Can hyperelastic material parameters be uniquely determined from indentation experiments?

Abstract

Indentation experiments have been widely adopted to evaluate the hyperelastic material properties of soft polymers and biological tissues. However, due to the combination of constitutive, geometric and contact nonlinearities, the hyperelastic indentation problem has not been solved analytically. The hyperelastic parameters extracted from the indentation load–displacement curve are poorly defined. Even the uniqueness of these parameters remains unknown. This work determines the uniqueness of hyperelastic material parameters by conducting dimensional analysis based on different combinations of hyperelastic models (which include constitutive nonlinearity) and the shapes of indenters (which reflect geometric and contact nonlinearities). It is proved that the unique determination of hyperelastic material parameters depends on a simple criterion: whether the dimensionless hyperelastic parameters are coupled with indentation displacement or not. Furthermore, for the indentation problems in which unique hyperelastic material parameters cannot be guaranteed, a unique measure is constructed and validated by experiments on biomedical polymers.

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1. Introduction

An indentation experiment is a popular engineering method to evaluate the mechanical properties of a material.^1–4 By impressing an indenter into the surface of a solid material, an f–d curve is generated, where f and d denote the load applied and the indentation depth marched by the indenter, respectively. Mechanical properties of the material are then extracted from the curve with various mathematical and computational methods.^3–7 Since the late 1990's^3,8–12 till recently,^13–17 the uniqueness of linear elastic and elasto-plastic material parameters obtained from indentation f–d curves has been investigated. It was proved that linear elastic parameters can be uniquely determined regardless of the shape of the indenter,¹⁸ whereas the unique determination of elasto-plastic parameters depends on the shape of the indenter.^11,15,16

In recent decades, accompanying the blossom of biomaterials research¹⁹ and soft robotics,²⁰ indentation experiment has been widely adopted to determine material properties of biomedical polymers/gels^21,22 and biological tissues.^23,24 These materials are much softer than the traditional engineering materials and commonly exhibit hyperelastic constitutive behaviors, which include both nonlinear stress–strain relationship and finite elastic deformation. Combining with the nonlinearity introduced by the contact problem itself, it's challenging to determine hyperelastic material parameters from indentation f–d curve. To authors' knowledge, whether hyperelastic material parameters can be uniquely determined from indentation f–d curve has not yet been investigated. If this question is not answered, reliability of the parameters could be easily challenged.

This work is aimed to determine the uniqueness of hyperelastic material parameters obtained from indentation f–d curve. Furthermore, for the hyperelastic indentation problems from which unique parameters are not able to be determined, a new measure will be defined. The work includes mathematical analysis, experimental validation and computational interpretation. Indentation problems in which a small indenter is impressed into a semi-infinite hyperelastic solid will be investigated (Fig. 1a–c). The hyperelastic solid is isotropic and incompressible. Shape of the rigid indenter could be conical, spherical or flat. Conical and spherical indenters are studied for their wide application on materials of all types. Flat indenter is for its popular usage on soft materials.^25,26

Fig. 1 (a–c) Schematics of conical, spherical and flat indenters impressed into the surface of a semi-infinite hyperelastic solid. (d) Comparison of f–d curves obtained from experiments, FEA and fitting of eqn (2). Black line is the average of five individual f–d curves obtained from conical indentation experiments on PEGDA hydrogel (shown as inset). Colored hollow dots represent the f–d curves generated by FEA with different combinations of hyperelastic parameters (G₀ and γ₁) based on Arruda–Boyce model. The orange line is generated from the first equation of eqn (2).

2. Material and methods

2.1. Material

A popular biomedical hydrogel, poly(ethylene glycol) diacrylate (PEGDA) hydrogel was fabricated with following procedure. First, 0.7 wt% of photoinitiator, 2-hydroxy-4′-(2-hydroxyethoxy)-2-methylpropiophenone (Sigma-Aldrich Co., St. Louis, MO, USA), was added into deionized (DI) water. The mixture was heated at 80 °C for 20 minutes in a sealed opaque bottle and cooled down to room temperature. 5 wt% of PEGDA (Sigma-Aldrich Co., St. Louis, MO, USA) was then added and stirred for uniform dispersion. The precursor was injected into Petri dish until filled up and cured by ultraviolet (UV) light. The cured PEGDA hydrogel sample has a diameter of 35 mm and a height of 10 mm. Samples were stored in DI water for one day before testing.

2.2. Methods

Conical indentation experiment. Indentation experiment was conducted with a conical indenter on a Bose ElectroForce 3220 mechanical tester (ElectroForce Systems Group, Eden Prairie, MN, USA) with displacement rate of 0.5 mm s⁻¹. The conical indenter (shown as inset of Fig. 1d) has a half angle, θ, of 45° and a tip radius of 100 μm. The tests were repeated for 5 times on 5 individual samples. The obtained f–d curve represents the average of five individual curves. The error bar denotes the standard deviation.

Finite element method (FEM) to extract hyperelastic material parameters. Nonlinear finite element simulations were conducted using Abaqus 6.12 (Simulia, Inc., Providence, RI, USA). Shown in Fig. 2, an axisymmetric model was created employing 4082 CAX4R (four-point axisymmetric reduced integration) elements. The mesh was refined at contact region. Arruda–Boyce hyperelastic model²⁷ was adopted as constitutive law. A rigid conical indenter with a half angle of 45° was impressed into the hyperelastic solid for 3 mm. For the boundary conditions, bottom surface was fixed at all degrees of freedom. f–d curves were generated by implementing given hyperelastic parameters (G₀ and γ₁) into the Arruda–Boyce constitutive imbedded in Abaqus.

Fig. 2 Finite element mesh to simulate conical indentation experiment.

Dimensional analysis. Buckingham Π theorem²⁸ was applied to construct f–d relationship in terms of Π function for conical, spherical and flat indenters. In this work, the analysis was conducted on a multiple-parameter Ogden model. Analysis on models with fewer parameters is similar. The detailed procedure is shown in Section I of ESI.†

3. Results and discussion

3.1. Exhibition of non-uniqueness

Black line in Fig. 1d shows the f–d curve generated by conical indentation experiment on PEGDA hydrogels. The hyperelastic material parameters based on Arruda–Boyce hyperelastic model²⁷ were extracted by using finite element method (FEM). Shown as colored hollow dots in Fig. 1d, multiple pairs of hyperelastic material parameters could fit the same experimental data well, which indicates that the hyperelastic parameters obtained from the conical f–d curve are not unique and suggests that the investigation of uniqueness is necessary.

3.2. Determination of uniqueness

Hyperelastic materials exhibit versatile constitutive behaviors. Some can be described by single-parameter model,^29,30 whereas some have to be captured by multiple-parameter ones.^26,31,32 Hence, people constructed many hyperelastic constitutive models which are distinct in both the number of parameters and the general form. Table 1 listed several classic models.^27,33–36 In this work, we categorize the models by the number of parameters and unify the parameters of different models into a general parameter set {G₀, γ_j}, where G₀ is the initial hyperelastic shear modulus with a unit of Pascal and γ_j is a series of dimensionless hyperelastic parameters. The positive integer j takes the value from 1 to χ − 1 where χ is the total number of material parameters of a given hyperelastic model. For the single-parameter model, γ_j is eliminated. The unified parameters are also shown in Table 1.

Table 1 List of hyperelastic models in energy form and their hyperelastic material parameters. λ_i (i = 1, 2, 3) is the principal stretch. I₁ is the first invariant of strain tensor which equals λ₁² + λ₂² + λ₃². I₂ is the second invariant of strain tensor which equals λ₁²λ₂² + λ₂²λ₃² + λ₁²λ₃². I₃ equals 1 due to incompressible assumption. Superscripts of the energy density function, NH, MR, FU, AB and OG, denote the names of the models of neo-Hookean,³³ Mooney–Rivlin,³⁴ Fung,³⁵ Arruda–Boyce,²⁷ and Ogden,³⁶ respectively. C₁, C₂, G₀, λ_m, μ_p and α_p (p = 1, …, M) are hyperelastic parameters. Their relationships with unified hyperelastic parameters G₀ and γ_j are listed in the last two columns of the table. The value of 2N is the total number of material parameters in Ogden model

Number of hyperelastic parameters	Name of hyperelastic model	Hyperelastic model in energy form	Initial hyperelastic shear modulus G0	Dimensionless hyperelastic parameter γj
One	neo-Hookean	W^NH = C1(I1 − 3)	2C1	None
Two	Mooney–Rivlin	W^MR = C1(I1 − 3) + C2(I2 − 3)	2(C1 + C2)
	Fung		C1	C2
	Arruda–Boyce		G0	λm
Multiple	Ogden

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To determine the uniqueness of parameter set {G₀, γ_j}, the f–d relationship obtained from dimensional analysis based on a multiple-parameter Ogden model relationship is shown as,

where superscript OG denotes the name of Ogden model. Subscript capital letters, C, S and F, denote conical, spherical and flat indenters, respectively. For Ogden model, χ takes the value of 2N. ρ_l is geometry factor in which l takes the value of C, S or F. Obviously, ρ_C = θ, ρ_S = r and ρ_F = a, where θ is the half angle of conical indenter, r is the radius of spherical indenter, and a is the radius of the cross-section of cylindrical-shaped flat indenter. Δ_l is the ratio of d to ρ_l.

For single-parameter hyperelastic materials, by eliminating dimensionless parameters γ_j in eqn (1), there is only one material parameter G₀ to be determined. Hence, regardless of the shape of indenter, once given ρ_l, the sole material parameter G₀ can be uniquely determined for all three shapes of indenters.

For multiple-parameter hyperelastic materials, the uniqueness depends on the shape of indenter. For conical indenter, observing the first equation in eqn (1), although only one is physically reasonable, infinite sets of {G₀, γ_j} could satisfy the same equation. Hence, the hyperelastic material parameters G₀ and γ_j cannot be uniquely determined. This conclusion was demonstrated in the previous section in Fig. 1d. For spherical or flat indenter, we transform the f–d relationships shown as the second and third equations in eqn (1) into a normalized form of Ψ − Δ_l (l = S or F), where Ψ denotes f_S/rdd for spherical indenter and f_F/ad for flat indenter. The f–d relationships of spherical and flat indenters are then rewritten as Ψ^OG_l = G₀Π^OG_l(γ₁, …, γ_2N−1, Δ_l). The uniqueness depends on the form of Π^OG_l function. If Π^OG_l can be decoupled into the form of Π_l^OG_γ_j(γ₁, …, γ_2N−1)Π_l^OG_Δ_l(Δ_l), the material parameter set cannot be uniquely determined because there are infinite sets of {G₀, γ_j} to constitute the same value of G₀Π_l^OG_γ_j(γ₁, …, γ_2N−1). In contrast, if γ_j and Δ_l are coupled in Π^OG_l function, {G₀, γ_j} is conditionally unique. This conclusion is explained in the Section II of ESI.†

Above analysis is schematically illustrated in Fig. 3a as a 3-layer binary tree, in which, the first layer considers the number of parameters, the second layer considers the shape of indenter, and the third layer considers whether γ_j and Δ_l are coupled in Π function or not. The layers are named as number (of parameter), shape (of indenters) and coupling (between γ_j and Δ_l). The “Number layer” suggests that, for an indentation experiment, if the hyperelastic material can be described by single-parameter hyperelastic model, the hyperelastic parameter can be uniquely determined no matter the indenter is conical, spherical or flat. The “Shape layer” suggests that, if the hyperelastic material has to be described by multiple parameters, the uniqueness of parameters depends on the shape of the indenter. If the indenter is conical, hyperelastic parameters can't be uniquely determined. The “Coupling layer” suggests that, if the indenter is spherical or flat, the uniqueness depends on whether γ_j and Δ_l are coupled in Π function. If yes, they could be uniquely determined but cannot be guaranteed (conditionally unique); otherwise, cannot be uniquely determined.

Fig. 3 (a) Analysis logic of the uniqueness question. (b) The unified criterion to determine uniqueness.

3.3. A simple criterion to determine uniqueness

Although Fig. 3a has three layers, the criterion of unique determination could be unified as one simple question: whether γ_j and d are coupled in f–d relationship. This criterion obviously works for the coupling layer in Fig. 3a but needs a little explanation for the number and shape layers.

For the shape layer in Fig. 3a, eqn (1) shows that γ_j and d are decoupled in conical solution which leads to non-unique sets of {G₀, γ_j}. Meanwhile, they could be coupled for spherical and flat indenters which in turn provides the possibility to uniquely determine {G₀, γ_j}. Therefore, the underneath criterion for the shape layer is still about the relationship between γ_j and d. For the number layer in Fig. 3a, absence of γ_j in the single-parameter model can be considered as a singular case for which γ_j is either coupled or decoupled with d. Because uniqueness is already been demonstrated, γ_j shall be treated as coupled with d.

The question whether the hyperelastic parameters could be uniquely determined from indentation f–d curve or not is finally answered with a simple criterion: whether γ_j and d are coupled in f–d relationship or not. As schematically shown in Fig. 3b, if the answer is yes, {G₀, γ_j} could be uniquely determined (absolutely or conditionally); otherwise, cannot. Although the answer is derived based on given hyperelastic constitutive models, the analysis is mathematically general.

3.4. Construction of a new unique measure

Once the uniqueness question is answered, it's natural to raise following question: for the hyperelastic indentation problem from which set of {G₀, γ_j} cannot be uniquely determined, is there another measure which is both unique and can govern the materials' hyperelastic behavior? Illustrated by Fig. 3a, the question could be rephrased as, for the situations shown as the two blue circles, can we define any unique measure?

The two blue circles represent decoupled hyperelastic indentation problems based on the multiple-parameter model with conical, spherical and flat indenters. Eqn (1) is then rewritten as:

where,

Eqn (3) shows that, the unit of Ḡ_l (l = C, S or F) is Pascal which agrees with that of the elastic modulus. Hence, it's reasonable to denote Ḡ_l as an equivalent hyperelastic shear modulus. The top-bar of Ḡ_l denotes its equivalency to shear modulus and the subscript l indicates that, for indenters with different shapes, conditions to define Ḡ_l are slightly different. For conical indenter, on top of the decoupling condition, θ needs to be given.

Then, observing eqn (2), following two conclusions are achieved. First, Ḡ_l can be uniquely determined from the f–d curve. Second, Ḡ_l is independent of the form of constitutive model.

Because closed-form solution is not available for the hyperelastic indentation problem, the hyperelastic material parameters can only be extracted with complex numerical methods such as inverse finite element method (FEM) (shown in Fig. 1d). To simplify the procedure, people either adopt point-wise parameters, which depend on the value of strain,^37,38 or simply ignore hyperelasticity.^39,40 Compared with the point-wise and approximate modulus, Ḡ_l shines for its uniqueness and model independency.

To demonstrate application of Ḡ_l, Ḡ_C was evaluated by fitting the f–d curve obtained from conical indentation experiment with the first equation of eqn (2). The quadratic function (shown as orange line in Fig. 1d) fits the experimental data well and yields the value of 3.46 kPa. Ḡ_C was also applied to a stiffer polymer polydimethylsiloxane (PDMS) which was fabricated and tested by Zhang and co-workers.⁴¹ The fitted Ḡ_C is 1.62 MPa.

The above verification demonstrated that Ḡ_C is effective in the modulus range from kPa to MPa, and could capture the difference between the stiffness of the two model materials PEGDA and PDMS. Compared with seeking help from FEM,^42,43 polynomial fitting^39,44 and phenomenological model,⁴⁵ adoption of Ḡ_C has following benefits. First, Ḡ_C is mathematically simpler. Second, derivation of the first equation in eqn (2) is actually not depending on the uniqueness of {G₀, γ_j}. Hence, Ḡ_C is unique regardless of the uniqueness of {G₀, γ_j}. Third, Ḡ_C is a constant number describing the curvature of f–d curve, which allows it to capture the overall hyperelastic indentation response.

3.5. Identification of hyperelasticity

By comparing the powers of d in the f–d relationships of hyperelastic and linear elastic solutions, a useful criterion to identify hyperelasticity was obtained. For isotropic and incompressible linear elastic solids, the power of d is 2, 1.5 and 1 for conical, spherical and flat indenters, respectively. For hyperelastic indentations problems, as shown in eqn (1), it is 2 for conical indenter and indeterminate for spherical and flat indenters. Based on above knowledge, Fig. 4 is constructed. It suggests that, for the soft polymer or tissue, when people are not sure whether it is hyperelastic or linear elastic, a flat indentation experiment shall be carried out. If the obtained f–d curve is not linear, the material is hyperelastic.

Fig. 4 Schematics of how to identify hyperelasticity by conducting flat indentation experiment.

Spherical indenter also produces f–d curves with different powers of d but was not preferred to provide judgement for following two reasons. First, its power of d for linear elastic material is 1.5 which is not easy to be identified. Second, Δ_S is bounded by 1 which limits the value of deformation and, in turn, weakens the exhibition of hyperelasticity.

3.6. Discussion

Main objective of this work is to determine the uniqueness of hyperelastic parameters obtained from indentation f–d curves. Uniqueness is often related with posedness, an important mathematical feature of nonlinear inverse problem. To provide a comprehensive perspective to readers, we would like to briefly discuss the implication of this work on the posedness of hyperelastic indentation problem.

As defined by Jacques Hadamard,⁴⁶ a well-posed inverse problem shall fulfill following conditions: (i) there exists a solution to the problem (existence), (ii) the solution is unique (uniqueness), and (iii) the solution continuously depends on the data (stability). This work shows, for hyperelastic indentation problems, the 1^st condition was naturally satisfied and the 2^nd condition could be determined. Based on the determination of the 2^nd condition, we can conclude that the hyperelastic indentation problems without unique solution are ill-posed. Whether those with unique solution are well-posed needs assessment on the 3^rd condition – stability.

Generally speaking, stability has always been an issue for hyperelastic problems. Due to the complex constitutive behavior of hyperelastic material, even the hyperelastic parameters extracted from uniaxial tensile tests can be easily influenced by experimental error.^47,48 For hyperelastic indentation problem, the difficulty exaggerated. Zhang et al.⁴⁹ investigated the stability of hyperelastic parameters extracted from spherical indentation test, and suggested that the stability of the initial shear modulus and parameters of the Π function depend on the value of Δ_S and material parameters such as locking stretch.

For the hyperelastic indentation problems with unique solutions, if the material could be described by the single-parameter constitutive model, the conditional number of the sole parameter stays as 1 for displacement-controlled experiment (proved in the Section III of ESI†). The stability of the parameter is, therefore, ensured for all three types of indenters. The problem is then determined as well-posed. If the material has to be described by two- or multiple-parameter constitutive model, the stability needs to be assessed, most likely, with the help of complex numerical methods, which could be a meaningful future study. The posedness of indentation problems due to different combinations of indenter shapes and hyperelastic models are summarized in ESI Table S2.†

4. Conclusion

In summary, combing mathematical analysis, experimental validation and computational interpretation, this work determines the uniqueness of hyperelastic parameters obtained from indentation f–d curve with a simple criterion (Fig. 3b): whether γ_j and d are coupled in f–d relationship or not. If the answer is yes, the parameters could be uniquely determined (absolutely or conditionally); otherwise, cannot.

To facilitate the practical usage of this criterion, the uniqueness of parameters due to different combinations of indenter shapes and hyperelastic models are summarized in Table 2. It's clear to see that, for single-parameter hyperelastic models, the uniqueness is ensured for all three types of indenters. For two- and multiple-parameter hyperelastic models, conical indentation leads to non-unique sets of hyperelastic parameters; the situations of spherical and flat indentations are indeterminate.

Table 2 The uniqueness of hyperelastic material parameters with different combinations of indenter shapes and hyperelastic models

Indenter shape	No. of parameter
	Single-parameter	Two-parameter	Multiple-parameter
Conical	Unique	Non-unique	Non-unique
Spherical	Unique	Indeterminate: depends on whether γj and Δl are coupled in Π function. If yes, conditionally unique; otherwise, non-unique
Flat	Unique

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Furthermore, for the hyperelastic indentation problems from which unique parameters are not able to be determined, a new unique measure Ḡ_l is defined and demonstrated its effectiveness by applying to the conical indentation tests on two types of polymers.

In addition, a useful criterion to identify hyperelasticity via indentation test was also obtained and illustrated in Fig. 4.

Acknowledgements

This work was supported by the National Natural Science Foundation of China (No. 11572227) and Hong Kong Research Grant Council Early Career Scheme Fund (No. CityU138713).

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Footnote

† Electronic supplementary information (ESI) available. See DOI: 10.1039/c6ra15747e

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