to a neo-Hookean material.

Computational model

Geometry considerations

3D finite element simulations of spherical indentation on cylindrical specimens were performed with the commercially available software Abaqus (Dassault Systèmes Simulia Corp.). The indentation was simulated as frictionless contact between a rigid sphere (with Rs varying between 2 and 100 μm) and a cylindrical specimen (Rc = 5μm) with either linear elastic or hyperelastic material properties. Figure 1c shows the computational model with the applied boundary conditions. Taking advantage of symmetry, a quartered section of the model, with approximately 200–400k ten-node quadratic tetrahedral elements, was utilized to reduce computational costs. The magnitude of indentation strain (δ/Rc) remained constant across all simulations, while a different Rs/Rc value was used for each simulation. A total of six indenter sizes were simulated, with Rs equal to 2, 3, 5, 7, 10, 20, 30, 50, 75, and 100 μm. These correspond to Rs/Rc ratios of 0.4, 0.6, 1, 1.5, 2, 4, 6, 10, 15, and 20, respectively. Consequently, the maximum value of δ/Rs will be different across each simulation.

The range of Rs/Rc values was determined by considering the appropriate amount of ellipticity. It was shown in the previous section that for Rs/Rc < 0.4, the ellipticity parameter \(k\rightarrow 1\), and the indentation can be approximated by a circular contact formulation, such as those provided by Lin et al. [27]. At the upper limit, a line contact exists for large values of k, or when Rs/Rc ≈ 20 (see Fig. 4). This range is also applicable to experimental data since commonly used silica bead indenter tips range from 5 to 100 microns while probing cellular components with cylindrical radii on the order of 1–10 microns.

Fig. 4

Example of finite element simulations which demonstrate the effect of Rs/Rc ratios on the ellipticity of the contact area. For Rs/Rc = 0.4 (top simulation), the contact area is almost circular, while for Rs/Rc = 10, the contact area has greater eccentricity (k)

For simulations with larger ellipticity in the contact area (larger indenters), the cylindrical specimens were made to be geometrically longer along the longitudinal direction (y-axis) so that the free boundary normal to the y-axis would not affect the solution. Across all models, the length of the specimen was made to be approximately twice as long as the major axis of the contact area during the maximum indentation depth.

Material parameters

For each of the seven indenter sizes, three different types of material models were implemented for the cylindrical specimen: linear elastic, neo-Hookean, and Mooney-Rivlin. For the linear elastic material model, we used a Young’s modulus of E = 100 Pa and assumed that the specimen was incompressible (ν = 0.5). Note that the correction factors for Ωellip, ΓNH, and ΓMR are dimensionless functions that do not depend on the specific material parameters. For the Mooney-Rivlin models, additional simulations with varying κ values (κ= 0.1, 0.2, 0.3, ...1.0) were performed for each ratio of Rs/Rc.

Mesh convergence

The mesh size of the cylindrical specimen was biased to be more refined near the area of contact with the indenter. A mesh convergence study was performed for the linear elastic model of the smallest Rs/Rc ratio, since this was the case that corresponded to the largest indentation strains. To find the necessary minimum element size, several simulations were performed with the smallest element size at the center of contact varying between 2.5 × 10− 5 mm and 5 × 10− 4 mm. Although all simulations (including the coarsest mesh) were able to accurately resolve the theoretical force-displacement curves at small strains (\(F_{ellip}^{LE}\) of Eq. 16), it was found that a minimum element size of 1 × 10− 4 mm was necessary for the convergence of the total internal energy.

Boundary conditions

The rigid sphere was displaced by δ = 700 μm in the negative x-direction in 100 ms. This loading rate is comparable to some AFM experiments [27]; however, it should be noted that viscoelastic effects are not considered in these computations. The interface between the indenter and specimen was assumed to be frictionless. The model was given symmetric boundary conditions along surfaces normal to the positive y- and z-directions and the negative x-direction, while it remained free to expand in the negative y-direction. A schematic of the fixed translational and rotational degrees of freedom which create the symmetric boundary conditions is shown in Fig. 1c. In summary, the negative x-normal plane of symmetry was not allowed to move in the x-direction, the positive y-normal plane was not allowed to move in the y-direction, and the z-normal plane of symmetry was not allowed to move in the z-direction.

Calculation of corrective functions

Since analytical expressions are known for FH and Ωellip, our aim is to utilize the results from the following sections to obtain expressions for the correction functions ΓNH and ΓMR. After obtaining these correction functions, simplistic algebraic expressions are provided and can be used to extract hyperelastic material properties from experiments of cylindrical specimens undergoing spherical indentation.

In order to obtain algebraic expressions for the corrective functions, we obtained the simulated corrective factors \(\widehat {\Gamma }_{NH}(\delta /R_{s})\) and \(\widehat {\Gamma }_{MR}(\delta /R_{s})\) from various Rs/Rc values over a useful range. Hereafter we will use the symbol (\(\widehat { - }\) ) to denote the values obtained with the finite element model. Each simulation provides a total reaction force (FNH or FMR ), which can be divided by the analytical expression of \(F_{ellip}^{LE}\) (16) in order to obtain the corrective factors \(\widehat {\Gamma }_{NH}(\delta /R_{s})\) and \(\widehat {\Gamma }_{MR}(\delta /R_{s})\), respectively. From \(\widehat {\Gamma }_{NH}(\delta /R_{s})\) and \(\widehat {\Gamma }_{MR}(\delta /R_{s})\) it is possible to deduce the final expressions for ΓNH(δ/Rs,Rs/Rc) and ΓMR(δ/Rs,Rs/Rc,κ).


Linear elastic specimen simulations

Although the aim of this study is to obtain corrective functions to describe the specimen with a nonlinear constitutive model, an analogous “corrective factor” (\(\widehat {\Gamma }_{LE}\)) can be computed which compares the FEM simulation of a linear elastic (LE) specimen with the theoretical curve for \(F_{ellip}^{LE}\). The analytical solution of \(F_{ellip}^{LE}\) can be divided by the total reaction force obtained during a given simulation (FLE) in order to obtain \(\widehat {\Gamma }_{LE}\) such that:

$$ \widehat{\Gamma}_{LE}=\frac{F_{ellip}^{LE}}{F_{LE}}=\frac{F_{H}{\Omega}_{ellip}^{LE}}{F_{LE}} $$

Due to the discretization of surface elements in the finite element model, the simulation will be inherently inaccurate for very small indentations. In order to find the minimum indentation depths that allow for an accurate FEM solution, we began by performing simulations with a linear elastic material model for the cylindrical specimen. At the indentation depth where the linear elastic simulation matches the elliptical contact theory, the simulation matches the theoretical curve for \(F_{ellip}^{LE}\), and we therefore only consider fitting the corrective functions beyond this indentation depth.

A comparison between the theoretical force-displacement relationship of Eq. 16 for various values of Rs/Rc and simulations with linear elastic specimens is shown in Fig. 5. In Fig. 5a, the nondimensional forces (for both the simulation and theoretical \(F_{ellip}^{LE}\)) are plotted against δ/Rs. For small values of Rs/Rc (approaching the behavior of spherical contact), the linear elastic theory for elliptical contact (16) predicts the behavior quite accurately, even for fairly large indentation (δ/Rc > 0.1). For larger values of Rs/Rc and larger indentations (δ/Rc > 0.1), the simulations deviate farther from Eq. 16, as shown more clearly in Fig. 5b. This is due to a decrease in accuracy of the algebraic expressions in Eqs. 79, and 10 for larger values of eccentricity k [42].

Fig. 5

a Nondimensional forces (both simulations and theoretical) plotted over nondimensional indentation depth for various values of Rs/Rc with a linear elastic (LE) specimen. Each curve corresponds to a value of Rs/Rc. Finite element simulations (dashed lines) are compared to the linear elastic theory of elliptical contact (solid lines) given in Eq. 16. b The calculated correction factors (\(\widehat {\Gamma }_{LE}\) ) from the simulations of a linear elastic specimen. \(\widehat {\Gamma }_{LE}\) measures the deviation from the linear elastic theory of elliptical contact, and is plotted over the nondimensional indentation δ/Rs

For a given Rs/Rc, when \(\widehat {\Gamma }_{LE}= 1\), the indentation is large enough for the simulation to be considered accurate enough for obtaining corrective functions. Figure 5b plots \(\widehat {\Gamma }_{LE}\) over the relative indentation δ/Rs. The divergence of \(\widehat {\Gamma }_{LE}\) from 1 is a result of two causes: (1) larger indentations creating a deviation from small strain theory and (2) larger ratios of Rs/Rc (and consequently, larger eccentricity values) impart less accuracy in Γellip. The values of δ/Rs at which \(\widehat {\Gamma }_{LE}= 1\) are smallest for the largest values of Rs/Rc, since a larger indenter will impart a larger area of contact onto the specimen (therefore, the limiting mesh size within the contact area becomes less significant).

Neo-Hookean specimen simulations

Figure 6a compares (16) to the force-displacement response obtained from simulations with neo-Hookean (NH) specimens across various ratios of Rs/RC. The neo-Hookean contact also shows increased deviation from the theoretical elliptical contact theory (\(F_{ellip}^{LE}\)) at higher values of Rs/Rc and at higher relative indentations (δ/Rs). For comparison, the elliptical contact theory curves used a Young’s modulus of E = 4C10(1 + ν). These results are in agreement with previous indentation studies on hyperelastic materials which have shown that the indentation response is stiffer than that predicted by linear elasticity [28]. As expected, FNH is similar to FLE at small indentation, but begins to differ from FLE at larger indentations (Fig. 6b). The simulated corrective factors \(\widehat {\Gamma }_{NH}\) were calculated by:

$$ \widehat{\Gamma}_{NH}=\frac{F_{ellip}^{LE}}{F_{NH}} $$
Fig. 6

a Nondimensional forces (both the simulations of the neo-Hookean specimen and the theoretical curve for \(F_{ellip}^{LE}\)) plotted over nondimensional indentation depth for various values of Rs/Rc. Finite element simulations (dashed lines) are compared to the linear elastic theory of elliptical contact (solid lines) given in Eq. 16. b The calculated correction factors (\(\widehat {\Gamma }_{NH}\) ) from the simulations of a neo-Hookean specimen. Each curve corresponds to a value of Rs/Rc. The correction factors calculated from the simulations (circles) are plotted alongside the corresponding fitted power-law functions (solid lines). Only selected values of Rs/Rc are shown for clarity. A complete list of the power-law function coefficients is provided in Table 1

The corrective factors for the neo-Hookean specimen (shown in Fig. 6b) are plotted as a function of indentation and follow a similar trend to the \(\widehat {\Gamma }_{LE}\) factors. That is, it is apparent that the neo-Hookean correction factors are a function of δ/Rs. A power-law function is used to fit the correction factor curves such that:

$$ \widehat{\Gamma}_{NH}(\delta/R_{s})=A(R_{s}/R_{c})\left( \frac{\delta}{R_{s}}\right)^{m(R_{s}/R_{c})} $$

For a given Rs/Rc, the power-law functions were fitted to \(\widehat {\Gamma }_{NH}\) for values of δ/Rs that were greater than the indentation at which \(\widehat {\Gamma }_{LE}\underline {<}1\). For clarity, only selected values of Rs/Rc are shown; however, a complete list of the fitted power-law function coefficients is provided in Table 1.

Table 1 Summary of power-law coefficients, A and m for the corrective functions of Eq. 28 (neo-Hookean model) for various values of Rs/Rc

Figure 7 shows that A(Rs/Rc) and m(Rs/Rc) logarithmically decay with increasing the size ratios Rs/Rc. For smaller ratios of Rs/Rc, the solution becomes analogous to circular contact with a half-space, and therefore \(A\rightarrow 1\), \(m\rightarrow 0\), and \(\widehat {\Gamma }_{NH}(\delta /R_{s})\rightarrow 1\) for all values of Rs/Rc. The corrective function for a neo-Hookean material can now be written in its final form as:

$$ {\Gamma}_{NH}=\left[ A_{1}\ln(\frac{R_{s}}{R_{c}})+A_{2} \right] \left( \frac{\delta}{R_{s}} \right)^{m_{1}\ln(\frac{R_{s}}{R_{c}})+m_{2}} \\ $$

A summary of the coefficients for the neo-Hookean corrective function is given in Table 2. Note that the function m(Rs/Rc) remains small and negative over the range of Rs/Rc.

Fig. 7

Logarithmic functions found for curve fitting of ΓNH (solid lines). The logarithmic functions fitted to the simulation results (circles) showed an excellent fit, with R2 values, or coefficients of determination, greater than 0.90

Table 2 Summary of coefficients for the generalized corrective functions of Eq. 28 (Neo-Hookean Model) and Eq. 35 (Mooney-Rivlin model)

Mooney-Rivlin specimen simulations

In order to obtain a correction function for the Mooney-Rivlin model (MR), simulations with varying parameter ratios (κ= 0,0.1, 0.2, 0.3, ...1.0) were performed for each size ratio of Rs/Rc. Since ten different size ratios were considered, this resulted in 110 total simulations analyzed for a Mooney-Rivlin specimen. Similar to the previous approach with the linear elastic and neo-Hookean materials, the corrective factors \(\widehat {\Gamma }_{MR}\) are calculated by the relationship:

$$ \widehat{\Gamma}_{MR}=\frac{F_{ellip}^{LE}}{F_{MR}} $$

For each value of κ, the calculated value of ΓMR plotted over δ/Rs still follows a power-law behavior; however, in contrast to the neo-Hookean simulations, the functions A and m now have an additional dependence on the parameter ratio κ such that:

$$ \widehat{\Gamma}_{MR}(\delta/R_{s},\kappa)=A(R_{s}/R_{c},\kappa)\left( \frac{\delta}{R_{s}}\right)^{m(R_{s}/R_{c},\kappa)} $$

We begin by finding an expression for the function A for the ten simulations (Rs/Rc =0.4, 0.6, 1, 1.5, 2, 4, 6, 10, 15, 20) for each specified value of κ. The simulations show that the function A can be rewritten as:

$$ A(R_{s}/R_{c},\kappa)=A_{1}(\kappa)\ln \left( \frac{R_{s}}{R_{c}} \right) +A_{2}(\kappa) $$

where A1(κ) and A2(κ) are functions determined after comparing the functional A(Rs/Rc,κ) across all values of κ. Similarly, the equation for the functional expression m, at a given value of κ, is described by:

$$ m(R_{s}/R_{c},\kappa)=m_{1}(\kappa)\ln \left( \frac{R_{s}}{R_{c}} \right) +m_{2}(\kappa) $$

Likewise, m1(κ) and m2(κ) are functions determined following a comparison of the functional m(Rs/Rc,κ) across all values of κ.

As shown in Fig. 8, the parameters A1 and m1 can be approximated as constant values across κ. The functions A2 and m2, however, show some dependence on κ, and can be described with the linear functions:

$$ A_{2}=.12\kappa+.64 $$
$$ m_{2}=.04\kappa-.12 $$
Fig. 8

Plot of ΓMR coefficients as functions of κ found in simulations (circles) and the corresponding fitted linear functions (solid lines). The R2 values, or coefficients of determination, are 0.50, 0.99, 0.81, and 0.99 for A1, A2, m1, and m2, respectively

The generalized form for the Mooney-Rivlin corrective function is now given by:

$$ {\Gamma}_{MR}=\left[ A_{1}\ln(\frac{R_{s}}{R_{c}})+A_{2}(\kappa) \right] \left( \frac{\delta}{R_{s}} \right)^{m_{1}\ln(\frac{R_{s}}{R_{c}})+m_{2}(\kappa)} $$

A summary of the calculated parameters for the Mooney-Rivlin corrective function is given in Table 2. In theory, the calculated model parameters for κ = 1 (the parameter C01 = 0) should be identical to the parameters found with the neo-Hookean form. One can see that while the coefficients for A2 and m2 show excellent agreement between the neo-Hookean and Mooney-Rivlin forms (when κ = 1), the coefficients A1 and m1 are slightly different. This is most likely due to the fact that the linear functions that describe A1 and m1 have a poorer fit to the data (R2 = 0.50 and 0.81, respectively) in comparison to the functions describing A2 and m2 (R2 > 0.99). Note that Eq. 35 can be further simplified by approximating m1 = 0, although m1 remains included in the assessment for accuracy.

From the corrective function, we see that \({\Gamma }_{MR} \propto \delta ^{-0.03\ln {\frac {R_{s}}{R_{c}}}+(.04\kappa -.12)} \). It follows that

$$ F_{MR} \propto \delta^{3/2}\delta^{0.03\ln{\frac{R_{s}}{R_{c}}}-0.04\kappa+ 0.12}. $$

Therefore, our nonlinear solution deviates from linear elastic theory (\(F_{ellip}^{LE}\propto \delta ^{3/2}\)) by increasing the indentation power term. For the range of Rs/Rc examined in this study, the additional indentation power term varies from 0.05 (Rs/Rc = 0.4, κ = 1) to 0.18 (Rs/Rc = 20, κ = 0). Consequently, the force (FMR) is proportional to the indentation within the range of δ1.55 to δ1.68.

Comparison of corrective functions

Smaller values of ΓMR (i.e., ΓMR << 1) indicate stronger deviation from the linear elastic elliptical contact theory. Figure 9 shows contour plots of the corrective functions for ΓMR against the nondimensional variables of size ratio (Rs/Rc) and indentation ratio (δ/Rs) for the two extreme cases: κ = 0 (Fig. 9a) and κ = 1 (Fig. 9b). For values of small Rs/Rc and δ/Rs, the corrective function approaches a value of 1 (ΓMR ≈ 1), and linear elasticity theory may be used. For a given set of δ/Rs and Rs/Rc, a comparison between Fig. 9a and b shows that when κ = 0, ΓMR is smaller (ΓMR deviates farther from 1). Therefore, the material demonstrates an increase in nonlinear behavior. As expected, the corrective function deviates farthest from linear elasticity (ΓMR = .025) when κ = 0, Rs/Rc is large, and δ/Rs is large.

Fig. 9

a Contour plot of corrective function ΓMR over the nondimensional variables (Rs/Rc and δ/Rs) for κ = 0.1. b ΓMR over the nondimensional variables (Rs/Rc and δ/Rs) for κ = 1. Both contour plots show Eq. 35 with the coefficients specified in Table 1

One can see the most variation in ΓMR across Rs/Rc. For a given value of Rs/Rc, the corrective function changes slowly with respect to the extent of indentation δ/Rs. This is explained by the fact that m1 and m2 are small across all values of κ, and therefore Eq. 35 will demonstrate greater dependence on Rs/Rc.


Effects of large deformations and accuracy of corrective functions

In this section, we discuss the accuracy of our corrective functions. For incompressible thin specimens under large deformation (large values of indentation strain, δ/Rc), the specimen experiences restricted deformation in the direction of applied loading, and will therefore experience more expansion along the sides and length of the cylinder. In the previous section, all simulations for computing ΓMR utilized an indentation strain of 0.14; however, in this section, we extend the analysis to larger indentation strains (δ/Rc > 0.30) to assess the accuracy of our solution.

Figure 10a shows the force-displacement curves calculated with the proposed corrective functions (35) in comparison to additional finite element simulations performed at a larger indentation depth. A plot of the corresponding sum-of-squares error (SSE) is shown in Fig. 10b (SSE values are given by \(SSE={\Sigma }(\widehat {y}-y)^{2}\), where \(\widehat {y}\) are the forces obtained through the new simulations, and y are the force values calculated using the corrective function of Eq. 35). One can see that for smaller values of Rs/Rc, the corrective function provides an excellent fit (SSE < 0.25) up until δ/Rc = 0.45. For larger ratios of Rs/Rc, the corrective function only seems suitable for indentation strains less than 0.25, after which the SSE values exceed 0.25.

Fig. 10

a Plot of normalized force over the nondimensional variables (δ/Rc) for the case of Rs/Rc = 1 (blue) and Rs/Rc = 20. Simulations per formed over large indentations (dashed lines) are compared to force-displacement curves calculated with the correction functions (κ = 1) found in this study (solid lines) and linear elastic theory (“+” lines). The approximation to the indentation of a cylinder by a half-space is also shown for comparison (circles). b Plot of the sum of squared errors for the correction functions found in this study (solid lines) and compared to linear elastic elliptical contact theory (“+” lines)

As expected, the theoretical curves from linear elastic theory (“+” curves) show a much larger error. For Rs/Rc = 1, the linear elastic theory provides a reasonable estimate for δ/Rc < 0.28. For Rs/Rc = 20, the linear elastic theory of elliptical contact shows significant errors (SSE > 0.25) for δ/Rc < 0.08.

Substrate effects

As mentioned in the preceding section, one way in which substrate interactions affect the solution is by restriction of deformation along the direction of loading. Another way in which the substrate interactions affect the solution is by modifying the boundary conditions between the specimen and substrate. For the ease of creating a symmetric model (Fig. 1c), we chose to use a boundary condition that allows the specimen to slip along the longitudinal direction, while being restricted in the direction normal to the loading (i.e., “non-adhered” boundary condition). Alternatively, one could also examine the effects of a “fully adhered” boundary condition, which is expected to to behave differently than the “non-adhered” boundary condition at large values of δ/Rc. In reality, the boundary condition between a cell and its substrate is not entirely fully adhered or entirely non-adhered. As described by Mahaffey et al. 2004, cell membranes will often have a higher abundance of focal adhesion sites around the edges. A combination of full-adhesion and no-adhesion modeling can be used to obtain material properties for different regions of the cell [46].

Regimes for R s/R c

The correction factors provided in this study were found using computational models with Rs/Rc between 0.4 and 20. This range of Rs/Rc was sufficient to encompass the lower bound of elliptic contact (rigid spherical indenter on a deformable half-space specimen) and the upper bound of elliptic contact (rigid half-space indenter on a deformable cylinder). The lower bound of elliptic contact becomes a circular contact area, while the upper bound becomes a rectangular/line contact area. As mentioned previously, for values of Rs/Rc < 0.4, the cylindrical specimen is relatively “flat,” and the conventional formulations for circular contact areas (spherical indentation on a half-space) can be utilized. Previous works have also considered the effects of large indentation and substrate interaction for spherical indentation on a half-space. For a spherical indentation on a flat cellular specimen (Rs/Rc < 0.4) of finite thickness h (h ≥ 0.1Rs), one can use the analytical expressions provided by Dimitriadis [47]. However, they assume linearity, and the result is meant for small indentations (δ ≤ 0.1h). For very thin flat specimens (Rs/Rc < 0.4, h ≤ 0.1Rs), an analytical expression is provided by Chadwick [48].

Here, we do not specify a theoretical upper bound for Rs/Rc, although Rs/Rc = 20 is the maximum value used in fitting the expression of the corrective function. For very large values of Rs/Rc, the behavior is expected to approach the theoretical limit for the indentation of a half-space on a cylinder. This is also equivalent to the case of two parallel cylinders in contact (although with one cylinder having a radius that tends to infinity). The two parallel cylinders brought into contact will result in a rectangular contact area. Precise equations for the contact between two cylinders are provided in previous works [24, 44] (although a major drawback of these models is that the contact force cannot be explicitly defined as a function of indentation in closed form [49]). The indentation depth for a rigid half-space with a linear elastic cylinder of length L is given by [44]:

$$ \delta=\frac{F}{\pi L E^{*}} \left( 1+\ln{\frac{\pi E^{*} L^{3}}{R_{cyl}F}} \right), $$

assuming that the half-space is rigid, \(\frac {1}{E^{*}}=\frac {1-\nu ^{2}}{E}\). Equation 37 is plotted in Fig. 10a (circles). It is observed that the theoretical force-displacement curve for the elliptical contact of Rs/Rc = 20 is very close to the upper bound where Rs approaches infinity (i.e., rigid half-space indenter), indicating that the range of Rs/Rc values we selected for our computational studies is sufficient for capturing both the lower and upper bounds of ellipticity.

Limitations and future considerations

This study examined two simple and well-known hyperelastic (neo-Hookean and the Mooney-Rivlin) models, although these are not the only possibilities for hyperelastic strain energy functions. More sophisticated material models, however, might require more material parameters and could result in more complex corrective functions.

For values of Rs/Rc less than 0.4, one can expect ΓMR ≈ 1 for fairly large indentation depths (until δ/Rc = 0.45). For larger size ratios (Rs/Rc > 20), the correction factors are precise until values of δ/Rc > 0.25. Errors at larger indentation depths should be considered, as discussed in the previous section. Large values of δ/Rc could also result in large errors when applying the model to experimental data where the observed interaction between the specimen and substrate is fully adhered.

The work presented herein assumes that the cell is a homogeneous continuous solid in order to obtain an overall effective modulus. However, cells are heterogeneous in nature—comprising of a lipid membrane surrounding networks of biopolymer filaments with cytoplasmic fluid. Our formulation applies only to the case of an incompressible material, ignoring the possible time-dependent behavior of the cytoplasmic cell volume [50]. To expand this work further into applications of cell mechanics over various time scales, one could use a biphasic material model approach to capture the hyperelastic solid network (comprised of the cytoskeleton and organelles) surrounded by cytoplasmic fluid. Furthermore, there is the caveat that we have not factored in the effects of strain rate into the indentation formulations, which would also be important when considering the effective mechanical properties of cells.

Potential applications

In order to obtain the hyperelastic material properties from the experimental data, one could use the following procedure:

  1. 1.

    Multiply the experimental data (Fexp) by the corrective function for the neo-Hookean model (FexpΓNH).

  2. 2.

    Set \(F_{exp} {\Gamma }_{NH}=\frac {2\pi }{3}\frac {4C_{10}(1+\nu )}{1-\nu ^{2}}\sqrt {2R}\delta ^{3/2}k \overline {\mathcal {F}}^{-3/2}\overline {\mathcal {E}}^{1/2}\) where, \({\Gamma }_{NH}=\left [ A_{1}\ln (\frac {R_{s}}{R_{c}})+A_{2} \right ] \left (\frac {\delta }{R_{s}} \right )^{m_{1}\ln (\frac {R_{s}}{R_{c}})+m_{2}}\) ν = 0.5, and the constants are given by: A1 = − 0.16, A2 = 0.76, m1 = − 0.03, and m2 = − 0.07.

  3. 3.

    Calculating k, \(\overline {\mathcal {F}}\), and \(\overline {\mathcal {E}}\) with the known dimensions of Rs and Rc, and assuming ν = 0.5, use the preceding expression to obtain C10.

An equivalent result can also be obtained by using ΓMR (setting κ = 1) instead of ΓNH. Once C10 is obtained, one can use ΓMR and find an appropriate value for κ by numerically solving for the minimization of squared errors between the estimated FMR curves and the experimental results. Therefore, an initial value of C10 should be determined first, otherwise, κ cannot be uniquely determined [51].

The formulations provided in this study can be applied to several cylindrical-shaped specimens that appear in biological materials. In Fig. 1, we showed an example of spherical indentation on a neuronal axon (cylinders with diameters on the order of a few microns). In addition to axons, the vast majority of brain cells contain cylindrical-like morphologies in their processes (glial cell processes, neurites, dendrites, etc.).

Examples of other cells that exhibit a stellate-like morphology (i.e., have processes) include osteocytes, mesenchymal stem cells, pericytes, and fibroblasts. Due to the heterogeneity of the mechanical properties of cells, it is important to obtain properties with respect to specific cellular compartments [9, 52] (and not just measuring the response of the spherical somal body with classical Hertz formulations). Aside from probing fibrous cellular components, other uses of this work include probing other types of cylindrical-shaped biological specimens such as certain strains of bacteria (i.e., bacilli), as well as fibrous materials of the extracellular matrix (such as fibrin, collagen, and elastin). With respect to non-biological applications, this work can be applied to measuring hyperelastic properties of cylindrical synthetic polymer fibers and natural fibers that can be used in fiber-reinforced composites (provided that they can be approximated as incompressible).

An example of the formulation applied to the experimental data of AFM indentation on a single collagen fibril is shown in Fig. 11. The experimental data is taken from Andriotis et al. [53], with Rs = 5 nm and \(R_{c}\sim 50\) nm. Using the procedure outlined above, and performing a minimization of root-mean-square error to a range of C10 and κ, we obtained hyperelastic parameters of C10 = 6.8 × 108 Pa and κ = 0.6. In the linear elastic regime, this corresponds to a Young’s modulus of E = 6.8 × 109 Pa, which is within the range of values found by Andriotis et al. (\(E\sim 6-14\) GPa). Although we can reasonably approximate the Young’s modulus with our formulation, the usefulness in the formulation relies in its ability to estimate the nonlinear properties as well.

Fig. 11

Application of Mooney-Rivlin formulation presented in this study with experimental data on indentation of a collagen fibril (Andriotis et al. [53]). Fitted hyperelastic properties were found to be C10 = 6.8 × 108 Pa, κ = 0.6, C01 = 4.6 × 108 Pa


Our approach provides a generalized formulation that can be used to extract mechanical properties from indentation into soft cylindrical bodies. The generalized corrective function ΓMR of Eq. 35 with the corresponding coefficients of Table 2 can be applied to force-displacement relationships for cylindrical incompressible hyperelastic materials subjected to spherical indentation. These corrective functions are valid for a wide range of specimen and indenter sizes: with small values of Rs/Rc approaching the limit of a flat specimen subjected to spherical indentation and large values of Rs/Rc approaching the limit of a flat indenter with a cylindrical specimen. For large values of Rs/Rc and large indentations (δ/Rs), the force-displacement behavior deviates farther from linear elastic elliptical contact theory.

Although the motivation of this study was to provide a theoretical framework for obtaining useful mechanical properties of cylindrical cellular bodies subjected to large indentation and deformation, the results can be applied to experimental data of non-cellular soft specimens (collagen, fibrin, biomimetic polymer strands, etc.) where a hyperelastic model is also required. Given the availability of indentation experiments over various length scales, the corrective functions provided in this study are a promising tool for measuring the hyperelastic properties of soft materials in a wide range of applications.


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This work was funded by the DoD SMART Scholarship Program and the US Army Research Lab (Aberdeen Proving Ground, MD), under Cooperative Agreement Number W911NF-12-2-0022.

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Dagro, A.M., Ramesh, K.T. Nonlinear contact mechanics for the indentation of hyperelastic cylindrical bodies. Mech Soft Mater 1, 7 (2019).

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  • Indentation
  • Hyperelastic
  • Nonlinear contact