Indentation creep tests to assess the viscoelastic properties of soft materials: Theory, method and experiment

International Journal of Non-Linear Mechanics xxx (xxxx) xxxx

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Indentation creep tests to assess the viscoelastic properties of soft materials: Theory, method and experiment Xiao Zhang 1 , Yang Zheng 1 , Guo-Yang Li, Yan-Lin Liu, Yanping Cao ∗ AML, Institute of Biomechanics and Medical Engineering, Department of Engineering Mechanics, Tsinghua University, Beijing 100084, China

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Keywords: Reduced creep function Scaling law Finite element simulations Experiments Portable indentation instrument

ABSTRACT Determining time-dependent mechanical properties of soft materials is essential in understanding their deformation behaviors under various stimuli. This paper investigates the use of indentation creep tests to measure the viscoelastic properties of soft materials at local areas. A simple scaling law between the reduced creep function and the creep displacement of the indenter is revealed in this paper based on a theoretical analysis. We show that the scaling relation can be used to interpret indentation creep tests of viscoelastic soft solids with arbitrary surface profile provided that the contact area does not change. Both numerical and practical experiments have been performed to validate the theory and the analytical solution. In our experiments, a low-cost portable indentation system is proposed to measure the reduced creep function. Our results show that the low-cost instrument and the analytical solution to interpret the experimental data reported here represent a useful testing method to deduce the intrinsic viscoelastic properties of soft materials in a non-destructive manner.

1. Introduction Understanding the mechanical properties of compliant materials (e.g. soft tissues, polymeric gels and soft elastomers) is essential in a variety of fields, including not only medicine, but also biology, materials science, tissue engineering and soft matter physics. The conventional testing methods, e.g., uniaxial tension, biaxial tension and compression tests are not applicable in measuring regional properties of compliant materials in a non-destructive manner [1–4]. Therefore, the development of a reliable testing method to characterize the mechanical properties of soft materials across different length scales has received considerable interest over the years [5–20]. This study is concerned with indentation tests which have been frequently used to measure the linear elastic [5], viscoelastic [5–11], poroelastic [12–15] or hyperelastic properties [16,17] of soft materials at a local area. In most indentation tests, indentation load–depth curves can be recorded with a high level of accuracy. However, inferring the mechanical properties of soft materials from the indentation responses represents a challenging inverse problem and is a central issue in the practical use of indentation tests. In this paper, we investigate the determination of intrinsic viscoelastic properties of soft materials, i.e., the reduced creep function, using indentation creep tests. A scaling law is derived to reveal the correlation between the indentation responses and the reduced creep function. We show that the scaling law derived here is applicable to the cases in which the indenter has arbitrary surface profile and the indented soft solid has arbitrary geometry provided that the contact

area does not change in the indentation creep tests. The scaling law leads to a simple inverse method to deduce the reduced creep function of a viscoelastic material; moreover, it inspires us to develop a lowcost portable indentation instrument to perform indentation creep tests. The paper is organized as follows. In Section 2, we derive a simple scaling law between the reduced creep function and indentation responses based on dimensional analysis and elastic–viscoelastic correspondence principle. We prove that this scaling law is independent of the geometric parameters of both the indenter and the tested soft material provided that the contact area is basically constant in indentation creep tests. Based on the scaling relation, a simple inverse method is proposed to interpret the indentation creep tests. In Section 3, we validate our method based on the scaling law using finite element (FE) simulations. Experiments are performed on phantom gels in Section 4 to demonstrate the usefulness of the theory and method. Inspired by our theoretical analysis and numerical studies, a portable indentation instrument has been developed in this study. In Section 5, limitations in the present theory and the portable indentation instrument have been discussed. Section 6 gives the concluding remarks. 2. Theory and method A scaling law revealing the relationship between the reduced creep function of a soft material and the indentation responses is derived in this section. The indentation creep test of an elastic compliant

∗ Corresponding author. E-mail address: [email protected] (Y. Cao). 1 These authors made equal contributions to this study.

https://doi.org/10.1016/j.ijnonlinmec.2018.12.005 Received 8 March 2018; Received in revised form 21 November 2018; Accepted 5 December 2018 Available online xxxx 0020-7462/© 2018 Elsevier Ltd. All rights reserved.

Please cite this article as: X. Zhang, Y. Zheng, G.-Y. Li et al., Indentation creep tests to assess the viscoelastic properties of soft materials: Theory, method and experiment, International Journal of Non-Linear Mechanics (2018), https://doi.org/10.1016/j.ijnonlinmec.2018.12.005.

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the contact area does not change in indentation tests. Then Eq. (4) can be further rewritten as ( ) 𝜉2 𝜉𝑀 𝜁1 𝜁2 𝜁𝑁 𝑃 ℎ= 𝛱 𝑣, 𝛽 , … , 𝛽 , 𝛾 , 𝛾 , … , 𝛾 , (5) 1 𝐸𝜉1 I 𝜉2 𝜉𝑁 𝜉 2 𝜉 𝑀 𝜉 1

1

1

1

1

where 𝛱I is a dimensionless function. We then proceed to the indentation of linear viscoelastic materials following the theoretical analysis in the literature [25]. The constitutive law of a linear viscoelastic material can be written as 𝜎𝑖𝑗 (𝑡) = 𝜕𝜀𝑘𝑙 (𝜏) 𝑡 ∫0 𝐶𝑖𝑗𝑘𝑙 (𝑡 − 𝜏) 𝜕𝜏 d𝜏, where 𝐶𝑖𝑗𝑘𝑙 is the elastic tensor. For an isotropic viscoelastic material ] 𝑡[ 𝜕𝜀𝑖𝑗 (𝜏) 𝜕𝜀 (𝜏) 2𝐺 (𝑡 − 𝜏) 𝜎𝑖𝑗 (𝑡) = d𝜏, (6) + 𝜆 (𝑡 − 𝜏) 𝛿𝑖𝑗 𝑘𝑘 ∫0 𝜕𝜏 𝜕𝜏 where the two Lamé constants 𝐺 and 𝜆 in the time domain are related to the relaxation modulus 𝐸 (𝑡) and Poisson’s ratio 𝑣 (𝑡) by

Fig. 1. Schematic plot of a flat punch indenter indenting into a soft solid.

material as shown in Fig. 1 is first considered, and then the elastic– viscoelastic correspondence principle [21,22] is adopted to deal with the indentation of viscoelastic materials. The present boundary value problem shares the similar feature with the pipette aspiration creep tests addressed in our previous study [23] though the indentation and pipette aspiration are completely two different testing methods. Inspired by the derivations performed in [23], we assume that the geometries of the tested material and the indenter are arbitrary but the contact area between them does not change in indentation tests. This condition can be satisfied when a flat punch indenter (no matter the cross-section geometry) comes into contact with a local flat surface (Fig. 1). For the indentation of a linear elastic solid with the constitutive law given by 𝜎𝑖𝑗 =

𝐸 𝐸𝑣 𝜀 + 𝛿 𝜀 , 1 + 𝑣 𝑖𝑗 (1 + 𝑣)(1 − 2𝑣) 𝑖𝑗 𝑘𝑘

(7a)

𝜆 (𝑡) =

𝐸 (𝑡) 𝑣 (𝑡) . (1 + 𝑣 (𝑡)) (1 − 2𝑣 (𝑡))

(7b)

ℎ∗ (𝑠) =

𝑠𝐽 ∗ (𝑠) 𝑃 ∗ (𝑠) 𝑃 ∗ (𝑠) 𝛱I = 𝛱I , ∗ 𝜉1 𝑠𝐸 (𝑠) 𝜉1

(8)

where 𝐸 ∗ (𝑠) and 𝐽 ∗ (𝑠) are the Laplace transform of the relaxation modulus and the creep function, respectively, and s is the transform ( ) variable. 𝐽 ∗ (𝑠) is related to 𝐸 ∗ (𝑠) by 𝐽 ∗ (𝑠) = 1∕ 𝑠2 𝐸 ∗ (𝑠) [21]. 𝑃 ∗ (𝑠) and ℎ∗ (𝑠) represent the Laplace transform of the indentation load and depth, respectively. Elastic–viscoelastic correspondence principle is applicable here when the contact area does not change. The inverse Laplace transform of Eq. (8) gives

(1)

ℎ (𝑡) =

𝑡 𝛱I d𝑃 𝐽 (𝑡 − 𝜏) d𝜏. 𝜉1 ∫0 d𝜏

(9)

The creep function of a viscoelastic soft material can be written as 𝐽 (𝑡) = 𝐽 (0)𝐽̃(𝑡),

(10)

where the instantaneous creep compliance 𝐽 (0) defines the elastic deformation behavior of the material. The reduced creep function 𝐽̃ (𝑡) determines the intrinsic time-dependent deformation behavior of a linear viscoelastic soft material, which is the main concern of this study. For a generalized Maxwell model, the reduced creep function 𝐽̃ can be written in the form of Prony series, i.e.,

where 𝑃 is the indentation load, The parameters 𝜉1 , 𝜉2 , … , 𝜉𝑀 and 𝜁1 , 𝜁2 , … , 𝜁𝑁 define the profile of the indenter and tested material, respectively, with M and N being finite integers. Dimensional analysis is adopted in this study to characterize the correlation between the experimental responses and the geometric and physical parameters of the system. The geometric parameters 𝜉𝑚 (1 ≤ 𝑚 ≤ 𝑀), 𝜁𝑛 (1 ≤ 𝑛 ≤ 𝑁) should have the following dimensions [ ] 𝜉𝑚 = [ℎ]𝑅𝑚 , (3a)

𝐽̃ (𝑡) = 1 +

𝐾 ∑

[ ( )] 𝑔𝑖 1 − exp −𝑡∕𝜏𝑖 ,

(11)

𝑖=1

where 𝑡 is time, 𝑔𝑖 and 𝜏𝑖 are material constants. In indentation creep tests, the indentation load is described with the Heaviside step function. { 𝑃 = 𝑃max (𝑡 ≥ 0) 𝑃 = , (12) 𝑃 = 0 (𝑡 < 0)

(3b)

where 𝑅𝑚 , 𝑟𝑛 are real numbers. Applying the Buckingham Pi theorem in dimensional analysis [24] to Eq. (2) gives ) ( 𝜉 𝜉 𝜁 𝜁 𝜁 𝑃 , 𝑣, 𝛽2 , … , 𝛽𝑀 , 𝛾1 , 𝛾2 , … , 𝛾𝑁 , (4) ℎ = 𝜉1 𝛱0 1 2 𝐸𝜉12 𝜉1𝑁 𝜉 2 𝜉 𝑀 𝜉1 𝜉1 1

𝐸 (𝑡) , 2 (1 + 𝑣 (𝑡))

Here we assume that the Poisson’s ratio of the tested material is timeindependent, e.g., many soft elastomers are usually assumed to be incompressible with 𝑣 = 0.5. Invoking the elastic–viscoelastic correspondence principle [21,22], the 𝑃 − ℎ relation can be obtained from Eq. (5) by replacing the elastic modulus 𝐸 with 𝐸 ∗ (𝑠) 𝑠

where 𝜎𝑖𝑗 and 𝜀𝑖𝑗 are the components of the stress and strain tensors, respectively, 𝜀𝑘𝑘 is the dilatational strain, 𝜆 is Lamé constant and G is the shear modulus. 𝐸 is Young’s modulus, and 𝑣 is Poisson’s ratio. 𝛿𝑖𝑗 represents the Kronecker delta. The indentation depth must be a function of the following independent parameters ( ) ℎ = 𝑓 𝑃 , 𝐸, 𝑣, 𝜉1 , 𝜉2 , … , 𝜉𝑀 , 𝜁1 , 𝜁2 , … , 𝜁𝑁 , (2)

[ ] 𝜁𝑛 = [ℎ]𝑟𝑛 ,

𝐺 (𝑡) =

where 𝑃max is the creep load. The differential of the Heaviside step function gives the Dirac delta function; therefore, inserting Eq. (12) into (9) gives

1

ℎ (𝑡) =

where 𝜉1 and 𝐸 have the independent dimensions, 𝛽𝐾 (𝐾 = 1, 2, … , 𝑀) and 𝛾𝐿 (𝐿 = 1, 2, … , 𝑁) are real numbers. 𝜉1 has the length unit. 𝛱0 is a dimensionless function. Eq. (4) represents a general expression of the 𝑃 − ℎ curve for the indentation tests of a linear elastic material. Theory of linear elasticity predicts a linear relationship between 𝑃 and ℎ when

𝑡 𝛱I 𝐽 (𝑡 − 𝜏) 𝑃max 𝛿 (𝜏) d𝜏. 𝜉1 ∫0

(13)

Based on the integral property of the Dirac delta function, from Eq. (13) we have 𝛱 (14) ℎ (𝑡) = I 𝑃max 𝐽 (𝑡) , 𝜉1 2

Please cite this article as: X. Zhang, Y. Zheng, G.-Y. Li et al., Indentation creep tests to assess the viscoelastic properties of soft materials: Theory, method and experiment, International Journal of Non-Linear Mechanics (2018), https://doi.org/10.1016/j.ijnonlinmec.2018.12.005.

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Fig. 2. (a) Computational model for a flat punch indenter indenting into a viscoelastic solid, 𝑟i is the radius of the flat punch. (b) Deformation of the indented solid. (c) A comparison of the reduced creep function identified using Eq. (16) (points) with the actual reduced creep function input in the FE analysis.

In Eq. (14), all the geometric parameters are included in the term of which is time-independent. Eq. (14) gives ℎ (0) =

𝛱I 𝑃 𝐽 (0) , 𝜉1 max

𝛱I , 𝜉1

Fig. 3. Simulations of indentation creep tests. (a) The indenter has a square cross-section and (c) the indenter has a triangle cross-section. (b) and (d) show the deformation of the indented solid. (e) A comparison of identified reduced creep function using √ Eq. (16) (points) with the actual reduced creep function input in the FE analysis. 𝑟i = 𝑆∕𝜋 is the equivalent radius of non-cylindrical indenters, where 𝑆 is the cross-section area of the indenter.

(15)

Dividing Eq. (14) by (15) gives 𝐽̃ (𝑡) = ℎ̃ (𝑡) ,

(16) soft material. Fig. 2(a) illustrates the computational model. A total of about 40,000 eight-node quadrilateral axisymmetric elements are used to model the substrate. The creep load is taken as 𝑃max = 40 N. Fig. 2(b) presents the deformed configuration. A comparison of the real solution, i.e., the reduced creep function input in our finite element simulations, with the identified result (points in Fig. 2(c)) based Eq. (16) indicates that the scaling law derived here is valid in this example.

where ℎ̃ (𝑡) = ℎ (𝑡) ∕ℎ (0). Here ℎ (0) is the indentation depth at the starting point of the creep test. Eq. (16) reveals that the evolution of the indentation depth with time is determined by the reduced creep function and has nothing to do with other geometric and physical parameters when the indented solid is linear viscoelastic and the contact area does not change in indentation creep tests. Moreover, Eq. (16) provides us a simple inverse method to deduce 𝐽̃(𝑡) from ℎ̃ (𝑡) which is a directly measurable quantity in indentation creep tests.

3.2. Indentation creep tests of a solid with a non-cylindrical indenter

3. Numerical simulations

To validate the effect of the indenter profile, a 3D computational model has been built and the indenter has square and triangle crosssection, respectively. Figs. 3(a) and (c) show the mesh used in the computation model. Approximately 150,000 ten-node quadrilateral tetrahedron elements are used to model the substrate. Figs. 3(b) and (d) show the deformed configuration. The reduced creep functions extracted from the indentation creep displacements are shown as square and triangle points in Fig. 3(e), which indeed match the real creep functions input in the finite element model well.

To validate the scaling law given by Eq. (16), FE simulations are carried out to in this section. The general purpose finite element softwareABAQUS [26] is used in our numerical experiments; we simulate the indentation creep tests of the soft solids with different geometries. In our simulations, the instantaneous modulus 𝐸 of the soft solid is taken as 2.0 MPa, and its Poisson’s ratio 𝑣 = 0.48. For all simulations, mesh density is decided by comparing the computed indentation load–depth curves with those given by a refined mesh and check whether the difference between normalized indentation depths at the same indentation loads are less than 1%.

3.3. Indentation creep tests of a solid with irregular geometry This example dedicates to verify the applicability of Eq. (16) to the circumstances under which the tested solid is not a flat half-space and its typical dimension is comparable with the contact radius. The axisymmetric computational models are given by Figs. 4(a), (c) and (e). A total of about 25,000 eight-node quadrilateral axisymmetric

3.1. Indentation creep tests of a solid with regular surface profile We first consider indentation creep tests of a viscoelastic half-space, where the contact area is much smaller than the size of the tested 3

Please cite this article as: X. Zhang, Y. Zheng, G.-Y. Li et al., Indentation creep tests to assess the viscoelastic properties of soft materials: Theory, method and experiment, International Journal of Non-Linear Mechanics (2018), https://doi.org/10.1016/j.ijnonlinmec.2018.12.005.

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Fig. 4. Indentation of a viscoelastic solid with its typical dimension comparable with the contact radius. (a), (c) and (e) show the FE models for three different cases. (b), (d) and (f) show the deformed indented solids. (g) represents the comparison of the identified reduced creep functions using Eq. (16) with the real solution input in the direct FE analysis.

elements are used to model the tested material. The creep load is taken as 𝑃max = 40 N and the corresponding deformed configurations are shown in Figs. 4(b), (d) and (f), respectively. Fig. 4(g) shows the reduced creep function determined using Eq. (16) based on the creep displacement ℎ (𝑡) of the indenter given by the FE simulations. The identified solution match the actual reduced creep function input in the FE analysis remarkably well, indicating that the Eq. (16) is valid when the indented solid has irregular geometry. The error in the identified solution for the example given by Fig. 4(a) is greater than those for the examples of Fig. 4(c) and (e), which may be attributed to the effect of finite rotation (because of the comparable 𝑟𝑖 and 𝑟𝑠 for the example of Fig. 4(a)) on the relationship between 𝑃 and ℎ. 3.4. Indentation creep tests of a soft film resting on a rigid substrate In this example, we explore indentation creep tests of a soft film resting on a rigid substrate, in which the substrate effect will come into play. The computational model is given by Fig. 5(a) and a total of about 10,000 eight-node quadrilateral axisymmetric elements are used to model the soft film. Since the contact area is comparable with the film thickness, the tested solid cannot be assumed as a half-space. The creep load is taken as 𝑃max = 40 N. Fig. 5(b) presents the deformed configuration of the film. Same as the numerical examples above, the variation of the indentation depth with time in the creep procedure is recorded, which gives ℎ̃ (𝑡). Then the reduced creep function 𝐽̃ (𝑡) is determined from Eq. (16) and shown in Fig. 5(c) (points). The actual 𝐽̃ (𝑡) input in our finite element analysis is included in Fig. 5(c) for comparison (lines). The results in Fig. 5(c) show that although in this example the effect of substrate is significant, the error in the identified solution is rather small (smaller than 3%), indicating that Eq. (16) is applicable to the case where the tested material has finite thickness, which is comparable or even smaller than the dimension of the contact area.

Fig. 5. Simulation of the indentation creep tests of a soft layer with finite thickness. (a) and (b) show the FE model and the deformed soft layer, respectively. (c) shows the comparison of the identified creep function (points) with the actual solution (solid line).

4

Please cite this article as: X. Zhang, Y. Zheng, G.-Y. Li et al., Indentation creep tests to assess the viscoelastic properties of soft materials: Theory, method and experiment, International Journal of Non-Linear Mechanics (2018), https://doi.org/10.1016/j.ijnonlinmec.2018.12.005.

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International Journal of Non-Linear Mechanics xxx (xxxx) xxxx Table 1 Geometrical parameters and the weights of the four indenters.

4. Experiments on phantom gels Experiments are performed on phantom gels to demonstrate the usefulness of the theory and method presented in Section 2. Inspired by our theoretical analysis, we develop a low cost and portable indentation instrument as described below.

4.1. Materials and methods

Section shape

Circumscribed circle diameter/mm

Height/mm

Weight/g

Square Pentagon Hexagon Circle

30 30 30 30

30 30 30 30

106.06 127.57 139.07 167.33

4.4. Results and analysis Based on the indenter displacement–time curves measured in our experiments, the reduced creep function 𝐽̃ (𝑡) was determined using Eq. (16). The results in Fig. 7 shows that the samples with different agarose concentrations exhibit different creep behavior. Although the greater agarose concentration leads to stiffer samples and smaller creep deformation, the reduced creep functions in Fig. 7 show that creep extent of the sample with greater agarose concentration is slightly higher. To examine the effects of sample thickness, reduced creep functions of the samples with different thicknesses were determined using the present method. The three thicknesses are 40 mm, 30 mm, 20 mm, respectively. The results in Fig. 8 indicate that the thickness of the sample has rather weak effects on the determination of the reduced creep function, consistent with our theoretical prediction. Our theory also shows that under described conditions the geometry of the indenters basically has no effects on the determination of the reduced creep function of a linear viscoelastic material. To justify this conclusion, experiments were further performed using indenters with different geometries, i.e., four indenters with different cross-section geometries (Table 1) were adopted. Fig. 9 shows the reduced creep function determined with Eq. (16). The results indicate that although the indenters with different geometries lead to deformation states, which may affect the linear relationship between 𝑃 and ℎ, as we assumed in the theoretical analysis, indeed the identified creep functions determined using the indenters with different geometries are close to each other, i.e. relative deviations between four tests are less than 8%. In this sense, the experimental results support our theoretical analysis and numerical simulations in Sections 2 and 3.

Homogeneous cylindrical phantoms of different dimensions, with the diameter being 125 mm and thicknesses varying from 20, 30 to 40 mm, were prepared. The phantoms were composed of agarose solution (Sigma Aldrich) and Dulbecco’s modified Eagle’s Medium glucose (DMEM) [27]. Agarose powder was dissolved in distilled water, heated at 300 ◦ C via stirring and left to cool until 80 ◦ C. DMEM was then added to agarose solution to obtain agarose concentrations of 4 mg/ml and 5 mg/ml, respectively. The mixture was stirred until it was homogeneous, then it was poured into cylindrical glass molds and refrigerated at 4 ◦ C until the phantoms solidified. The concentration difference of agarose leads to the difference in viscoelastic properties.

4.2. A portable indentation instrument and experimental set-up Inspired by our theoretical analysis and numerical simulations, we develop a portable indentation instrument to perform indentation creep tests. Fig. 6(a) illustrates the key idea underlying the portable creep indenter and the detailed experimental set-up. Fig. 6(b) shows the practical portable creep indentation instrument consisting of a flat punch indenter and a simple optical system to measure the indenter displacement. A Laser Rangefinders (ZLDS11X; ZSY Group Ltd) was used to determine the accurate displacement of the sample surface, the resolution is 2 μm. Flat punch indenters with different cross-section geometries were used in indentation creep tests. Electromagnet (ELE-P20, Elecall Ltd) attached to digital altimeter (the resolution is 10 μm) was used to control the position of the indenter. The indenter was released when it came into contact with the indented solid. The displacement(mm) as a function of time(s) was recorded at a frequency of 2 Hz, and the indentation depth– time relation was used to determine the reduced creep function from Eq. (16).

5. Discussion In our analysis, the effect of indenter inertia is not considered. Here we address this effect using FE simulations. We simulate the indentation of a viscoelastic soft solid based on a dynamic mechanical model, in which inertia effects are included. Figs. 10(a) and (b) shows the identified creep function with the actual solution input in the FE simulations. The results show that the indentation displacement–time curve exhibits an oscillation at the beginning stage and the indenter reaches the equilibrium position in a short time (after approximately 0.2 s). This indicates that the present method is applicable when the creep time is greater than the oscillation time of the indenter, e.g., greater than 1 s for the present example. However, it should be pointed out that when the oscillation time is comparable or even greater than the characteristic creep time of the material, the present method is inapplicable. The linear relationship between 𝑃 and ℎ during loading procedure forms the basis of the use of Boltzmann superposition and elastic– viscoelastic correspondence principle. However, in practical loading and creep procedure, both material and geometry nonlinearities may be involved. For spherical indentation of soft materials, MacManus et al. (2017) have reported an interesting method to evaluate linear and nonlinear strains using indentation depth and indenter radius [28]. Zhang et al. [29] have investigated the indentation of hyperelastic materials described with the neo-Hookean, Mooney–Rivlin and Arruda–Boyce models; the computational results show that the linear relationship between 𝑃 and ℎ holds well for a flat punch indentation when the ratio

4.3. Measurements Indentation creep tests were performed on the phantoms using the indenters with different cross-section geometries. Samples were placed right below the laser beam. Initial height of sample surface was measured before loading. The indenter was then attracted to an electromagnet, which moved right above the sample and its position was controlled by a digital altimeter as shown in Fig. 6(a). The indenter was released when it came into contact with the sample surface. Indenter displacement–time curves were recorded for a time period of 300 s in each test and used to evaluate the reduced creep function. Six samples were prepared with the thicknesses varying from 20 mm, 30 mm to 40 mm, and the agarose concentration being 4 mg/ml and 5 mg/ml, respectively. Four indenters (Table 1) with square, pentagonal, hexagonal and circular cross-sections were adopted. The weights of the indenters were controlled and the ratio between maximum actual indentation depth and indenter radius is smaller than 0.15. 5

Please cite this article as: X. Zhang, Y. Zheng, G.-Y. Li et al., Indentation creep tests to assess the viscoelastic properties of soft materials: Theory, method and experiment, International Journal of Non-Linear Mechanics (2018), https://doi.org/10.1016/j.ijnonlinmec.2018.12.005.

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Fig. 6. (a) Schematic of the portable creep indenter and the detailed experimental set-up. (b) The practical portable creep indentation instrument consisting of a flat punch indenter and an optical system to measure the indenter displacement.

of indentation depth to indenter radius is smaller than 0.3 [29]. To further confirm this conclusion, FE simulations were performed using a nonlinear viscoelastic model with the hyperelastic deformation behavior described with the neo-Hookean model and the Arruda–Boyce model, respectively (see Appendix for the description of material models). The FE model is the same as that described in Section 3.1. Fig. 10(c) shows the identified reduced creep functions for different ratios of the indentation depth to indenter radius. It can be seen from the results that indeed the effects of finite deformation are negligible when the ratio of maximum indentation depth to indenter radius is up to 0.3. The portable indentation system presented in this study is limited to the measurement at macroscale. The instrument should be refined when experiments are performed at the microscale. 6. Concluding remarks Fig. 7. A comparison of the reduced creep functions of two samples with different concentrations of 4 mg/ml and 5 mg/ml, respectively. Flat punch indenter with circular cross-section was applied in experiments, hs = 2.67ri .

This study investigates indentation creep tests for determining the reduced creep function 𝐽̃ (𝑡), which defines the intrinsic time-dependent 6

Please cite this article as: X. Zhang, Y. Zheng, G.-Y. Li et al., Indentation creep tests to assess the viscoelastic properties of soft materials: Theory, method and experiment, International Journal of Non-Linear Mechanics (2018), https://doi.org/10.1016/j.ijnonlinmec.2018.12.005.

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International Journal of Non-Linear Mechanics xxx (xxxx) xxxx

the creep behavior of viscoelastic materials in indentation creep tests, but also leads to simple method to infer the reduced creep function from indentation responses. Second, finite element simulations have been carried out to validate the analytical solution. The results show that the scaling relation applies to the cases in which indented solids have irregular geometries and/or finite dimensions. In addition, the effects of material and geometry nonlinearities on the practical use of the scaling law have been discussed. The computational results indicate that the effects of finite deformation are negligible when the ratio of maximum indentation depth to indenter radius is up to 0.3. Third, practical experiments have been performed to demonstrate the usefulness of the analytical solution given by the theoretical analysis. The experiments are performed with a portable indentation system for which detailed experimental set-up has been presented. The merits in the proposed portable indentation instrument lie in its low-cost and simple set-up in comparison with other commercialized indentation systems. Our experiments support the conclusions drawn from the theoretical and numerical analysis, i.e., the reduced creep function ̃ can be simply determined from analytical solution 𝐽̃(𝑡) = ℎ(𝑡), when ̃ ℎ(𝑡) is measured in experiments. The effect of the indenter inertia on the application of the portable indentation instrument in practical measurements has been addressed using FE simulations. The results help understand the extent to which the portable indentation instrument is applicable. Finally, it is worth mentioning that this study focuses on the determination of the reduced creep function. We show that determining 𝐽̃ does not require the knowledge of geometric parameters under described conditions. However, this conclusion is not applied to the determination of 𝐽 (0). When 𝐽 (0) is the main concern of a study, one has to use an indenter with regular geometry to simplify the data analysis. Another obvious limitation in this method is that significant errors may occur when the oscillation time of an indenter is comparable to the characteristic creep time of a tested material.

Fig. 8. A comparison of reduced creep functions for the samples with different thicknesses. Experiments were performed on phantoms with (a) the agarose concentration of 4 mg/ml, and (b) the agarose concentration of 5 mg/ml, the flat punch indenter with circular cross-section was adopted.

Acknowledgments We acknowledge support from the National Natural Science Foundation of China (Grant Nos. 11572179, 11172155 and 11432008). Appendix The nonlinear viscoelastic model used in our finite element simulations is described as follows, which assumes that the viscous response is characterized by a linear rate constitutive equation. This model was originally suggested by Simo [30]; and was included in some commercial finite element software later on, such as ABAQUS and ANSYS with slight modifications. For example, in ABAQUS [26], the Simo’s model [30] is amended by taking into account the viscous volumetric response and gives the following set of equations in the spatial configuration for the Kirchhoff stress ) ( ) ( ) 𝜕𝑔 −𝟏 ( 𝑭 𝑡 − 𝑡′ 𝝉 𝐷 𝑡 − 𝑡′ 𝑭 −𝐓 𝑡 − 𝑡′ d𝜏 ′ , 𝐭 0 ∫0 𝜕𝜏 ′ 𝐭 𝑡

Fig. 9. A comparison of reduced creep functions determined with the indenters with different cross-section geometries. Experiments were performed on the phantom with the agarose concentration of 5 mg/ml. The samples have same thickness, ℎs = 40 mm.

𝝉 𝐷 (𝑡) = 𝝉 𝐷 (𝑡) + dev 0

(A.1a) 𝑡

𝝉 𝐻 (𝑡) = 𝝉 𝐻 (𝑡) + 0

behavior of a linear viscoelastic material. In summary, the following contributions have been made.

∫0

) 𝜕𝑘 𝐻 ( 𝝉 𝑡 − 𝑡′ d𝜏 ′ , 𝜕𝜏 ′ 0

(A.1b)

where 𝝉 𝐷 and 𝝉 𝐻 are the deviatoric and the hydrostatic parts of the d𝑡′ Kirchhoff stress. d𝜏 ′ = 𝐴 (𝜃(𝑡 ′ )) , where 𝜃 is the temperature and 𝐴𝜃 the 𝜃 ( ) ′ shift function. 𝑭 𝐭 𝑡 − 𝑡 represents the isochoric part of the deformation gradient of the state at 𝑡 − 𝑡′ relative to the state at 𝑡. 𝑔 and 𝑘 are the shear and bulk viscoelastic kernel functions, respectively. Here, we focus on the case where the Poisson’s ratio is time-independent, and hence 𝑔 = 𝑘. 𝝉 𝐷 and 𝝉 𝐻 are determined from the initial elastic stored energy 0 0

First, a scaling law based on dimensional analysis and elastic– viscoelastic correspondence principle has been derived. The analysis reveals that the reduced creep function is merely determined by the normalized indentation creep depth and the scaling relation is independent of the geometric parameters of the system when the contact area does not change. The analytical solution not only provides insight into 7

Please cite this article as: X. Zhang, Y. Zheng, G.-Y. Li et al., Indentation creep tests to assess the viscoelastic properties of soft materials: Theory, method and experiment, International Journal of Non-Linear Mechanics (2018), https://doi.org/10.1016/j.ijnonlinmec.2018.12.005.

X. Zhang, Y. Zheng, G.-Y. Li et al.

International Journal of Non-Linear Mechanics xxx (xxxx) xxxx

Fig. 10. (a) and (b) Effects of the indenter inertia on the determination of the reduced creep function. Indenter oscillation occurs at the beginning stage of the creep and the indenter reaches the equilibrium state quickly. (c) Creep functions identified using Eq. (16) from the indentation creep tests of a nonlinear viscoelastic solid for which the hyperelastic deformation behavior is described with the neo-Hookean model and Arruda–Boyce model, respectively, the results match the actual solution (solid line) very well when maximum indentation depth to indenter radius is smaller than 0.3.

function 𝛹 0 . In this paper, 𝛹 0 corresponding to the neo-Hookean model and Arruda–Boyce hyperelastic model [31] are used, i.e., 𝛹0 =

𝜇0 1 (𝐼 − 3) + (𝐽𝑒𝑙 − 1)2 , 2 1 𝐷

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(A.2)

and 𝛹0 = 𝜇 + +

{

1 (𝐼 2 1

− 3) +

19 (𝐼 4 7000𝜆6𝑚 1 1 𝐽 2 𝑒𝑙 −1 ( 2 𝐷

1 (𝐼 2 20𝜆2𝑚 1

− 81) +

− 9) +

11 (𝐼 3 1050𝜆4𝑚 1

519 (𝐼 5 673750𝜆8𝑚 1

− 243)

− 27) }

(A.3)

− ln 𝐽𝑒𝑙 )

where 𝜇, 𝜆𝑚 and 𝐷 are material parameters, 𝜆𝑚 is the locking stretch at which upturn of the stress–strain curve would rise significantly. 𝐽𝑒𝑙 is the elastic volume ratio. The initial shear modulus 𝜇0 is related to the shear modulus 𝜇 by 𝜇0 = 𝜇(1 +

99 513 42039 3 + + + ) 5𝜆2𝑚 175𝜆4𝑚 875𝜆6𝑚 67375𝜆8𝑚

(A.4)

The initial bulk modulus 𝐾0 is related to 𝐷 with the expression 𝐾0 = 𝐷2 , where 𝐷 is zero for incompressible materials. The viscoelastic kernel function 𝑔 can be defined in the following form 𝑔 = 1 − 𝜒 (𝑡) ,

(A.5)

where 𝜒 (𝑡) is a function with 𝜒 (0) = 0 and 0 ≤ 𝜒 (𝑡) < 1, which is usually defined as Prony series 𝜒 (𝑡) =

𝑀 ∑

[ ( )] g𝛼 1 − exp −𝑡∕𝜏𝛼 ,

(A.6)

𝛼=1

where g𝛼 is a positive real number defining the extent of the relaxation of the material, 𝜏𝛼 represents the characteristic relaxation time, and M is an integer. 8

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[16] S.C. Meliga, J.W. Coffey, M.L. Crichton, C. Flaim, M. Veidt, M.A. Kendall, The hyperelastic and failure behaviors of skin in relation to the dynamic application of microscopic penetrators in a murine model, Acta Biomater. 48 (2017) 341–356. [17] G. Rauchs, J. Bardon, D. Georges, Identification of the material parameters of a viscous hyperelastic constitutive law from spherical indentation tests of rubber and validation by tensile tests, Mech. Mater. 42 (11) (2010) 961–973. [18] J. Hu, S. Jafari, Y. Han, A.J. Grodzinsky, S. Cai, M. Guo, Size-and speed-dependent mechanical behavior in living mammalian cytoplasm, Proc. Natl. Acad. Sci. (2017) 201702488. [19] H. Ozawa, T. Matsumoto, T. Ohashi, M. Sato, S. Kokubun, Comparison of spinal cord gray matter and white matter softness: measurement by pipette aspiration method, J. Neurosurg. 95 (2) (2001) 221–224. [20] R. Zhao, K.L. Sider, C.A. Simmons, Measurement of layer-specific mechanical properties in multilayered biomaterials by micropipette aspiration, Acta Biomater. 7 (3) (2011) 1220–1227. [21] R.S. Lakes, Viscoelastic Materials, Cambridge University Press, New York, 2009. [22] R.M. Christensen, Theory of Viscoelasticity: An Introduction, Academic, New York, 1982. [23] Y.P. Cao, G.Y. Li, M.G. Zhang, X.Q. Feng, Determination of the reduced creep function of viscoelastic compliant materials using pipette aspiration method, J. Appl. Mech. 81 (7) (2014) 071006.

9

Please cite this article as: X. Zhang, Y. Zheng, G.-Y. Li et al., Indentation creep tests to assess the viscoelastic properties of soft materials: Theory, method and experiment, International Journal of Non-Linear Mechanics (2018), https://doi.org/10.1016/j.ijnonlinmec.2018.12.005.