A Comprehensive Review of Indentation of Gels and Soft Biological Materials

He, Dongjing; Malu, Deep; Hu, Yuhang

doi:10.1115/1.4065434

Abstract

Indentation measurement has emerged as a widely adapted technique for elucidating the mechanical properties of soft hydrated materials. These materials, encompassing gels, cells, and biological tissues, possess pivotal mechanical characteristics crucial for a myriad of applications across engineering and biological realms. From engineering endeavors to biological processes linked to both normal physiological activity and pathological conditions, understanding the mechanical behavior of soft hydrated materials is paramount. The indentation method is particularly suitable for accessing the mechanical properties of these materials as it offers the ability to conduct assessments in liquid environment across diverse length and time scales with minimal sample preparation. Nonetheless, understanding the physical principles underpinning indentation testing and the corresponding contact mechanics theories, making judicious choices regarding indentation testing methods and associated experimental parameters, and accurately interpreting the experimental results are challenging tasks. In this review, we delve into the methodology and applications of indentation in assessing the mechanical properties of soft hydrated materials, spanning elastic, viscoelastic, poroelastic, coupled viscoporoelastic, and adhesion properties, as well as fracture toughness. Each category is accomplished by the theoretical models elucidating underlying physics, followed by ensuring discussions on experimental setup requirements. Furthermore, we consolidate recent advancements in indentation measurements for soft hydrated materials highlighting its multifaceted applications. Looking forward, we offer insights into the future trajectory of the indentation method on soft hydrated materials and the potential applications. This comprehensive review aims to furnish readers with a profound understanding of indentation techniques and a pragmatic roadmap of characterizing the mechanical properties of soft hydrated materials.

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Figure 3

1 Introduction

Indentation measurement, also referred to as depth-sensing indentation, has been widely recognized as a valuable technique for characterizing the mechanical properties of materials. In an indentation test, a designated indenter is controlled to make contact with and indent into the material. Under the external load applied by the indenter, the material undergoes local deformation. Throughout this process, the displacement and reaction force of the indenter are recorded. By analyzing these recorded responses using established contact mechanics theories, the intrinsic mechanical properties of the material can be obtained [1–3].

Indentation has emerged as a crucial tool for characterizing the mechanical properties of soft hydrated materials, such as gels [4–7], cells [8–10], and biological tissues [11–15], over the past two decades. The unique characteristics of the indentation method make it particularly practical for characterizing these materials. First, gels and biological materials are often extremely soft, fragile, and challenging to handle, requiring maintenance in a liquid environment. The indentation method can be seamlessly performed under liquid with minimum sample preparation. Moreover, indentation tests can be conducted on small volumes of material, enabling the characterization of microscopic subjects like single cells and microgel particles [16–18]. Additionally, the indentation method allows for measurements across different locations of a heterogeneous material, facilitating the determination of the local properties—a crucial aspect in various applications involving gels and biological tissues, such as studying the properties of different constituents in a biological material [13,14], mapping the properties of a synthesis patterned gels [19–21] and investigating the sample heterogeneity [22–24]. Another significant advantage of the indentation method is its adaptability to measure mechanical properties across different length scales by altering the size of the indenter or adjusting the indentation depth, thereby modifying the contact size [5–7,25,26]. Hence, the indentation method proves to be a practical and versatile approach for characterizing soft hydrated materials.

Despite the aforementioned advantages, extracting intrinsic material parameters from indentation tests can be challenging. The stress and deformation fields of the material induced by indentation are nonuniform. For soft hydrated materials, factors such as viscoelastic deformation of the polymeric network and solvent transport contribute to time-dependent force–displacement responses [27–31]. Consequently, analyzing these responses to identify material parameters requires a comprehensive understanding of the complex constitutive modeling and contact mechanics theories. Additionally, accurate identification of intrinsic material properties requires careful experimental setup design, including considerations such as indenter geometry, indenter and sample sizes, loading profiles, and time scales. Determining these factors relies on a clear understanding of the physical mechanisms governing soft hydrated material deformation. These requirements raise a barrier to applying the indentation method to the characterization of soft hydrated materials.

In this review, we aim to provide a comprehensive overview of applying the indentation method to characterize soft hydrated materials encompassing gels, cells, and biological tissues. We cover the characterization of both bulk and interfacial properties in literature, including elastic, viscoelastic, poroelastic, viscoporoelastic, and adhesion properties, as well as fracture toughness of soft hydrated materials. For each category, we delve into theoretical models of the contact problems and outline the requirements for experimental setups and methodologies to identify the intrinsic material properties. Additionally, we review the existing measurements on gels and biological materials to demonstrate the applicability of indentation methods to specific material systems. Finally, we summarize existing works on soft hydrated material indentation, pinpoint the challenges and gaps in the literature, and offer a prospective outlook for future studies. This review is intended to serve as a practical guide for using indentation technique to characterize soft hydrated materials.

2 Elastic Property Measurements

When subjected to an external load, a material deforms. If the deformation is recoverable, once the external load is removed, this response of the material is termed as elastic deformation. The degree of deformation of the material under a specific external load is related to its elastic properties. In an indentation test, the force–displacement curve can be utilized to obtain elastic properties. In this section, we will review the contact mechanics models used to obtain elastic parameters from indentation measurements. Then, we will examine the literature where the indentation method is employed to characterize the elastic behavior of certain gels and biological materials.

2.1 Indentation on a Linear Elastic Half-Space.

We consider an elastic, isotropic material occupying an infinite half-space, with a rigid indenter vertically loaded and brought into contact with the upper surface of the material. The indenter, being rigid, maintains its shape postcontact while the material beneath undergoes inhomogeneous deformation. In cases of small displacements, the material's strain is infinitesimal, allowing for its description through linear elasticity. For a linear elastic, isotropic material, the mechanical properties can be described by two independent material parameters, shear modulus $G$ and Poisson's ratio $ν$ ⁠. The shear modulus $G$ represents the material's ability to resist shape change, while the Poisson's ratio $ν$ reflects its ability to resist the volume change of the material. For gels and biological materials, they are typically considered to be incompressible, so their Poisson's ratio $ν$ is close to 0.5 [9,13,14,16,32–34]. Another commonly used linear elastic parameter is Young's modulus $E$ ⁠, which can be expressed in terms of shear modulus and Poisson's ratio, as $E = 2 G (1 + ν)$ ⁠. For incompressible materials, $E = 3 G$ ⁠. In this review, we use shear modulus $G$ and Poisson's ratio $ν$ to represent the linear elastic property of the material for consistency, and one can easily calculate $E$ from the value of $G$ and $ν$ ⁠.

For a linear elastic material, the stress–strain relationship is linear, but the force–displacement curve of the indentation test is not necessarily linear due to the geometric nonlinearity. The force–displacement relation depends on the geometry of the indenter. Common shapes of the indenters include sphere, cylindrical punch, plane-strain cylindrical indenter, and conical indenter. For these geometries, analytical solutions have been given for the reaction force of the indenter in terms of displacement of the indenter, geometric parameters of the indenter, and the linear elastic property of the material [35]. Table 1 summarizes the analytical solutions of several widely used shapes of indenters. It can be observed that the reaction force $F$ is proportional to the elastic component $G / (1 - ν)$ for all the geometries while the dependency on the indentation depth $h$ is geometry specific. For cylindrical flat punch, it is $F \propto h$ ⁠; for spherical indenter, $F \propto h^{1.5}$ ⁠; and for conical indenter, $F \propto h^{2}$ ⁠. For other geometries of sharp indenter, such as Berkovich, Vickers, Knoop, cube corner, the reaction force also satisfies the scaling relation of the conical indenter, $F \propto h^{2}$ ⁠, with a geometry correction factor needed for calculating the reaction force [3,42]. By fitting experimental force–displacement curves to theoretical models, the linear elastic properties of the material can be determined. Scaling relations can validate the experimental data if they conform to the expected power law.

Table 1

Analytical solutions for indenters of common geometries

Geometry	Reaction force, $F$	Contact radius, $a$
Spherical indenter, radius $R$ [35–37]	$F = \frac{8 G h \sqrt{R h}}{3 (1 - ν)}$	$a = \sqrt{R h}$
Conical indenter, half opening angle $θ$ [35,38,39]	$F = \frac{4 G h^{2}}{π (1 - ν)} tan θ$	$a = \frac{2}{π} h tan θ$
Cylindrical flat punch, radius $R$ [35,40]	$F = \frac{4 G h R}{1 - ν}$	$a = R$
Plane-strain cylindrical indenter, radius $R$ [35,41]	$F = \frac{π G a^{2}}{2 R (1 - ν)}$	$h = \frac{a^{2}}{4 R} [2 \ln (\frac{4 R}{a}) - 1]$

Geometry	Reaction force, $F$	Contact radius, $a$
Spherical indenter, radius $R$ [35–37]	$F = \frac{8 G h \sqrt{R h}}{3 (1 - ν)}$	$a = \sqrt{R h}$
Conical indenter, half opening angle $θ$ [35,38,39]	$F = \frac{4 G h^{2}}{π (1 - ν)} tan θ$	$a = \frac{2}{π} h tan θ$
Cylindrical flat punch, radius $R$ [35,40]	$F = \frac{4 G h R}{1 - ν}$	$a = R$
Plane-strain cylindrical indenter, radius $R$ [35,41]	$F = \frac{π G a^{2}}{2 R (1 - ν)}$	$h = \frac{a^{2}}{4 R} [2 \ln (\frac{4 R}{a}) - 1]$

The indenters are assumed to be rigid. Here, $G$ is shear modulus, $ν$ is Poisson's ratio, and $h$ is indentation depth. Inserted illustrations reproduced with permission from Ref. [5].

For soft materials, spherical indenter is the most widely used geometry because it does not induce significant stress concentration and does not require stringent alignment between the indenter and the sample surface. The force–displacement relation shown in Table 1 is an extension from the original Hertzian solution for two spheres in contact [35,36]

F = \frac{8 G h^{*} \sqrt{R^{*} h^{*}}}{3 (1 - ν)}

(1)

where $h^{*}$ is the relative displacement of the center of the spherical indenter and the center of the elastic sphere, and $R^{*}$ is the effective radius, $R^{*} = R_{1} R_{2} / (R_{1} + R_{2})$ ⁠, where $R_{1}$ is the radius of the indenter and $R_{2}$ is the radius of the elastic sphere. If the size of the elastic sphere is much larger than that of the indenter, the solution becomes the indentation of the elastic half space. Here, it is noted that for the indentation of an elastic sphere, the relative displacement of the centers $h^{*}$ is not necessarily the displacement of the indenter. For a soft sphere resting on a rigid substrate, during the indentation, the bottom part of the sphere was also deformed [43,44]. In this case, correction needs to be made by considering the contact deformation due to both the indenter and the substrate [43,44].

To utilize the analytical formulation outlined in Table 1, several prerequisites must be met for the indentation experiment. First, the material is assumed to be an infinite half-space, requiring its dimensions to be significantly larger than the contact radius. Second, the strains within the material must be infinitesimal, necessitating the indentation depth to be much smaller than the indenter's size [35]. Lastly, adhesion and friction between the indenter and the material surface are considered negligible. In subsequent sections, we will also discuss the scenarios where these assumptions are not met in the indentation setup.

2.2 Indentation on Thin Layer of Materials.

In previous discussions, we considered that the thickness of the material is large enough for it to be considered an infinite half-space. However, in reality, this assumption may not always hold true. Instead, the thickness of the material could be comparable to the contact radius. In such cases, the presence of the supporting substrate significantly influences the material's deformation during indentation. Consequently, the analytical solutions presented in Table 1 may overestimate the modulus value, necessitating modifications to accurately describe the force–displacement relation and determine the material parameters.

The finite thickness layer indentation problem has been investigated both analytically and numerically by various studies, leading to the proposed practical force–displacement expressions. For instance, Dimitriadis et al. examined the spherical indentation of finite-thickness elastic poly(vinyl-alcohol) gels placed on a rigid substrate [45]. By solving the linear elasticity boundary value problem, they found the reaction force is the product of the Hertzian contact solution and a correction factor

f (χ)

[45]

F = \frac{8 G h \sqrt{R h}}{3 (1 - ν)} \cdot f (χ)

(2)

where

G

is the shear modulus,

ν

is the Poisson's ratio,

R

is the radius of the spherical indenter, and

h

is the indentation depth. Here, the correction factor

f (χ)

is a function of a dimensionless geometrical parameter

χ = \sqrt{R h} / d

⁠, where

d

is the thickness of the sample. The analytical expression of

f (χ)

was also provided by Dimitriadis et al. [45]. It was shown that for

h / d \leq 0.1

⁠, and if the sample is fixed to the substrate, the correction function can be expressed as [45]

f (χ) = 1 + 1.133 χ + 1.283 χ^{2} + 0.769 χ^{3} + 0.0975 χ^{4}

(3)

Since the solution was derived under the framework of linear elasticity, Eq. holds true only within a limited range of indentation depths where

h / d \leq 0.1

[45]. However, for larger indentation depths, nonlinear elastic effects may come into play, necessitating a modification to the correction function. Long et al. extended the correction factor function to encompass a broader range of indentation depths, using the neo-Hookean model for polyacrylamide gels [46]. In this case, the correction function becomes dependent not only on

χ = \sqrt{R h} / d

but also on

R / d

⁠. The expression of the correction factor was obtained by fitting the calculation results obtained from finite element simulations, as shown in Eq. . It's important to note that the sample is assumed to be fixed to a rigid substrate. The validity range for the correction factor expression is defined as

0.5 \leq R / d \leq 12.7

and

h / d \leq \min (0.6, R / d)

[46]

f (χ, \frac{R}{d}) = \frac{1 + 1.15 χ + [8.94 - 0.89 {(d / R)}^{2}] χ^{3} + 9.288 χ^{6}}{1 + 2.3 χ^{3}}

(4)

Apart from the reaction force, the contact radius for spherical indentation of the thin layer also deviates from the Hertzian contact theory. Similar to the reaction force, the contact radius can also be expressed using a correction factor [47]

a = \sqrt{R h} \cdot l (χ)

(5)

where

\sqrt{R h}

is the contact radius predicted by the Hertzian model for elastic half space. Yu et al. solved both reaction force correction function

f (χ)

and contact radius correction function

l (χ)

for indentation of thin layer materials and expressed the correction functions in terms of the Fredholm integral equation [47]. Later, Hu et al. provided explicit expressions of

f (χ)

and

l (χ)

for spherical indentation [48] for ease to use. It was shown that for

χ \leq 7.0

[48]

f (χ) = \frac{0.46 + 0.82 χ + 2.36 χ^{2}}{0.46 + χ}

(6)

l (χ) = \frac{0.5 + 0.57 χ + 1.41 χ^{2}}{0.5 + 0.49 χ + χ^{2}}

(7)

Another widely used geometry is conical indenter. An example is that people use atomic force microscope (AFM) with a sharp tip to probe a single cell, where both topological information and mechanical properties are obtained [49]. Santos extended the correction function solution to conical indentation, proposing an asymptotic expression based on numerical simulations [49]. Similarly, for conical indentation of a thin elastic layer, the reaction force can be expressed as a product of the half-space contact solution and a correction factor

f (χ),

given by [49]

F = \frac{4 G h^{2}}{π (1 - ν)} tan θ \cdot f (χ)

(8)

where

θ

is the half-opening angle of the conical indenter,

h

is the indentation depth, and the geometrical parameter

χ

for a conical indenter is defined as

χ = (h tan θ) / d

⁠, where

d

is the thickness of the sample [49]. Santos suggested that the correction factor for a conical indenter can be calculated by Eq. for

χ \leq 0.8

[49]

f (χ) = 1 + 0.6298 χ + 0.7236 χ^{2} + 1.249 χ^{3} - 0.3556 χ^{4}

(9)

It is important to note that these expressions assume the thin elastic layer is fixed to the substrate. If the sample and the substrate have frictionless interaction, the correction factors will differ [45,46]. Furthermore, for a more general case, a numerical approach should be employed to fit the force–displacement response of the indentation test and obtain the elastic properties of the material [50,51].

2.3 Large Deformation.

For soft gels and biological tissues, large deformations are common occurrences. Consequently, the stress–strain relation becomes nonlinear, rendering linear elasticity theory inadequate for describing their mechanical behaviors comprehensively. In previous discussions, indentation measurements were limited to small indentation depth, allowing for the extraction of linear elastic properties only. However, to explore the large deformation behavior, deeper indentation depths are necessary, resulting in force–displacement relations that deviate from linear elastic models. In such cases, the material must be characterized by a hyperelastic constitutive relation, and a nonlinear contact mechanics model becomes essential for interpreting the force–displacement response of the indenter.

Generally, calculating the reaction force of indentation under large indentation depth necessitates numerical methods, such as the inverse finite element method (FEM) [52,53]. Here, a hyperelastic constitutive relation is assumed and implemented into the finite element model. The material parameters in the hyperelastic model are determined by fitting the FEM results to the experimental data. However, compared to fitting analytical expressions, the inverse finite element method typically incurs significantly higher computational costs. Instead, attempts have been made to interpret the force–displacement curves using approximate formulations. Miyazaki and Hayashi proposed an exponential force– displacement relation for sharp probe AFM indentation of cells [54], while Jaasma et al. suggested both a similar exponential and quadratic expressions for spherical indentation of cells [55]. These models allow the stiffness of cells to be described by constant parameters, determined through fitting to experimental data [54,55]. While these phenomenological methods provide insights into the relative stiffness of cells, the fitting parameters may not be intrinsic. Efforts have also been made to obtain the nonlinear force–displacement relations through hyperelastic formulations. Lin et al. derived an approximation equation for nonlinear spherical indentation by expressing the hyperelastic relation in terms of mean stress and mean strain, calculated from the Hertzian contact predictions [56,57]. Additionally, Zhang et al. extended the Hertzian contact solution for nonlinear spherical indentation by dimensional analysis methods [58]. For instance, for an incompressible neo-Hookean solid, the reaction force

F

can be expressed as [58]

F = \frac{16 μ_{0} h \sqrt{R h}}{3} \cdot Π (\frac{h}{R})

(10)

where

h

is the indentation depth,

R

is the radius of the indenter,

μ_{0}

is the material parameter in the neo-Hookean model representing the shear modulus, and

Π (h / R)

is a dimensionless function accounting for the material's nonlinearity. The dimensionless function

Π (h / R)

can be calculated from FEM simulation and fitted with an empirical expression. It has been shown that for

h / R < 1

[58]

Π (\frac{h}{R}) = 1 - 0.15 \frac{h}{R}

(11)

By combining Eqs. (10) and (11), the force–displacement curve can be expressed in terms of the neo-Hookean parameter $μ_{0}$ ⁠, determined through experimental data fitting. Similar methods apply to other nonlinear hyperelastic models, such as Mooney–Rivlin, Arruda-Boyce, and Fung models, where numerical methods are essential for determining the dimensionless functions, as demonstrated by Zhang et al. [58].

2.4 Application of Indentation on Gels, Cells, and Biological Tissues.

Utilizing the analytical solutions detailed in the preceding sections, one can deduce the elastic properties from the indentation force–displacement response. Table 2 presents a summary of the shear modulus values obtained for various soft hydrated gels and biological tissues using indentation method. Notably, indentation tests have been adapted for samples spanning a wide range of scales, from microscopic single cells [8,9,34,74–82] to macroscopic polymeric gels [4,5,16–18,33,50,83–89] and biological tissues, such as cartilages [11,32,59–61], brain tissues [14,62–64], corneas [13,65–67], lungs [68,69], and livers [70]. The elastic properties of materials hold significant importance in various applications. For instance, the modulus of biological tissues serves as an indicator of several diseases, including glaucoma [66], cholestatic [70], and breast cancer [71], among others. At the cellular level, studies have demonstrated that cancer cells can exhibit significant variations in stiffness compared to normal cells, depending on the specific cell types and cancers [80–82]. Moreover, the mechanical and biological behaviors of cells are notably influenced by the stiffness of the substrate they inhabit [90–96]. Indentation tests also play a crucial role in investigating spatially heterogeneous biological tissues. For example, Darling et al. revealed through AFM indentation tests on articular cartilage that the pericellular matrix is considerably softer than the extracellular matrix [60]. Similarly, Budday et al. performed indentation tests on brain tissue to study the individual mechanical properties of gray and white matter, reporting that the white matter of the brain tissue tends to be slightly stiffer than gray matter [14].

Table 2

Shear modulus $G$ of various soft hydrated gels and biological tissues measured by indentation method

Material	References and comment	Shear modulus, $G$	Indentation geometry
Cartilage	Hori and Mockros [11], human articular cartilage	0.4–3.5 MPa	Sphere, $R = 10 mm, 20 mm, and 30.4 mm$
	Korhonen et al. [59], bovine humeral, patellar, and femoral cartilages	0.16–0.42 MPa	Sphere, $R = 0.5 mm and 1.5 mm$
	Park et al. [32], bovine humeral cartilage	15.3±6.26 kPa	Sphere, $R = 2.5 μ m$
	Darling et al. [60], articular cartilage	Pericellular matrix: 6.3–98 kPa; extracellular matrix: 22–275 kPa	Sphere, $R = 2.5 μ m$
	Li et al. [61], murine meniscus	3.0–4.6 MPa	Sphere, $R = 5 μ m$
Brain	Miller et al. [62], human brain, in vivo	1.08 kPa	Cylindrical flat punch, $R = 5 mm$
	Dommelen et al. [63], porcine brain	0.67–1.2 kPa	Sphere, $R = 1 mm$
	Budday et al. [14], bovine brain	White matter: 0.631±0.197 kPa; gray matter: 0.456±0.096 kPa	Cylindrical flat punch, $R = 0.375 - 0.75 mm$
	Antonovaite et al. [64], mouse hippocampus	0.5–0.8 kPa	Sphere, $R = 60 - 105 μ m$
Cornea	Last et al. [13], human cornea	Anterior corneal basement membrane: 2.5±1.4 kPa; Descemet's membrane: 16.7±5.9 kPa	Sphere, $R = 1 μ m$
	Last et al. [65], human cornea	Bowman's layer: 36.6±4.4 kPa; anterior stroma: 11.0±2.0 kPa	Sphere, $R = 1 μ m$
	Last et al. [66], human Trabecular Meshwork (HTM)	Normal 0.56–2.93 kPa, glaucoma: 26–83 kPa	Sphere, $R = 1 μ m$
	Mundo et al. [67], human Descemet's membrane	0.23–2.6 kPa	Tetrahedral probe
Lung	Salerno and Ludwig [68], rat lung parenchyma	0.2–1 kPa	Cylindrical flat punch, $R = 2.25 mm$
	Shkumatov et al. [69], lung airway tissue	0.7–16 kPa	Sphere, $R = 2.5 μ m$
Liver	Nava et al. [70], human liver in vivo	Normal: 0.09 MPa; cholestatic: 0.25 MPa	Sphere, $R = 2.25 mm$
Breast	Plodinec et al. [71], human breast biopsies	Normal 0.37–0.61 kPa, Benign 0.63–1.23 kPa, Cancer 0.51–0.66 kPa.	Pyramidal tip
Blood vessel	Peloquin et al. [72], bovine carotid artery	0.83±0.63 kPa	Sphere, $R = 4.5 μ m$
	Valk et al. [73], human heart aortic valve leaflet layers	Fibrosa layer: 6.77–18.9 kPa, spongiosa layer: 4.27–8.93 kPa	Conical indenter
Cell	Hoh and Schoenenberger [8], MDCK cells	∼1 kPa	Conical indenter
	Radmacher et al. [9], human platelet	0.33–16 kPa	Conical indenter
	Domke et al. [34], osteoblasts	0.7–3 kPa	Conical indenter
	Rotsch and Radmacher [74], fibroblasts	∼ 1.5–4 kPa	Conical indenter
	Collinsworth et al. [75], C2C12 cells	3–15 kPa	Conical indenter
	Ng et al. [76], chondrocytes	0.25–1.1 kPa	Sphere, $R = 2.5 μ m$
	Li et al. [77], breast cancer cells	0.07–0.47 kPa	Sphere, $R = 2.25 μ m$
	Maloney et al. [78], human mesenchymal stem cells (hMSCs)	0.6–3 kPa	Sphere, $R = 25 nm$
	McKee et al. [79], human trabecular meshwork (HTM) cell	0.5–3.8 kPa	Pyramid tip
	Viswanathan et al. [80], human thyroid cells	Primary 0.74–2.29 kPa, cancer 0.39–0.45 kPa	Conical indenter
	Bahwini et al. [81], human brain cells	Normal 1.27±0.60 kPa, cancer 0.25±0.11 kPa	Conical indenter
	Pei et al. [82], human liver cells	Normal 0.036–0.126 kPa, hepatoma Bel7402 0.035–0.085 kPa, hepatoma HepG2 0.010–0.031 kPa	Conical indenter
Biopolymer network	Kwon et al. [33], actin network	0.09–0.27 Pa	Sphere, $R = 50 μ m$
	Nowatzki et al. [50], extracellular matrix (aECM) protein thin film	0.1–0.3 Mpa	Sphere, $R = 300 nm$
	Soofi et al. [83], matrigel	440±250 Pa	Sphere, $R = 0.5 μ m$
	You et al. [84], Heparin Gel	3.8–38.6 kPa	Pyramid tip
	Hopkins et al. [85], silk fibroin hydrogels	1.3–11 kPa	Sphere, $R = 6 μ m$
	Dingle et al. [18], cortical cells self-assembled into three-dimensional spheroid	25–100 Pa	Sphere, $R = 2.5 μ m$
	Welsch et al. [86], fibrin gel	460±260 Pa	Sphere, $R = 1.75 μ m$
Synthesis gel	Radmacher et al. [4], gelatin hydrogel	6.7 kPa–33 MPa	Conical indenter
	Wiedemair et al. [16], (pNIPAm-co-AAc) hydrogel particle	33.9 Pa (25 °C)–505 Pa (35 °C)	Conical indenter
	Hashmi and Dufresne [17], NIPAM microgel	2.9–32 kPa	Sphere, $R = 0.5 μ m$
	Henderson et al. [87], ionic cross-linking PMMA hydrogel	1.5 kPa–7 MPa	Cylindrical flat punch, $R = 220 μ m$
	Phelps et al. [88], maleimide cross-linked PEG hydrogel	0.18–0.75 kPa	Pyramidal tip
	Shin et al. [89], CNT-GelMA hydrogels	3.3–10.6 kPa	Sphere, $R = 1 mm$

Material	References and comment	Shear modulus, $G$	Indentation geometry
Cartilage	Hori and Mockros [11], human articular cartilage	0.4–3.5 MPa	Sphere, $R = 10 mm, 20 mm, and 30.4 mm$
	Korhonen et al. [59], bovine humeral, patellar, and femoral cartilages	0.16–0.42 MPa	Sphere, $R = 0.5 mm and 1.5 mm$
	Park et al. [32], bovine humeral cartilage	15.3±6.26 kPa	Sphere, $R = 2.5 μ m$
	Darling et al. [60], articular cartilage	Pericellular matrix: 6.3–98 kPa; extracellular matrix: 22–275 kPa	Sphere, $R = 2.5 μ m$
	Li et al. [61], murine meniscus	3.0–4.6 MPa	Sphere, $R = 5 μ m$
Brain	Miller et al. [62], human brain, in vivo	1.08 kPa	Cylindrical flat punch, $R = 5 mm$
	Dommelen et al. [63], porcine brain	0.67–1.2 kPa	Sphere, $R = 1 mm$
	Budday et al. [14], bovine brain	White matter: 0.631±0.197 kPa; gray matter: 0.456±0.096 kPa	Cylindrical flat punch, $R = 0.375 - 0.75 mm$
	Antonovaite et al. [64], mouse hippocampus	0.5–0.8 kPa	Sphere, $R = 60 - 105 μ m$
Cornea	Last et al. [13], human cornea	Anterior corneal basement membrane: 2.5±1.4 kPa; Descemet's membrane: 16.7±5.9 kPa	Sphere, $R = 1 μ m$
	Last et al. [65], human cornea	Bowman's layer: 36.6±4.4 kPa; anterior stroma: 11.0±2.0 kPa	Sphere, $R = 1 μ m$
	Last et al. [66], human Trabecular Meshwork (HTM)	Normal 0.56–2.93 kPa, glaucoma: 26–83 kPa	Sphere, $R = 1 μ m$
	Mundo et al. [67], human Descemet's membrane	0.23–2.6 kPa	Tetrahedral probe
Lung	Salerno and Ludwig [68], rat lung parenchyma	0.2–1 kPa	Cylindrical flat punch, $R = 2.25 mm$
	Shkumatov et al. [69], lung airway tissue	0.7–16 kPa	Sphere, $R = 2.5 μ m$
Liver	Nava et al. [70], human liver in vivo	Normal: 0.09 MPa; cholestatic: 0.25 MPa	Sphere, $R = 2.25 mm$
Breast	Plodinec et al. [71], human breast biopsies	Normal 0.37–0.61 kPa, Benign 0.63–1.23 kPa, Cancer 0.51–0.66 kPa.	Pyramidal tip
Blood vessel	Peloquin et al. [72], bovine carotid artery	0.83±0.63 kPa	Sphere, $R = 4.5 μ m$
	Valk et al. [73], human heart aortic valve leaflet layers	Fibrosa layer: 6.77–18.9 kPa, spongiosa layer: 4.27–8.93 kPa	Conical indenter
Cell	Hoh and Schoenenberger [8], MDCK cells	∼1 kPa	Conical indenter
	Radmacher et al. [9], human platelet	0.33–16 kPa	Conical indenter
	Domke et al. [34], osteoblasts	0.7–3 kPa	Conical indenter
	Rotsch and Radmacher [74], fibroblasts	∼ 1.5–4 kPa	Conical indenter
	Collinsworth et al. [75], C2C12 cells	3–15 kPa	Conical indenter
	Ng et al. [76], chondrocytes	0.25–1.1 kPa	Sphere, $R = 2.5 μ m$
	Li et al. [77], breast cancer cells	0.07–0.47 kPa	Sphere, $R = 2.25 μ m$
	Maloney et al. [78], human mesenchymal stem cells (hMSCs)	0.6–3 kPa	Sphere, $R = 25 nm$
	McKee et al. [79], human trabecular meshwork (HTM) cell	0.5–3.8 kPa	Pyramid tip
	Viswanathan et al. [80], human thyroid cells	Primary 0.74–2.29 kPa, cancer 0.39–0.45 kPa	Conical indenter
	Bahwini et al. [81], human brain cells	Normal 1.27±0.60 kPa, cancer 0.25±0.11 kPa	Conical indenter
	Pei et al. [82], human liver cells	Normal 0.036–0.126 kPa, hepatoma Bel7402 0.035–0.085 kPa, hepatoma HepG2 0.010–0.031 kPa	Conical indenter
Biopolymer network	Kwon et al. [33], actin network	0.09–0.27 Pa	Sphere, $R = 50 μ m$
	Nowatzki et al. [50], extracellular matrix (aECM) protein thin film	0.1–0.3 Mpa	Sphere, $R = 300 nm$
	Soofi et al. [83], matrigel	440±250 Pa	Sphere, $R = 0.5 μ m$
	You et al. [84], Heparin Gel	3.8–38.6 kPa	Pyramid tip
	Hopkins et al. [85], silk fibroin hydrogels	1.3–11 kPa	Sphere, $R = 6 μ m$
	Dingle et al. [18], cortical cells self-assembled into three-dimensional spheroid	25–100 Pa	Sphere, $R = 2.5 μ m$
	Welsch et al. [86], fibrin gel	460±260 Pa	Sphere, $R = 1.75 μ m$
Synthesis gel	Radmacher et al. [4], gelatin hydrogel	6.7 kPa–33 MPa	Conical indenter
	Wiedemair et al. [16], (pNIPAm-co-AAc) hydrogel particle	33.9 Pa (25 °C)–505 Pa (35 °C)	Conical indenter
	Hashmi and Dufresne [17], NIPAM microgel	2.9–32 kPa	Sphere, $R = 0.5 μ m$
	Henderson et al. [87], ionic cross-linking PMMA hydrogel	1.5 kPa–7 MPa	Cylindrical flat punch, $R = 220 μ m$
	Phelps et al. [88], maleimide cross-linked PEG hydrogel	0.18–0.75 kPa	Pyramidal tip
	Shin et al. [89], CNT-GelMA hydrogels	3.3–10.6 kPa	Sphere, $R = 1 mm$

3 Time-Dependent Elasticity Determination and Measurement

In the preceding sections, we explored the measurement of elasticity of soft hydrated materials. However, conventional elasticity theory has long been acknowledged as inadequate for fully describing the mechanical properties of gels and biological tissues. These materials exhibit time-dependent responses, meaning their stress–strain relationship can change over time [27–31]. Consequently, the force and displacement responses of indentation tests depend on time scales relating to loading rate, holding time, and frequency [97–99]. To accurately characterize the time-dependent behavior and obtain the corresponding intrinsic material properties, a comprehensive understanding of the physical mechanisms of the materials' time-dependent responses is necessary, and the experimental setup needs to be designed properly based on the theoretical formulation.

The time-dependent behaviors of soft hydrated materials stem from their microscopic structures. These materials consist of a polymeric network and interstitial solvent molecules. Depending on the mechanism of the motion of these components, the time-dependent elasticity can be categorized into viscoelasticity and poroelasticity [31,100–102]. Viscoelasticity arises from short-range configurational changes of the polymeric networks while the long-range transport of solvent molecules in the polymeric network induces poroelastic effects [31,102]. In this section, we first review the established theories of linear viscoelasticity and linear poroelasticity applied to the indentation problem and then explore the viscoporoelasic indentation for the coupled conditions. For each aspect, we describe the methodology and applications of using indentation to identify the intrinsic parameters.

3.1 Linear Viscoelasticity.

For the indentation of a viscoelastic material, the reaction force on the indenter depends on time and frequency scales. Figure 1 shows three typical testing methods for characterizing the time-dependent behaviors of materials: relaxation, creep, and oscillation [103,104]. In relaxation tests, the indenter is driven to press onto the material until reaching a certain depth $h$ ⁠. Subsequently, the force is measured as a function of time while the depth is kept constant. During the holding period, the reaction force of the indenter gradually relaxes due to viscoelasticity and eventually reaches a plateau $F (\infty)$ in the long-time limit (Fig. 1(a)). In creep tests, the force on the indenter is maintained constant while the displacement of the indenter is measured. Due to viscoelasticity, the indentation depth gradually increases until reaching a plateau $h (\infty)$ (Fig. 1(b)). These two tests are performed in the time domain. Alternatively, the viscoelastic behavior can be shown in the frequency domain by oscillation test. In this test, the indenter is first pressed onto the material. Then a small amplitude oscillation with controlled frequencies is added (Fig. 1(c)). By comparing the force spectrum and displacement spectrum, a phase lag value $δ$ can be obtained, which can be used to determine the viscoelastic properties of the material.

Fig. 1

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Three types of indentation measurements typically used in characterizing the time-dependent behaviors of materials: (a) relaxation, (b) creep, and (c) oscillation

3.1.1 Viscoelastic Creep and Relaxation Indentation.

To interpret the intrinsic viscoelastic parameters from the indentation tests, it is essential to employ a constitutive model capable of describing the viscoelastic behavior of a material. For a linear viscoelastic solid, its deformation involves two mechanisms, solid-like elastic behavior and fluid-like viscous behavior. For solid-like elastic deformation, the stress is proportional to the strain, whereas in the fluid-like part, the stress correlates with the changing rate of strain [103,104]. This physical understanding leads to the formulation of a stress–strain relation within a rheological model, which consists of two kinds of idealized rheological components, the springs and the dashpots. The spring embodies the elastic part, characterized by a shear modulus, while the dashpot represents the viscous property, described by a viscosity. The combination of these components constructs a rheological model. Table 3 summarizes the commonly utilized rheological models, including the Maxwell model, the Kelvin-Voigt model, the standard linear solid model, and the generalized Maxwell model [103,104]. In each rheological model, the shear moduli of the springs and the viscosities of the dashpots serve as the intrinsic material parameters, governing the stress–strain relations within the rheological model.

Table 3

Viscoelastic rheological models for soft hydrated materials

Model	Relaxation modulus $G (t)$	Storage modulus $G^{'} (ω)$	Loss modulus $G^{″} (ω)$
Maxwell model [103,104]	$G (t) = G_{0} e^{- t / τ}$ ⁠, $τ = η / G_{0}$ ⁠, $G_{\infty} = 0$ ⁠.	$G^{'} (ω) = \frac{η^{2} ω^{2} G_{0}}{G_{0}^{2} + η^{2} ω^{2}}$	$G^{″} (ω) = \frac{η ω G_{0}^{2}}{G_{0}^{2} + η^{2} ω^{2}}$
Kelvin-Voigt model [103,104]	$G (t) = G_{\infty} + η δ (t)$ ⁠, $G_{0} = + \infty$ ⁠, $δ (t) :$ Dirac delta function	$G^{'} (ω) = G_{\infty}$	$G^{″} (ω) = η ω$
Standard linear solid model [104]	$G (t) = G_{\infty} + G_{1} e^{- t / τ_{1}}$ ⁠, $τ_{1} = η_{1} / G_{1}$ ⁠, $G_{0} = G_{\infty} + G_{1}$	$G^{'} (ω) = G_{\infty} + \frac{η_{1}^{2} ω^{2} G_{1}}{G_{1}^{2} + η_{1}^{2} ω^{2}}$	$G^{″} (ω) = \frac{η_{1} ω G_{1}^{2}}{G_{1}^{2} + η_{1}^{2} ω^{2}}$
Generalized Maxwell model (Prony series) [104,105]	$G (t) = G_{\infty} + \sum_{i = 1}^{N} G_{i} e^{- t / τ_{i}}$ ⁠, $τ_{i} = η_{i} / G_{i}$ ⁠, $G_{0} = G_{\infty} + \sum_{i = 1}^{N} G_{i}$	$G^{'} (ω) = G_{\infty} + \sum_{i = 1}^{N} \frac{η_{i}^{2} ω^{2} G_{i}}{G_{i}^{2} + η_{i}^{2} ω^{2}}$	$G^{″} (ω) = \sum_{i = 1}^{N} \frac{η_{i} ω G_{i}^{2}}{G_{i}^{2} + η_{i}^{2} ω^{2}}$

For a particular rheological model, the time-dependent stress–strain relations can be described as follows [103,106,107]:

σ_{i j}^{*} (t) = 2 \int_{- \infty}^{t} G (t - s) \frac{\partial ε_{i j}^{*} (s)}{\partial s} d s

(12)

ε_{i j}^{*} (t) = \frac{1}{2} \int_{- \infty}^{t} J (t - s) \frac{\partial σ_{i j}^{*} (s)}{\partial s} d s

(13)

where

σ_{i j}^{*} (t)

and

ε_{i j}^{*} (s)

denote the deviatoric parts of the stress component

σ_{i j} (t)

and strain component

ε_{i j} (t)

⁠, respectively. Specifically,

σ_{i j}^{*} (t) = σ_{i j} (t) - σ_{k k} (t) / 3

⁠, and

ε_{i j}^{*} (t) = ε_{i j} (t) - ε_{k k} (t) / 3

⁠. Here,

t

is the current time,

G (t)

is the time-dependent relaxation modulus, and

J (t)

is the time-dependent creep compliance. It is assumed that for gels and biological tissues, the material is incompressible, and hence the stress–strain relations are expressed in terms of the deviatoric part of the tensors in Eqs. and . The convolution integration originates from the linear superposition of the infinitesimal changes in the stress field brought by the infinitesimal changes of the strain field. In other words, Eqs. and are applicable for linear viscoelasticity. The relaxation modulus

G (t)

corresponds to the shear modulus in elasticity. In some literature, the relaxation modulus is represented by Young's modulus [107,108]. However, in this review, the term “relaxation modulus” refers to the shear modulus. Similar considerations apply to creep compliance

J (t)

⁠. Mathematically, the creep compliance and the relaxation modulus are related by a convolution integral [103]

\int_{0}^{t} G (t - s) J (s) d s = t

(14)

\int_{0}^{t} J (t - s) G (s) d s = t

(15)

By solving these integration equations, the creep compliance can be determined for a given relaxation modulus, and vice versa.

The analytical expressions of the relaxation modulus

G (t)

depend on the specific rheological models being used. The specific forms for common rheological models are summarized in Table 3. As an example, in the generalized Maxwell model, the relaxation modulus can be represented by a series of exponential decays, known as the Prony series [105]

G (t) = G_{\infty} + \sum_{i = 1}^{N} G_{i} e^{- t / τ_{i}}

(16)

where $G_{\infty}$ is the equilibrium shear modulus, $N$ is the number of dashpots in the generalized Maxwell segments, $G_{i}$ is the shear modulus of the spring connected to the ${i th}^{}$ dashpot, and $τ_{i}$ is the ${i th}^{}$ time scale with $τ_{i} = η_{i} / G_{i}$ ⁠, where $η_{i}$ is the viscosity of the ${i th}^{}$ dashpot. The instantaneous modulus can also be calculated, $G_{0} = G_{\infty} + G_{1} + G_{2} + \dots + G_{N}$ ⁠. The Prony series is commonly employed for its simplicity and ability to describe viscoelastic deformation across a wide range of timescales [105,109,110]. In practice, the number of dashpots $N$ is chosen such that the viscoelastic behavior can be described by a minimal set of fitting parameters [111]. When $N = 1$ ⁠, the model reduces to standard linear solid model.

Based on the stress–strain behaviors described above, the indentation force–displacement response has been quantified. In the case of linear viscoelastic, isotropic material occupying an infinite half space, compressed by a rigid indenter, the reaction force of the indenter can be calculated with the convolution integration [35,106]. For example, for spherical indentation, the time-dependent reaction force

F (t)

and displacement

h (t)

of the indent can be expressed as (considering

ν = 0.5

⁠) [35,106]

F (t) = \frac{16 \sqrt{R}}{3} \int_{- \infty}^{t} G (t - s) \frac{\partial {[h (s)}^{\frac{3}{2}}]}{\partial s} d s

(17)

h {(t)}^{\frac{3}{2}} = \frac{3}{16 \sqrt{R}} \int_{- \infty}^{t} J (t - s) \frac{\partial F (s)}{\partial s} d s

(18)

where $R$ is the radius of the spherical indenter. Here, the convolution integration originates from the fact that in spherical indenation of linear viscoelastic materials, the following quantities are linearly related: the reaction force $F (t)$ ⁠, the stress field, the strain field, and the displacement to the 3/2 power, $h {(t)}^{3 / 2}$ ⁠. As a result, the reaction force can be calculated by considering the superpostion of the infinitesimal changes in the reaction force brought by the infinitesimal changes of the displacement to the 3/2 power [35,106]. Similar convolution expression can be obtained for indentation of linear viscoelastic materials using other geometries of indenters. Detailed derivation can be found in references [35,106].

In general, Eqs. and can be used to fit the spherical indentation experimental data of arbitrary

F (t)

and

h (t)

to obtain the viscoelastic properties. Under certain special conditions, the force–displacement relation can be further simplified. In relaxation and creep experiments, if the loading rate is very fast, it can be assumed that the viscoelastic effect has not yet occurred when reaching the maximum applied displacement or force [35,106]. For spherical indentation relaxation experiment with fast loading, an explicit form can be obtained to relate the relaxation modulus

G (t)

to the measured relaxation force, obtained by simply replacing the shear modulus in Hertzian contact with the relaxation modulus [35,106]

F (t) = \frac{16 G (t) h \sqrt{R h}}{3}

(19)

where

R

is the radius of the spherical indenter and

h

is the indentation depth (Fig. 1(a)). Similarly, for the creep experiment with fast loading, the creep compliance

J (t)

can be directly related to the measured indentation depth measurement by [35,106]

h (t) = {[\frac{3 F}{16 \sqrt{R}} J (t)]}^{\frac{2}{3}}

(20)

where $F$ is the holding force (Fig. 1(b)). Similar considerations apply to other geometries [35]. By substituting the specific form of $G (t)$ or $J (t)$ for the chosen rheological model into Eqs. (19) or (20) and fitting the fast-loading spherical indentation response during the holding period, the intrinsic viscoelastic properties of the material can be determined.

3.1.2 Viscoelastic Oscillation Indentation.

Viscoelastic behavior can also be analyzed in the frequency domain. When a viscoelastic material is subjected to cyclic loading, the response varies depending on the frequency of the loading. At high frequencies, the material doesn't have enough time to relax within one cycle, resulting in a response akin to the short-time behavior in the time domain. Conversely, at very low frequencies, the material's response reflects long-time behavior. Thus, viscoelastic properties can be equivalently represented by a frequency-dependent function called the complex modulus

G^{*} (ω)

⁠, which relates to the time-dependent relaxation modulus

G (t)

as [103]

G^{*} (ω) = j ω \int_{0}^{\infty} G (t) e^{- j ω t} d t

(21)

where $ω$ is the angular frequency of the cyclic loading and $j$ is the imaginary unit. Since $G^{*} (ω)$ is a complex function, it can be written in the form of $G^{*} (ω) = G^{'} (ω) + j G^{″} (ω)$ ⁠, where $G^{'} (ω)$ is the storage modulus and $G^{″} (ω)$ is the loss modulus. These moduli correspond to the shear modulus in linear elasticity. Similar arguments can be made for Young's modulus. Physically, the storage modulus $G^{'} (ω)$ is associated with the solid-like elastic behavior during the periodic deformation of the angular frequency $ω$ ⁠, while the loss modulus $G^{″} (ω)$ ⁠, on the other hand, describes the fluid-like viscous dissipation of the energy during the deformation [103]. The analytical expressions of $G^{'} (ω)$ and $G^{″} (ω)$ for different rheological models are listed in Table 3. By measuring the storage modulus and loss modulus and fitting the experimental data to the analytical expressions in Table 3, the intrinsic viscoelastic properties can be determined.

To determine the storage modulus and loss modulus of linear viscoelastic material, the indentation oscillation test is employed. Initially, the indenter is loaded onto the material and held at the indentation depth

h

until the equilibrium force

F (\infty)

is reached (Fig. 1(c)). The equilibrium modulus

G_{\infty}

can be calculated from Table 1. Subsequently, a small oscillation displacement

h_{a} e^{j ω t}

is applied relative to the holding position, where

h_{a}

is the amplitude of the oscillation (⁠

h_{a} ≪ h

⁠) and

ω

is the angular frequency of the oscillation. In this case, the reaction force also oscillates at the same frequency

ω

but with a phase lag

δ

compared to the input, as shown in Fig. 1(c). The phase lag is determined by the intrinsic viscoelastic properties of the material, independent of the probe geometry. It can be calculated corresponding to a rheological model by

tan δ = G^{″} (ω) / G^{'} (ω)

⁠. By fitting the experimental values of phase lag

δ

to the analytical form based on a specific rheological model (Table 3), along with the value of equilibrium modulus

G_{\infty}

⁠, the intrinsic viscoelastic properties of the material can be determined. Alternatively, the storage modulus

G^{'} (ω)

and loss modulus

G^{″} (ω)

can be directly calculated from the oscillating force response. For instance, for spherical indentation tests, if

h_{a} ≪ h

⁠, the expressions are [97,112]

G^{'} (ω) = \frac{1}{8 \sqrt{R h}} \frac{F_{a}}{h_{a}} cos δ

(22)

G^{''} (ω) = \frac{1}{8 \sqrt{R h}} \frac{F_{a}}{h_{a}} sin δ

(23)

where $F_{a}$ is the amplitude of the oscillational force (Fig. 1(c)), $and R$ is the radius of the indenter. Similar expressions can be obtained for different geometries [112]. The expression in Table 3 can be used to fit the storage modulus and loss modulus from experiment data, and the intrinsic viscoelastic properties can be obtained.

3.1.3 Application of Viscoelastic Indentation on Gels, Cells, and Biological Tissues.

The relaxation indentation, creep indentation, and oscillation indentation methods have been applied extensively on soft hydrated materials to determine their viscoelastic properties, as summarized in Table 4. Typically, biological materials such as brain tissues [14,64,109], cartilages [98], ocular tissues [114], kidneys [98], and biopolymer networks [99,113] exhibit considerable viscoelastic behavior, often characterized by the ratio of the equilibrium modulus to instantaneous modulus, $G_{\infty} / G_{0}$ ⁠. Moreover, the viscoelastic responses of biological tissues occur across multiple time scales, necessitating the utilization of multiple terms in Prony series to appropriately capture the viscoelastic behaviors [14,98,99,109,113–116]. In investigations focusing on single cells, Mahaffy et al. utilized a spherical probe to assess the dynamic modulus of fibroblast cells within a frequency range of 50–200 Hz, yielding storage shear modulus ranging from 0.7 to 1.0 kPa [97]. Subsequently, Alcaraz et al. explored the dynamic modulus of lung epithelial cells over frequencies spanning 0.1–100 Hz observing a power law relationship governing the storage modulus [117], a model widely adopted in describing cellular viscoelasticity [121,123–125]. Materials following a power law expression typically exhibit viscoelastic behavior across a wide range of time scales. Conversely, Darling et al. demonstrated that for various cell types, the indentation relaxation spectrum could be adequately described by the one-timescale standard linear solid model, a widely used one for modeling the viscoelasticity of cells [10,118,119].

Table 4

Viscoelastic properties of soft-hydrated materials measured by indentation method

Material	References	Geometry	Rheological model	Initial shear modulus $G_{0}$	Ratio of relaxation $G_{\infty} / G_{0}$	Time scale/frequency
Porcine brain	Gefen et al. [109]	Sphere, $R = 2 mm$	Prony series, 2 time scales	1.1–2.0 kPa	∼0.44	1.4–42 s
Bovine brain	Budday et al. [14]	Cylindrical flat punch, $R = 0.75 mm$	Prony series, 2 time scales	white matter 0.63±0.20 kPa, gray matter 0.46±0.10 kPa	0.3–0.6	4–160 s
Mouse hippocampus	Antonovaite et al. [64]	Sphere, $R = 60 - 105 μ m$	—	0.8±0.1 kPa	∼0.63	1–10 Hz
Porcine kidney	Mattice et al. [98]	Sphere, $R = 2.78 mm$	Prony series, 3 time scales	5–12 kPa	0.21±0.03	0.04–40 s
Meniscus extracellular matrix	Li et al. [113]	Sphere, $R = 5 μ m$	Prony series, 2 time scales	29.2–89.5 kPa	∼0.6	0.2–15 s
Bovine and human ocular tissues	Yoo et al. [114]	Sphere, $R = 1 - 3 mm$	Prony series, 3 time scales	0.89–40 kPa	0.05–0.53	0.65–111 s
Collagen-agarose cogel	Lake et al. [99]	Plane strain bar	Prony series, 2 time scales	1.3–4.0 kPa	∼0.04	0.2–15 s
Human cervical tissue	Yao et al. [115]	Sphere, $R = 3 mm$	Prony series, 2 time scales	1.0–5.1 kPa	0.43–0.48	6.8–112 s
Costal cartilage	Mattice et al. [98]	Sphere, $R = 1.58 mm$	Prony series, 3 time scales	0.8–3.0 MPa	0.14±0.03	0.4–100 s
Breast cancer tissue	Qiu et al. [116]	Cylindrical flat punch, $R = 1 mm$	Prony series, 2 time scales	1.72–2.27 kPa	0.15–0.30	1.5–60 s
Fibroblast	Mahaffy et al. [97]	Sphere, $R = 0.5 - 6 μ m$	—	0.66–1.03 kPa	0.60–0.68	50–300 Hz
Human lung epithelial cell	Alcaraz et al. [117]	Conical indenter	Power law	∼1.5 kPa	∼0.3	0.1–100 Hz
Chondrocyte	Darling et al. [118]	Sphere, $R = 2.5 μ m$	Standard linear viscoelastic model	0.11–0.20 kPa	∼0.57	∼2.1 s
Chondrosarcoma cell	Darling et al. [118]	Sphere, $R = 2.5 μ m$	Standard linear viscoelastic model	0.16–0.58 kPa	0.5–0.8	4.4–8.0 s
Stem cell	Darling et al. [119]	Sphere, $R = 2.5 μ m$	Standard linear viscoelastic model	0.83–1.07 kPa	0.14–0.7	7.3–10.1 s
Osteoblast	Darling et al. [119]	Sphere, $R = 2.5 μ m$	Standard linear viscoelastic model	0.87–2.2 kPa	0.23–0.69	6.9–15.4 s
Adipocyte	Darling et al. [119]	Sphere, $R = 2.5 μ m$	Standard linear viscoelastic model	0.3–0.83 kPa	0.14–0.68	9.6–31.1 s
MOSE cells	Ketene et al. [120]	Sphere, $R = 5 μ m$	Standard linear viscoelastic model	0.10–0.15 kPa	0.72–0.86	1.1–1.4 s
Fibroblasts	Sousa et al. [121]	Sphere, $R = 3 μ m$	Power law	∼10 kPa	∼0.5	3 ms–41 ms
MDCK cell	Guan et al. [122]	Sphere, $R = 3.5 - 9 μ m$	Combined exponential and power law	0.25–4.8 kPa	<0.02	2.61–6.80 ms
HeLa cell	Guan et al. [122]	Sphere, $R = 3.5 - 9 μ m$	Combined exponential and power law	0.32–1.07 kPa	<0.1	2.91–4.30 ms

Material	References	Geometry	Rheological model	Initial shear modulus $G_{0}$	Ratio of relaxation $G_{\infty} / G_{0}$	Time scale/frequency
Porcine brain	Gefen et al. [109]	Sphere, $R = 2 mm$	Prony series, 2 time scales	1.1–2.0 kPa	∼0.44	1.4–42 s
Bovine brain	Budday et al. [14]	Cylindrical flat punch, $R = 0.75 mm$	Prony series, 2 time scales	white matter 0.63±0.20 kPa, gray matter 0.46±0.10 kPa	0.3–0.6	4–160 s
Mouse hippocampus	Antonovaite et al. [64]	Sphere, $R = 60 - 105 μ m$	—	0.8±0.1 kPa	∼0.63	1–10 Hz
Porcine kidney	Mattice et al. [98]	Sphere, $R = 2.78 mm$	Prony series, 3 time scales	5–12 kPa	0.21±0.03	0.04–40 s
Meniscus extracellular matrix	Li et al. [113]	Sphere, $R = 5 μ m$	Prony series, 2 time scales	29.2–89.5 kPa	∼0.6	0.2–15 s
Bovine and human ocular tissues	Yoo et al. [114]	Sphere, $R = 1 - 3 mm$	Prony series, 3 time scales	0.89–40 kPa	0.05–0.53	0.65–111 s
Collagen-agarose cogel	Lake et al. [99]	Plane strain bar	Prony series, 2 time scales	1.3–4.0 kPa	∼0.04	0.2–15 s
Human cervical tissue	Yao et al. [115]	Sphere, $R = 3 mm$	Prony series, 2 time scales	1.0–5.1 kPa	0.43–0.48	6.8–112 s
Costal cartilage	Mattice et al. [98]	Sphere, $R = 1.58 mm$	Prony series, 3 time scales	0.8–3.0 MPa	0.14±0.03	0.4–100 s
Breast cancer tissue	Qiu et al. [116]	Cylindrical flat punch, $R = 1 mm$	Prony series, 2 time scales	1.72–2.27 kPa	0.15–0.30	1.5–60 s
Fibroblast	Mahaffy et al. [97]	Sphere, $R = 0.5 - 6 μ m$	—	0.66–1.03 kPa	0.60–0.68	50–300 Hz
Human lung epithelial cell	Alcaraz et al. [117]	Conical indenter	Power law	∼1.5 kPa	∼0.3	0.1–100 Hz
Chondrocyte	Darling et al. [118]	Sphere, $R = 2.5 μ m$	Standard linear viscoelastic model	0.11–0.20 kPa	∼0.57	∼2.1 s
Chondrosarcoma cell	Darling et al. [118]	Sphere, $R = 2.5 μ m$	Standard linear viscoelastic model	0.16–0.58 kPa	0.5–0.8	4.4–8.0 s
Stem cell	Darling et al. [119]	Sphere, $R = 2.5 μ m$	Standard linear viscoelastic model	0.83–1.07 kPa	0.14–0.7	7.3–10.1 s
Osteoblast	Darling et al. [119]	Sphere, $R = 2.5 μ m$	Standard linear viscoelastic model	0.87–2.2 kPa	0.23–0.69	6.9–15.4 s
Adipocyte	Darling et al. [119]	Sphere, $R = 2.5 μ m$	Standard linear viscoelastic model	0.3–0.83 kPa	0.14–0.68	9.6–31.1 s
MOSE cells	Ketene et al. [120]	Sphere, $R = 5 μ m$	Standard linear viscoelastic model	0.10–0.15 kPa	0.72–0.86	1.1–1.4 s
Fibroblasts	Sousa et al. [121]	Sphere, $R = 3 μ m$	Power law	∼10 kPa	∼0.5	3 ms–41 ms
MDCK cell	Guan et al. [122]	Sphere, $R = 3.5 - 9 μ m$	Combined exponential and power law	0.25–4.8 kPa	<0.02	2.61–6.80 ms
HeLa cell	Guan et al. [122]	Sphere, $R = 3.5 - 9 μ m$	Combined exponential and power law	0.32–1.07 kPa	<0.1	2.91–4.30 ms

At the tissue level, indentation technique enables the determination of local viscoelastic properties of different structures of the tissue. For instance, Budday et al. revealed that gray matter in brain tissue exhibits greater viscosity compared to white matter [14]. Antonovaite et al. mapped the dynamic modulus of the mouse hippocampus using the oscillation indentation method, highlighting variations in storage modulus within regions characterized by differing nuclei sizes across the frequency range of 1–10 Hz [64]. At the cellular level, the viscoelastic behavior of the cells obtained from indentation test can serve as an indicator of cancer malignancy. Darling et al. observed significant decreases in the instantaneous modulus and viscosity of the chondrosarcoma cells with decreasing malignancy [118]. Similar trends were reported for mouse ovarian cancer cells [120] and human kidney cells [126]. Furthermore, it has been demonstrated that the stiffness of the substrate influences the viscoelastic properties of individual cells [127,128].

3.2 Linear Poroelasticity.

In soft hydrated materials, aside from viscoelastic deformation, the transport of solvent through the polymeric network can also contribute to time-dependent mechanical behaviors, giving rise to poroelasticity [30,100,129–131]. As shown in Fig. 2, unlike the viscoelasticity that arises from the short-range motion of the polymeric network, poroelasticity involves the migration of solvent molecules over a long range through the matrix [30,130]. Consequently, poroelasticity exhibits distinct characteristics compared to viscoelasticity. While the viscoelastic time scale is intrinsic to the materials, the poroelastic time scale is determined by the transport rate which is intrinsic to the material and also the transport length scale which is determined by specific boundary value problems [30]. In essence, while both viscoelasticity and poroelasticity contribute to time-dependent mechanical responses in soft hydrated materials, they arise from different physical mechanisms and exhibit unique behaviors dictated by distinct sets of parameters and length scales [27,28,31,102].

Fig. 2

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Viscoelastic and poroelastic deformations of a soft hydrated material. The viscoelastic deformation results from the short-range motion of the polymeric network. The poroelastic deformation is due to the solvent molecules migrating over a long range through the matrix.

In soft hydrated materials, it is commonly assumed that both the solid and liquid phases are incompressible, leading to volume changes of the material exclusively due to solvent migration—the alteration in solvent amounts [5,30]. To describe a poroelastic material where the individual solid and liquid components are incompressible, three independent material parameters are necessary [5]. The resistance of the polymeric matrix to deformation is quantified by the shear modulus $G$ ⁠. As solvent molecules migrate into or out of the polymeric matrix, causing expansion or contraction, the extendability of the network in accommodating the change of solvent content is captured by the drained Poisson's ratio $ν$ ⁠. Moreover, the kinetics of the solvent migration is governed by a diffusion equation with the diffusivity $D$ (SI unit: $m^{2} / s$ ⁠) as a material parameter. Alternatively, the kinetics of solvent migration can also be characterized by the permeability $k$ (SI unit: $m^{2}$ ⁠), also known as hydraulic permeability or intrinsic permeability, which can be related to the poroelastic properties [5], $k = \frac{1 - 2 ν}{2 (1 - ν)} \frac{D η_{s}}{G}$ ⁠, where $η_{s}$ is the viscosity of the solvent. Another alternative measure of the solvent migration kinetic is the hydraulic conductivity $K$ ⁠, defined as $K = k / η_{s}$ (SI unit: $m^{4} / (N \cdot s)$ ⁠), also referred to as permeability or hydraulic permeability in certain contexts [129,132].

Similar to linear poroelasticity, there was another theory called biphasic model that was also developed to describe the coupled deformation and transport behavior of soft hydrated materials, particularly cartilages [15,129]. Mathematically, the two theories are equivalent but developed based on different microstructural models. Poroelasticity is based on a composite picture while biphasic model considers the separate solid and liquid components and their interactions. A detailed comparison of the two theories can be found in Ref. [133].

3.2.1 Poroelastic Indentation.

Based on linear poroelasticity and biphasic model, the creep, relaxation, and oscillation indentation methods have been developed for extracting the poroelastic parameters. Unlike in viscoelastic indentation, the creep, relaxation, and oscillation tests are equivalent, they are not equally convenient for poroelastic characterization. While the viscoelastic time scale of a material is independent of any length scale, poroelastic time scales with the square of the diffusion length. For indentation problems, the length it takes for the solvent to migrate out of the hydrogel is the contact radius. Except for the cylindrical punch, for all the other shapes of indenters, the contact radius changes as indentation depth. Therefore, in creep tests, the displacement of the indenter keeps changing, and so does the contact radius, resulting in complex time characteristics. It is not feasible to get a simple solution for creep indentation to extract material parameters. Numerical simulations are often needed. On the contrary, in the relaxation indentation, the contact radius remains constant, exhibiting simple time-dependent behavior that can be described by master curves [5]. Therefore, the relaxation method is generally preferred over the creep method in poroelastic material characterization. Similarly, master curve solutions are also obtained for oscillation indentation methods as long as the oscillation amplitude is much smaller than the initial indentation depth. Under this condition, the contact radius can be considered not changing during oscillation. Sections 3.2.2–3.2.4 elaborate on the theoretical methodologies of poroelastic relaxation indentation and oscillation indentation, along with their applications in soft hydrated materials.

3.2.2 Poroelastic Relaxation Indentation.

For formula derivation, we consider an isotropic linear poroelastic material occupying an infinite half space. It is submerged in a solvent and the state of equilibrium is reached. Then, a rigid indenter is rapidly pressed onto the material to a predetermined depth

h

and held in position. The force on the indenter is measured as a function of time. Initially, when the indenter reaches the designated depth, solvent transport has yet to occur, rendering the material's behavior akin to an incompressible elastic material [5]. Consequently, the initial reaction force

F (0)

can be used to determine the shear modulus

G

of the material, as shown in Table 5 by setting the Poisson's ratio in the initial state as 0.5. During the holding period, solvent molecules migrate out of the material, causing the force of the indenter to relax until a new state of equilibrium is reached. After that, the relaxation force reaches a plateau

F (\infty)

⁠, signifying the completion of solvent transportation. Compared with the initial state, now the gel acts as a compressible material with a so-called drained Poisson's ratio

ν

smaller than 0.5. Therefore, the long-time reaction force

F (\infty)

and the initial reaction force

F (0)

can be related as

ν = 1 - \frac{1}{2} \frac{F (0)}{F (\infty)}

(24)

Table 5

Indentation of poroelastic materials for indenters of common geometries [5]

Geometry	Initial reaction force $F (0)$ and size of contact $a$	Relaxation function $g (τ)$
Spherical indenter, radius $R$	$\begin{matrix} F (0) = \frac{16 G h \sqrt{R h}}{3} \\ a = \sqrt{R h} \end{matrix}$	$g (τ) = 0.491 \exp (- 0.908 \sqrt{τ}) + 0.509 \exp (- 1.679 τ)$
Conical indenter, half opening angle $θ$	$\begin{matrix} F (0) = \frac{8 G h^{2}}{π} tan θ \\ a = \frac{2 h}{π} tan θ \end{matrix}$	$g (τ) = 0.493 \exp (- 0.822 \sqrt{τ}) + 0.507 \exp (- 1.348 τ)$
Cylindrical flat punch, radius $R$	$\begin{matrix} F (0) = 8 G h R \\ a = R \end{matrix}$	$g (τ) = 1.304 \exp (- \sqrt{τ}) - 0.304 \exp (- 0.254 τ)$
Plane-strain cylindrical indenter, radius $R$	$\begin{matrix} F (0) = \frac{π G a^{2}}{R} \\ h = \frac{a^{2}}{4 R} [2 \ln (\frac{4 R}{a}) - 1] \end{matrix}$	$g (τ) = 0.791 \exp (- 0.213 \sqrt{τ}) + 0.209 \exp (- 0.95 τ)$

Geometry	Initial reaction force $F (0)$ and size of contact $a$	Relaxation function $g (τ)$
Spherical indenter, radius $R$	$\begin{matrix} F (0) = \frac{16 G h \sqrt{R h}}{3} \\ a = \sqrt{R h} \end{matrix}$	$g (τ) = 0.491 \exp (- 0.908 \sqrt{τ}) + 0.509 \exp (- 1.679 τ)$
Conical indenter, half opening angle $θ$	$\begin{matrix} F (0) = \frac{8 G h^{2}}{π} tan θ \\ a = \frac{2 h}{π} tan θ \end{matrix}$	$g (τ) = 0.493 \exp (- 0.822 \sqrt{τ}) + 0.507 \exp (- 1.348 τ)$
Cylindrical flat punch, radius $R$	$\begin{matrix} F (0) = 8 G h R \\ a = R \end{matrix}$	$g (τ) = 1.304 \exp (- \sqrt{τ}) - 0.304 \exp (- 0.254 τ)$
Plane-strain cylindrical indenter, radius $R$	$\begin{matrix} F (0) = \frac{π G a^{2}}{R} \\ h = \frac{a^{2}}{4 R} [2 \ln (\frac{4 R}{a}) - 1] \end{matrix}$	$g (τ) = 0.791 \exp (- 0.213 \sqrt{τ}) + 0.209 \exp (- 0.95 τ)$

The indenters are assumed to be rigid. Inserted illustrations reproduced with permission from [5].

Here, $ν$ is an intrinsic material parameter describing the volume change of the material due to solvent migration in or out of a linear poroelastic material, and Eq. (24) can be used to obtain its value from indentation measurements. It is noted that this equation holds for any shape of indenters.

For the gel to evolve from the short-time limit toward the long-time limit, the solvent in the gel under the indenter must migrate. The relevant length in this diffusion-type problem is the radius of contact,

a

⁠, and the normalized time takes the form

τ = D t / a^{2}

⁠. The function

F (t)

obeys

\frac{F (t) - F (\infty)}{F (0) - F (\infty)} = g (τ)

(25)

The functional form of $g (τ)$ is only different for different shapes of indenters. Hu et al. solved the poroelastic relaxation indentation problem using finite element methods [5] and presented the results as continuous explicit functions for different shapes of indenters, as shown in Table 5. By fitting the experimental result to Eq. (25), the diffusivity of the material can be obtained.

This method has been proven correct and effective. It has been applied to various soft hydrated materials. One set of experimental results on relaxation indentation of polydimethylsiloxane swollen in organic solvents is shown in Fig. 3 [134]. In this experiment, a spherical indenter with a radius of 20 mm was pressed onto a swollen gel for three different indentation depths. The force relaxation curve for each indentation depth is measured. When the force is normalized by $a h$ ⁠, it can be seen that for larger indentation depth (i.e., larger contact radius), the relaxation time is longer (Fig. 3(b)). When the time $t$ is further normalized by $a^{2}$ ⁠, the three curves converge (Fig. 3(c)). This scaling relation proved that the time-dependent behavior of the gel is primarily poroelastic. With a proper value of fitting parameter $D$ ⁠, the normalized curve of $[F (t) - F (\infty)] / [F (0) - F (\infty)]$ versus $D t / a^{2}$ can fit with the theoretical curve very well.

Fig. 3

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Poroelastic relaxation indentation. A spherical indenter was pressed to three depths into a PDMS gel submerged in heptane. (a) Experimentally measured force as a function of time. (b) The force is normalized as $F / a h$ ⁠, but the time is not normalized. (c) The force is normalized as $F / a h$ ⁠, and the time is normalized as $t / a^{2}$ ⁠. (d) The force is normalized as $[F (t) - F (\infty)] / [F (0) - F (\infty)]$ and the time is normalized as $D t / a^{2}$ ⁠. (Republished with permission from [134]. Copyright Materials Research Society 2011).

Later, the method was also extended to poroelastic indentation of thin films. Hu et al. solved the spherical indentation relaxation problem of the thin layer of poroelastic materials [48]. First, similar to the case of half space, the shear modulus

G

of the materials can be determined from the initial reaction force

F (0)

⁠, but for thin layers of materials, a geometrical correction factor is needed [45,48]

F (0) = \frac{16 G h \sqrt{R h}}{3} \cdot f (χ)

(26)

where

R

is the radius of the spherical indenter,

h

is the indentation depth, and

f (χ)

is the dimensionless geometrical correction factor, and

χ = \sqrt{R h} / d

⁠, where

d

is the thickness of the sample. The expressions of

f (χ)

are shown in Fig. 4(b). Second, the drained Poisson's ratio

ν

can be determined by using the initial reaction force

F (0)

and long-time reaction force

F (\infty)

through Eq. , the same as for the half-space sample. Finally, the diffusivity

D

can be interpreted from the force relaxation curve. For poroelastic relaxation indentation of thin films, the normalized force relaxation curve depends not only on the normalized time

τ

but also on the normalized geometric factor

χ

⁠, [48]

\frac{F (t) - F (\infty)}{F (0) - F (\infty)} = g (τ, χ)

(27)

Fig. 4

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Spherical indentation relaxation of thin layer of linear poroelastic material. (a) Correction factor $l (χ)$ for contact radius.(b) Correction factor $f (χ)$ for initial reaction force. (c) Relaxation function $g (τ, χ)$ versus normalized time for different geometrical parameter $χ$ ⁠. (Republished with permission from [48]. Copyright AIP Publishing 2011).

It is noted that for spherical indentation of a thin film material, the radius of contact

a

is also influenced by the film thickness, that is [48]

a = \sqrt{R h} \cdot l (χ)

(28)

where $l (χ)$ is the correction factor for contact radius. Figure 4 summarizes the expressions for the correction factors, $f (χ)$ and $l (χ)$ ⁠, and the relaxation functions $g (τ, χ)$ for different $χ$ ⁠.

3.2.3 Poroelastic Oscillation Indentation.

To use the poroelastic relaxation indentation method, one has to apply the indentation depth rapidly and be able to measure the instantaneous force accurately. For situations, for instance, in the small-scale test when the poroelastic relaxation time is very short, this condition can be difficult to achieve. Motivated by this difficulty, Lai and Hu developed an alternative method for characterizing the materials' poroelastic properties based on oscillation indentation [6]. In this method, a rigid indenter is pressed into a sample originally equilibrated in a solvent solution to a certain depth

h

and then held in space for a period until the force reaches a plateau

F (\infty)

as illustrated in Fig. 1(c). Then an oscillational displacement with a small amplitude

h_{a}

is applied on top of the initial indent with a wide range of oscillation frequencies. Meanwhile, the force is measured as a function of time. Comparing the displacement spectrum and force spectrum, one can quantify the phase lag

δ

as a function of angular frequency

ω

due to poroelastic dissipation. Lai and Hu solved the poroelastic oscillation indentation problem and found a universal solution for data interpretation that is applicable to all the shapes of indenters for the condition that the oscillation amplitude

h_{a}

is much smaller than the initial indentation depth

h

[6]. The solutions are given as explicit functions. First, the drained Poisson's ratio

ν

determines the phase lag magnitude. For a smaller Poisson's ratio, more solvent can transport into or out of the network through one cycle of loading and unloading, resulting in bigger dissipation and thus bigger phase lag. Lai and Hu obtained the following explicit relation for calculating the drained Poisson's ratio from the experimentally measured maximum phase lag

δ_{c}

[6]:

ν = 0.5 - 0.027 δ_{c} - 0.00136 δ_{c}^{2}

(29)

Corresponding to the maximum phase lag, the critical frequency

ω_{c}

indicates how fast the solvent transports through the network. A quantitative relation was obtained as [6]

D = \frac{a^{2} ω_{c}}{1.975 + 0.872 ν + 1.605 ν^{2}}

(30)

which can be used to quantify the diffusivity of the solvent in the material. Here, $a$ is the radius of contact, which can be determined from Table 5 for different geometries. The maximum phase lag $δ_{c}$ and the critical frequency $ω_{c}$ can be determined from the plot of phase lag $δ$ versus normalized angular frequency $a^{2} ω$ ⁠, as shown in Fig. 5.

Fig. 5

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Indentation oscillation tests of polyacrylamide hydrogel. (a) The applied displacement and measured force under one frequency. (b) The phase lag is plotted against angular frequency. (c) The phase lag is plotted against normalized angular frequency. (Reproduced with permission from [6]. Copyright The Royal Society of Chemistry 2017).

Now, with a known Poisson's ratio, the shear modulus $G$ of the material can be calculated using the equilibrium reaction force $F (\infty)$ using the equations shown in Table 1 corresponding to the specific shapes of indenters. Contrary to the relaxation experiment, in this case, the equilibrium force is utilized to calculate the shear modulus instead of the instantaneous force. This eliminates the need for rapid loading, easing the challenge of small-scale measurement [6].

This method has been proven effective. One set of the experimental results on oscillation indentation of polyacrylamide hydrogel is shown in Fig. 5. This experiment was carried out using AFM with a spherical probe of 25 μm diameter. Three groups of measurements were taken with indentation depths of 200 nm, 350 nm, and 600 nm, respectively. The actuation frequency was taken from 0.4 Hz to 32 Hz. When the frequency is normalized as $ω a^{2}$ ⁠, the three curves overlap (Fig. 5(c)). Taking the peak phase lag value $δ_{c}$ and the corresponding critical normalized angular frequency $a^{2} ω_{c}$ ⁠, the hydrogel's drained Poisson's ratio and the diffusivity can be obtained. With the extracted Poisson's ratio and the equilibrium force as the plateau value in Fig. 5(a), the hydrogel's shear modulus can be calculated from $F (\infty)$ based on the equation listed in Table 1.

3.2.4 Application of Poroelastic Indentation on Gels, Cells, and Biological Tissues.

Relaxation and oscillation indentation tests have been applied to determine the poroelastic properties of various soft hydrated materials, including polymeric gels [5,6,134–136], cells [137,138], cartilage [12,139,140], among others. A summary is listed in Table 6. The poroelastic properties provide insights into the structural characteristics of the soft hydrated material. Specifically, the permeability value can be calculated from $k = \frac{1 - 2 ν}{2 (1 - ν)} \frac{D η_{s}}{G}$ ⁠, and it offers an estimation of the mesh size of the polymeric network. By conceptualizing the solvent transport pathways within the polymer network as cylindrical tubes, the pore size $ξ$ can be approximated by the diameter of these cylindrical tubes, which scales with the square root of the permeability, $ξ \sim k^{1 / 2}$ [110,141,142]. Consequently, the structural property of the matrix of the poroelastic material can be inferred from the solvent transport kinetic property. Nia et al. observed a significant increase in the permeability of glycosaminoglycan-depleted cartilage to normal cartilage [143]. Lai and Hu conducted indentation oscillation tests on polyacrylamide hydrogels under varying swelling ratios, revealing pore size changes ranging from 1 nm to 10 nm as the swelling ratio varies [141].

Table 6

Poroelastic properties of gels, cells, and biological tissues measured by indentation method

Material	References	Geometry	Shear modulus, $G$	Drained Poisson's ratio, $ν$	Diffusivity, $D$	Permeability, $k$
Osteoblast	Shin and Athanasiou [137]	Cylindrical flat punch, $R = 2.5 μ m$	0.51±0.17 kPa	0.37±0.03	(1.58±0.87)×10⁻⁷ m²/s	(1.18±0.65) × 10⁻¹³ m²
MDCK cell	Moeendarbary et al. [138]	Sphere, $R = 5 μ m$	0.9±0.4 kPa	∼0.3	(6.1±1.0) ×10⁻¹¹ m²/s	(1.9±0.3) × 10⁻¹⁷ m²
HeLa cell	Moeendarbary et al. [138]	Sphere, $R = 5 μ m$	0.4±0.1 kPa	∼0.3	(4.1±1.1) × 10⁻¹¹ m²/s	(2.9±0.8) × 10⁻¹⁷ m²
HT1080 cell	Moeendarbary et al. [138]	Sphere, $R = 5 μ m$	0.4±0.2 kPa	∼0.3	(4.0±1.0) ×10⁻¹¹ m²/s	(2.9±0.7) × 10⁻¹⁷m²
Alginate hydrogel	Hu et al. [5]	Conical indenter	27.9 kPa	0.28	3.24 × 10⁻⁸ m²/s	3.55 × 10⁻¹⁶ m²
Polyacrylamide gel	Lai and Hu [6]	Sphere, $R = 12.5 μ m$	16.0±1.0 kPa	0.32±0.02	(6.7±0.8) × 10⁻¹¹ m²/s	(1.1±0.1) × 10⁻¹⁸ m²
Polyacrylamide particle	Berry et al. [136]	Sphere, $R = 6.35 μ m$	6.0±2.2 kPa	0.37±0.03	(3.6±1.6) × 10⁻¹¹ m²/s	(1.4±1.1) × 10⁻¹⁸ m²
Mouse articular cartilage	Cao et al. [139]	Cylindrical flat punch, $R = 55 μ m$	2.0±0.3 MPa	0.20±0.03	(5.41±2.98) × 10⁻¹⁰ m²/s	(1.1±0.4) × 10⁻¹⁹ m²
Bovine femoral condylar cartilage	Mow et al. [12]	Cylindrical flat punch, $R = 0.75 mm$	0.16 MPa	0.39	3.9 × 10⁻¹⁰ m²/s	4.40 × 10⁻¹⁹ m²
Bovine patellar groove cartilage	Mow et al. [12]	Cylindrical flat punch, $R = 0.75 mm$	0.31 MPa	0.24	1.29 × 10⁻⁹ m²/s	1.42 × 10⁻¹⁸ m²
Bovine femoropatellar groove cartilage	Nia et al. [140]	Sphere, $R = 12.5 μ m$	0.19±0.04 MPa	0.1 (prescribed)	(4.28±0.49) × 10⁻⁹ m²/s	(1.00±0.11) × 10⁻¹⁷ m²
Human coagulation clot	He et al. [110]	Sphere, $R = 250 μ m$	0.059 kPa	0.12±0.05	4.8 × 10⁻¹² m²/s	(3.5±0.9) × 10⁻¹⁴ m²

Material	References	Geometry	Shear modulus, $G$	Drained Poisson's ratio, $ν$	Diffusivity, $D$	Permeability, $k$
Osteoblast	Shin and Athanasiou [137]	Cylindrical flat punch, $R = 2.5 μ m$	0.51±0.17 kPa	0.37±0.03	(1.58±0.87)×10⁻⁷ m²/s	(1.18±0.65) × 10⁻¹³ m²
MDCK cell	Moeendarbary et al. [138]	Sphere, $R = 5 μ m$	0.9±0.4 kPa	∼0.3	(6.1±1.0) ×10⁻¹¹ m²/s	(1.9±0.3) × 10⁻¹⁷ m²
HeLa cell	Moeendarbary et al. [138]	Sphere, $R = 5 μ m$	0.4±0.1 kPa	∼0.3	(4.1±1.1) × 10⁻¹¹ m²/s	(2.9±0.8) × 10⁻¹⁷ m²
HT1080 cell	Moeendarbary et al. [138]	Sphere, $R = 5 μ m$	0.4±0.2 kPa	∼0.3	(4.0±1.0) ×10⁻¹¹ m²/s	(2.9±0.7) × 10⁻¹⁷m²
Alginate hydrogel	Hu et al. [5]	Conical indenter	27.9 kPa	0.28	3.24 × 10⁻⁸ m²/s	3.55 × 10⁻¹⁶ m²
Polyacrylamide gel	Lai and Hu [6]	Sphere, $R = 12.5 μ m$	16.0±1.0 kPa	0.32±0.02	(6.7±0.8) × 10⁻¹¹ m²/s	(1.1±0.1) × 10⁻¹⁸ m²
Polyacrylamide particle	Berry et al. [136]	Sphere, $R = 6.35 μ m$	6.0±2.2 kPa	0.37±0.03	(3.6±1.6) × 10⁻¹¹ m²/s	(1.4±1.1) × 10⁻¹⁸ m²
Mouse articular cartilage	Cao et al. [139]	Cylindrical flat punch, $R = 55 μ m$	2.0±0.3 MPa	0.20±0.03	(5.41±2.98) × 10⁻¹⁰ m²/s	(1.1±0.4) × 10⁻¹⁹ m²
Bovine femoral condylar cartilage	Mow et al. [12]	Cylindrical flat punch, $R = 0.75 mm$	0.16 MPa	0.39	3.9 × 10⁻¹⁰ m²/s	4.40 × 10⁻¹⁹ m²
Bovine patellar groove cartilage	Mow et al. [12]	Cylindrical flat punch, $R = 0.75 mm$	0.31 MPa	0.24	1.29 × 10⁻⁹ m²/s	1.42 × 10⁻¹⁸ m²
Bovine femoropatellar groove cartilage	Nia et al. [140]	Sphere, $R = 12.5 μ m$	0.19±0.04 MPa	0.1 (prescribed)	(4.28±0.49) × 10⁻⁹ m²/s	(1.00±0.11) × 10⁻¹⁷ m²
Human coagulation clot	He et al. [110]	Sphere, $R = 250 μ m$	0.059 kPa	0.12±0.05	4.8 × 10⁻¹² m²/s	(3.5±0.9) × 10⁻¹⁴ m²

3.3 Viscoporoelastic Indentation.

As soft hydrated gels or biological materials consist of both polymeric networks and interstitial fluids, viscoelasticity and poroelasticity can be coupled in governing the materials' time-dependent behavior in certain conditions [27,31,100–102,144]. Theoretically, the viscoelastic and poroelastic behaviors of a material can be decoupled by choosing the right length scale and time scale of measurements. For instance, one can use a very large contact size to make the poroelastic time to be much longer than the viscoelastic time, and if the observation time is long, then the dominant behavior observed from the measurement will be poroelastic. A detailed discussion of the decoupling of poroelasticity and viscoelasticity can be found in [31]. However, there are situations when the choice of time and length scales of measurement is not free and the viscoelastic and poroelastic behaviors cannot be decoupled. In these cases, explicit solutions of viscoporoelastic indentation are often not possible, and finite element simulation is required to solve the problem and compare the results with experiments for parameter extractions [145–147]. Instead of solving the boundary value problem rigorously, several empirical expressions have been proposed for estimating the indentation relaxation force for viscoporoelastic materials. Chan et al. proposed a decomposition of the viscoporoelastic relaxation force into two additive terms for spherical indentation problem [148]

F (t) = F_{visco} (t) + F_{poro} (t)

(31)

where

F_{visco} (t)

represents the viscoelastic contribution, and

F_{poro} (t)

represents the poroelastic contribution. These two time-dependent contributions are calculated by [148]

F_{visco} (t) = [F (0) - F_{i}] \exp (- t / τ_{v})

(32)

F_{poro} (t) = F (\infty) + [F_{i} - F (\infty)] \exp (- α_{p} {(\frac{D t}{R h})}^{β_{p}})

(33)

where

F (0)

is the short-time reaction force,

F (\infty)

is the long-time equilibrium reaction force,

R

is the radius of the spherical indenter,

h

is the indentation depth,

τ_{v}

is the viscoelastic time scale,

D

is the diffusivity,

{and F}_{i}

is the intermediate load, which is a fitting parameter that describes the amount of viscoelastic relaxation, and

α_{p}

and

β_{p}

are geometric coefficients that are represented by geometrical parameter

χ = \sqrt{R h} / d

⁠,

d

is the thickness of the sample. For

0.11 \leq χ \leq 2.98

[148]

α_{p} (χ) = 1.15 + 0.44 χ + 0.89 χ^{2} - 0.42 χ^{3} + 0.06 χ^{4}

(34)

β_{p} (χ) = 0.56 + 0.25 χ + 0.28 χ^{2} - 0.31 χ^{3} + 0.1 χ^{4} - 0.01 χ^{5}

(35)

Here, the viscoelastic relaxation of the material is represented by a Maxwell model, and the poroelastic relaxation is described by a generalized exponential. In this decomposition model, the fitting parameters include $τ_{v}$ ⁠, $D$ and $F_{i}$ [148].

Alternatively, Strange et al. proposed a multiplication form of decomposition for the viscoporoelastic relaxation force [149]

F (t) = \frac{F_{visco} (t) F_{poro} (t)}{F (\infty)}

(36)

where

F (\infty)

is the long-time equilibrium reaction force. For indentation relaxation experiment with fast loading, the viscoelastic contribution

F_{visco} (t)

and the poroelastic contribution

F_{poro} (t)

are calculated by [149]

F_{visco} (t) = \frac{8 h \sqrt{R h}}{3 (1 + ν)} [G_{\infty} + \sum_{i = 1}^{n} G_{i} e^{- t / τ_{i}}]

(37)

F_{poro} (t) = F (\infty) + [F (0) - F (\infty)] \sum_{i = 1}^{m} A_{i} \exp (- {(\frac{c_{p} t}{R h})}^{β_{i}} / τ_{i}^{P E})

(38)

where $G_{\infty}$ is the long-time equilibrium shear modulus, $G_{i}$ and $τ_{i}$ are the viscoelastic Prony series parameters, $ν$ is the Poisson's ratio, $c_{p}$ is the consolidation coefficient, and $A_{i}$ ⁠, $β_{i}$ ⁠, and $τ_{i}^{P E}$ are the poroelastic parameters. Here, both viscoelasticity and poroelasticity of the material are described by the series of exponential decay.

It is noted that both the addition form and the multiplication form of decompositions postulated that viscoelasticity and poroelasticity can be separated, though the validity of this assumption is unclear. In general, a numerical method, such as FEM, is required to solve the problem. By fitting the indentation results with the FEM calculation, the intrinsic viscoporoelastic material parameters of a soft hydrated material can be determined. This process can be challenging, as the viscoelastic response and the poroelastic response may strongly intertwine, making the uniqueness of the fitting parameters difficult to guarantee [145–147,150,151]. To address this issue, complementary measurements are usually required to provide additional information on the materials' behaviors. He et al. proposed a strategy combining shear rheology measurement and the indentation test to determine the viscoelastic and poroelastic contributions of the blood clot separately, as shown in Fig. 6 [110]. Shear rheology measurements yield responses purely indicative of viscoelasticity since it does not induce volume change and thus only invokes viscoelastic responses, while indentation tests probe both viscoelastic and poroelastic responses of the materials. Since the viscoelastic parameters have been determined from the shear rheology, only the poroelastic parameters remain unknown, which can be obtained by fitting the indentation measurement results with the numerical simulation. This way, the uniqueness of the viscoporoelastic parameters can be ensured [110].

Fig. 6

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Combing shear rheology and indentation relaxation method to measure the viscoporoelastic properties of blood clot. (a) The viscoelastic parameters are determined by shear rheology. (b) Indentation relaxation of the blood clot. The predicted relaxation from pure viscoelastic part is not enough to account for the overall relaxation, indicating additional poroelastic effect. A coupled viscoporoeastic model is used to simulate the result and fitted with the experimental data to extract the poroelastic part of the material parameters. (Republished with permission from [110]. Copyright Elsevier Ltd. 2022).

4 Indentation Adhesion

Adhesion refers to the bond formed at the interface between two surfaces. In the context of an indentation test, the adhesion between the material being tested and the indenter affects the material's deformation, resulting in a deviation in the indenter's reaction force from previously discussed predictions [152,153]. Furthermore, adhesion plays a critical role in many applications of soft-hydrated materials, with indentation tests serving as a means to assess the intrinsic adhesion properties of the interface [7,154–161]. In this section, we reviewed the established theories related to the indentation adhesion technique. We also discussed the adhesion hysteresis and the influence of surface tension on the assessment of indentation adhesion.

4.1 Adhesion Energy Measurement Based on Traditional Models.

In an indentation test of an adhesive, linear elastic solid, the force–displacement relations can be altered from Table 1. The pull-off force $P$ which is the maximum negative force during the pulling process and the energy of separation $W$ are positively related to the adhesion of the interface and can be used to indicate the strength of adhesion (Fig. 7(b)), although these two values also depend on the size and geometry of the indentation test [7]. More rigorous methods have been developed for measuring the intrinsic adhesion properties of the interface. In general, the adhesion properties of the material can be described by an adhesion energy $γ$ ⁠, which describes the surface energy per unit contact area of two adhesive surfaces [35,162,163]. The widely used models to determine adhesion energy include the Johnson–Kendall–Roberts (JKR) model [162] and Derjaguin–Muller–Toporov (DMT) model [163]. Each model made their own assumptions, and thus the calculations of adhesion energy for the two models are different. Later, it was discussed that JKR and DMT models are suitable for different material systems. In general, the JKR model is more suitable for soft material with strong adhesion and large size of indenter, while DMT works better for hard material, low adhesion energy, and small radius of probe [164–167]. Maugis genderized the models for spherical indentation adhesion problem by introducing Dugdale's cohesive zone at the interface [164]. The Maugis–Dugdale model was able to connect the JKR and DMT solutions and describes the adhesion behavior materials with arbitrary stiffness, strength of adhesion, and indenter size [164]. Here, we introduce the JKR model and the Maugis–Dugdale model as these two models have been widely applied to soft hydrated material systems [7,152,153,168–171].

Fig. 7

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Indentation adhesion hysteresis under microscopic indentation. (a) Schematic of adhesive measurement during the retraction process. (b) The experimental force–displacement curve. The loading part follows Hertz contact prediction, while the retraction part shows significant pull-off force. (c) Analytical solution of the transition process from Hertz contact to adhesive contact. (d) Normalized pull-off force versus normalized initial contact radius for different $λ$ ⁠. (Reproduced with permission from [7]. Copyright Elsevier Ltd. 2019).

The JKR model describes the contact problem that a spherical indenter is in contact with a linear elastic, adhesive solid. Due to the adhesion between the two surfaces, the force–displacement relation and the size of contact are altered from the Hertz contact theory. According to the JKR model, for a rigid indenter, the reaction force

F_{JKR}

and the radius of contact

a_{JKR}

can be expressed implicitly in Eqs. and [162]

F_{JKR} = \frac{8 G {a_{JKR}}^{3}}{3 R (1 - ν)} - \sqrt{\frac{16 π G γ {a_{JKR}}^{3}}{1 - ν}}

(39)

h = \frac{{a_{JKR}}^{2}}{R} - \sqrt{\frac{π γ a_{JKR} (1 - ν)}{G}}

(40)

where

G

is the shear modulus of the material,

ν

is the Poisson's ratio,

R

is the radius of the spherical indenter,

h

is the displacement, and

γ

is the adhesion energy of the interface. For incompressible materials,

ν = 0.5

⁠. The adhesion energy

γ

only depends on the surfaces of the indenter and the material but not the geometry and size. While Eqs. and are expressed implicitly, the adhesion energy can be easily determined from only the pull-off force

P

⁠, as shown in the following equation [162]:

P = - \frac{3}{2} γ π R

(41)

It is noted that from Eq. (41), the pull-off force only depends on the radius of the indenter and the adhesion energy of the interface, but is independent of the indentation depth and the material elasticity. Using Eq. (41), the adhesion energy $γ$ can be easily determined from the pull-off force and by substituting the adhesion energy $γ$ back into Eq. (39), the elastic property of the material can be determined. Due to its simplicity, the JKR model has been applied to characterize the adhesion of various soft hydrated materials [152,153,168,169].

For Maugis–Dugdale model, instead of a single parameter of adhesion energy

γ

⁠, the intrinsic adhesive property of the interface is described by a rectangle shape of the cohesive zone, which requires two independent material parameters, namely, the cohesive strength

σ_{0}

and the separation distance

l_{0}

⁠. In this case, the adhesion energy

γ

can be expressed by

γ = σ_{0} l_{0}

⁠, representing the area surrounded by the traction–separation curve. For a rigid spherical indenter that is in contact with a linear elastic solid, according to Maugis–Dugdale model, the reaction force of the indenter

F_{M - D}

and the contact radius

a_{M - D}

can be calculated from the normalized following equations implicitly [164]:

{\bar{F}}_{M - D} = {({\bar{a}}_{M - D})}^{3} - λ {\bar{a}}_{M - D} (\sqrt{{(\frac{c}{a_{M - D}})}^{2} - 1} + {(\frac{c}{a_{M - D}})}^{2} \arctan \sqrt{{(\frac{c}{a_{M - D}})}^{2} - 1})

(42)

{\bar{h}}_{M - D} = {({\bar{a}}_{M - D})}^{2} - \frac{4}{3} λ {\bar{a}}_{M - D} \sqrt{{(\frac{c}{a_{M - D}})}^{2} - 1}

(43)

\frac{λ {({\bar{a}}_{M - D})}^{2}}{2} (\sqrt{{(\frac{c}{a_{M - D}})}^{2} - 1} + [{(\frac{c}{a_{M - D}})}^{2} - 2] \arctan \sqrt{{(\frac{c}{a_{M - D}})}^{2} - 1}) + \frac{4 λ^{2} {\bar{a}}_{M - D}}{3} (1 - \frac{c}{a_{M - D}} + \sqrt{{(\frac{c}{a_{M - D}})}^{2} - 1} \arctan \sqrt{{(\frac{c}{a_{M - D}})}^{2} - 1}) = 1

(44)

where $h$ is the indentation depth, $K = 8 G / [3 (1 - ν)]$ is the reduced modulus, $c$ is the apparent contact radius including the cohesive region, $R$ is the radius of the sphere, and $λ$ is the Maugis parameter that derives from normalization process. Here, ${\bar{F}}_{M - D} = \frac{F_{M - D}}{π γ R}$ is the normalized reaction force, ${\bar{h}}_{M - D} = h / {(\frac{π^{2} γ^{2} R}{K^{2}})}^{\frac{1}{3}}$ is the normalized indentation depth, ${\bar{a}}_{M - D} = {a_{M - D} / (\frac{π γ R^{2}}{K})}^{\frac{1}{3}}$ is the normalized contact radius, and $λ = 2 σ_{0} / {(\frac{π γ K^{2}}{R})}^{\frac{1}{3}}$ is the dimensionless Maugis parameter [164]. Combining Eqs. (42)–(44), the force–displacement relation can be calculated. It is worth noting that the Maugis–Dugdale model also predicts that the pull-off force is independent of indentation depth, the same as JKR model. However, different from JKR model that calculates the adhesion energy $γ$ of the material directly from the pull-off force value, the Maugis–Dugdale model requires fitting the whole force–displacement curve to determine the shear modulus $G$ ⁠, the cohesive strength $σ_{0}$ ⁠, the adhesion energy $γ$ ⁠, and the separation distance $l_{0} = γ / σ_{0}$ ⁠.

It has been shown that the JKR and DMT models are two extreme cases of the Maugis' solution. When $λ \to \infty$ ⁠, the Maugis' solution is close to the JKR approximation, and when $λ \to 0$ ⁠, it becomes DMT solution [164,166]. Similar dimensionless parameters have been proposed for cohesive behavior following Lennard–Jones potential [167]. In fact, it has been shown that the shape of the traction–separation curve of the cohesive zone does not significantly influence the overall force–displacement relation of the indentation problem if the cohesive strength $σ_{0}$ and the separation distance $l_{0}$ of the different cohesive relations are comparable [165]. Conversely, $λ$ can be used to define the applicable range of the JKR and DMT models. It has been shown that if $λ > 5$ ⁠, the JKR solution can be considered as valid [164–166].

4.2 Adhesion Hysteresis of Hydrogels.

For the indentation of hydrogels carried out in underwater conditions, the adhesion between the indenter and the hydrogel is often negligible unless specific interactions are designed. However, in a series of experiments conducted by Lai et al., it is shown that during the loading process, the force–displacement curve follows the Hertzian solution without an adhesion effect, but after the indenter is held in place for some time and then retracted, significant pull-off force is observed and the longer the holding time is, the bigger the pull-off forces [7]. Another new observation is that the pull-off force is length-dependent—the pull-off force increases first as the contact radius increases and then reaches a plateau. However, neither the adhesion hysteresis nor the length-dependent adhesion can be explained by the traditional theories discussed above including JKR, DMT, and Maugis–Dugdale models. Motivated by this difficulty, Lai et al. developed a theory [7]. The physical picture is illustrated in Fig. 7. During the loading period, no adhesion is formed, but during the holding period, cohesive bonds are formed within the region where contact has been made. As a result, the loading period follows Hertz contact prediction, and during the unloading, the force curves experience a transition from Hertz contact to Maugis–Dugdale adhesion contact (Fig. 7(c)). The reaction force during the transition period can be calculated by replacing the apparent contact radius $c$ with the initial contact radius $c_{0}$ into Eqs. (42) and (43). Where this transition curve intersects with the Maugis–Dugdale curve depends on the initial contact radius size. If the initial contact radius is big, the intersection point is before reaching the Maugis-Dugdale pull-off force, and the overall pull-off force value is always the Maugis-Dugdale predicted values. On the other hand, if the initial contact radius is small, the intersection point passes the Maugis–Dugdale pull-off force, and in this case, the pull-off force is smaller for a smaller initial contact radius [7]. The results are shown in Fig. 7(c). In general, the plateau value of the pull-off force is determined by the adhesion energy $γ$ ⁠, and the value of $λ$ influences the slope of the normalized pull-off forces (Fig. 7(d)). By fitting the pull-off force as a function of the initial contact radius, $γ$ and $λ$ can be obtained, and the cohesive strength $σ_{0}$ and the separation distance $l_{0}$ can also be calculated. With the ability to accurately and uniquely determine the adhesion properties, the adhesion hysteresis can be utilized to study the relation between the adhesion and structural properties of the material. Lai and Hu extracted the adhesion properties of the polystyrene and polyacrylamide hydrogel interface for various compositions of hydrogels [170]. The adhesion energy of the interface is in the range of $0.5 - 2.5 mJ / m^{2}$ ⁠, and the cohesive strength is in the range of $0.25 - 1.5 kPa$ [170]. It is found that both properties are positively correlated to the polymer concentration and the surface chain density of the hydrogels [170].

4.3 Surface Tension Effect on Indentation Adhesion Measurement.

For indentation carried out on extremely soft materials in air, the surface tension, also known as wetting or capillary force, could be significant enough to influence the strain field near the indenter [172]. As shown in Fig. 8, when the material is deformed by the indenter, the surface of the material is also stretched, and a surface tension $σ$ is generated along the tangential direction of the surface to resist the surface stretching. This additional traction contributes to the deformation of material, and therefore the mechanical responses can be altered from the prediction of traditional models [172,173]. Style et al. reported that for a spherical rigid ball resting on a silicone gel surface, when the size of the ball is smaller than the capillary length scale, the contact radius, and the indentation depth deviate from traditional JKR or Maugis predictions [172]. Later, models were proposed accounting for the transition from adhesion-dominated limit to surface tension-dominated limit [173–178]. It has been shown that the different regimes can be described by a dimensionless parameter $σ {(G R)}^{- 2 / 3} γ^{- 1 / 3}$ ⁠, where $σ$ is the surface tension, $G$ is the shear modulus of the material, $R$ is the radius of spherical indenter, and $γ$ is the adhesion energy [174,175]. For $σ {(G R)}^{- 2 / 3} γ^{- 1 / 3} ≪ 1$ ⁠, meaning that the surface tension is low, the material is relatively stiff with high adhesion energy, and the size of the indenter is large, the indentation is dominantly governed by adhesion [174,175]. On the other hand, if $σ {(G R)}^{- 2 / 3} γ^{- 1 / 3} ≫ 1$ ⁠, the indentation is dominantly governed by surface tension [174,175]. For example, for the indentation adhesion test for hydrogels in an underwater environment, the surface tension is significantly reduced, and the force–displacement behavior can be described by the models in Sec. 4.2 [7].

Fig. 8

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Indentation measurement of soft hydrated material with surface tension $σ$ contribution. Surface tension $σ$ is along the tangential direction of the surface.

Based on this physical picture, Hui et al. proposed the theoretical model for indentation adhesion problem with surface tension effect considered. For a rigid spherical indenter indenting on an incompressible linear elastic half-space material with shear modulus

G

⁠, adhesion energy

γ

⁠, and surface tension

σ

[175], the relation between the reaction force

F

and the indentation depth

h

can be expressed implicitly in terms of the contact radius

a

in a normalized form as [175]

{\bar{F}}_{σ} = {\bar{F}}_{H} (β) {({\bar{a}}_{σ})}^{3} - 2 \sqrt{\frac{{({\bar{a}}_{σ})}^{3}}{Λ (β)}}

(45)

{\bar{h}}_{σ} = 3 {\bar{h}}_{H} (β) {({\bar{a}}_{σ})}^{2} - 4 ϕ (β) \sqrt{\frac{{\bar{a}}_{σ}}{Λ (β)}}

(46)

where

{\bar{F}}_{H} (β)

⁠,

Λ (β), {\bar{h}}_{H} (β)

⁠, and

ϕ (β)

are dimensionless scalar functions of

β = σ / (2 G a)

⁠. The normalized parameters are defined as,

{\bar{F}}_{σ} = \frac{F}{3 π γ R / 2}, {\bar{h}}_{σ} = \frac{h}{\frac{1}{8} {(\frac{3 π^{2} γ^{2} R}{2 G^{2}})}^{1 / 3}},

and

{\bar{a}}_{σ} = \frac{a}{{\frac{1}{2} (\frac{9 π γ R^{2}}{4 G})}^{1 / 3}},

where

R

is the radius of the indenter. The dimensionless scalar functions

{\bar{F}}_{H} (β)

⁠,

Λ (β), {\bar{h}}_{H} (β)

⁠, and

ϕ (β)

can be calculated numerically. Hui et al. proposed the empirical expression of the functions, as follows [175]:

{\bar{F}}_{H} (β) = 1 + \frac{3 π β}{4} [\frac{β^{2} + 0.6016 β + 0.0171}{β^{2} + 0.3705 β + 0.0063}]

(47)

{\bar{h}}_{H} (β) = 1 + \frac{3}{4} \ln (1 + 6.582 β + 2.759 β^{2} + 0.3782 β^{3})

(48)

ϕ (β) = \frac{0.6425 β^{2} + 4.284 β + 1}{0.2039 β^{3} + 3.994 β^{2} + 6.011 β + 1}

(49)

Λ (β) = \frac{2 β^{2} + 7.4 β + 1}{π β^{3} + 15.78 β^{2} + 10.92 β + 1}

(50)

Combining Eqs. (45)–(50), the force–displacement relation can be determined. Here, for $β ≪ 1,$ meaning that the surface tension is neglectable, the solution becomes the JKR prediction [175]. The above solution can also be modified for compressible materials by simply changing the shear modulus $G$ to $G / [2 (1 - ν)]$ [175].

5 Indentation Penetration for Characterizing the Fracture Properties

The study of how soft materials are penetrated is significant across many fields, including those in biomedical applications such as invasive procedures and biomedical devices [179,180], as well as in the natural world among different biological systems [181,182]. Specifically, in the biomedical field, technologies like the design of microneedle patches [183] and the steering of needles [184,185] are practical applications where a sharp object or indenter breaks through tissues. For instance, the effectiveness of microneedle patches in delivering drugs is influenced by the needles' geometry, spacing, and material stiffness [183]. Hence, understanding the penetration force–displacement relation is essential for improving the performance of these technologies. Further, the penetration phenomenon is observed in nature, such as when insects/animals penetrate skin layers with their fangs, where the efficiency of the bite is affected by the shape of the fang, the speed of the bite, and the stiffness of the materials involved [186].

Material characterization is another important motivation for studying the penetration phenomenon [187–190]. Indentation–penetration tests are used for investigating fracture-related properties in soft hydrated materials. A penetration test usually entails deep indentation until the indenter ruptures the sample, forming a crack. Subsequently, upon further indenting, the crack propagates through the sample along the direction of indentation. Therefore, an indentation–penetration test can be divided into two basic regimes: the pre–rupture or deep indentation regime followed by a rupture point or critical point, and the post–rupture or crack propagation regime. Figure 9 shows a schematic indentation-penetration force–displacement response depicting these phases. At any indentation–penetration depth, the sample may be in either the indentation or the penetration regime. The transition from indentation to penetration state involves a tradeoff primarily between elastic energy and surface energy among other factors. For depths below the critical depth, the indentation phase is more energetically favorable, while for depths beyond the critical depth, the penetration phase is more energetically favorable. At the critical depth, the force–displacement curve often shows a kink and sudden drop in force (Fig. 9). The specific response of a sample depends on factors such as indenter geometry, indentation speed, and fracture-related length scales.

Fig. 9

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Schematic indentation penetration force–displacement response. Region A is the prerupture regime, region B represents the transition to postrupture regime, and region C is the postrupture regime.

The fracture mechanics of soft materials is a topic of current research, and the exact universal form of a material's intrinsic fracture property is yet to be established. Consequently, the estimation of such properties from an indentation–penetration test can be complicated. The estimated property can demonstrate a dependence on certain length scales and loading conditions. For a comprehensive discussion on the fracture of soft materials, refer to Long and Hui [191] and Long et al. [192]. Nevertheless, the fracture process in a penetration test can be moderately described by two material properties, the fracture nucleation energy, $Γ_{0}$ ⁠, and the fracture propagation energy also known as the critical energy release rate, $J_{IC}$ and $J_{IIC}$ ⁠, for mode- $I$ and mode- $II$ crack, respectively. The fracture nucleation energy represents the material's resistance against crack formation, while the fracture propagation energy denotes the resistance to crack propagation. Note that the terminologies, fracture energy and energy release rate, can be mistaken as the dimensions of both $Γ_{0}$ and $J_{C}$ are energy per area (⁠ $J / m^{2}$ ⁠). In this section, we review the theoretical formulations and applications of the penetration method for experimentally estimating $J_{C}$ and $Γ_{0}$ ⁠. Additionally, we also review the penetration-related applications in the literature, where penetration formulations are applied for predicting and analyzing the penetration force.

5.1 Indentation–Penetration Theory.

In an indentation–penetration test setup, the indenter or punch commonly takes a cylindrical shape with various tip shapes such as flat-bottom, sharp-tip (conical, prismatic, double-edged, beveled, lancet), sphere-ended, and hemispherical. Analytical expressions are available for some simple punch geometries and penetration conditions. For more complex cases, numerical simulations are necessary. In this section, the half-space assumption is made, where the sample size is much larger than the relevant fracture length scales, indenter size, and indentation depth, ensuring no boundary effect in the penetration test.

In the prerupture regime, the force–displacement response is typically elastic but is out of the bounds of the Hertzian solution limit due to the large indentation depths. Let

F

denote the force on the indenter,

h

the indentation depth,

R

the indenter radius. For flat-bottomed or sharp-tipped punch

R

is the cylinder radius, while for sphere-ended punch

R

is the spherical tip radius. Indentation force at large depth is given by the formula proposed by Fakhouri et al. [193]

F = k^{'} E h^{2} + \frac{8}{3} E R h

(51)

where

E

is the Young's modulus and

k^{'}

is a fitting parameter that describes the deep indentation response. For incompressible material,

E = 3 G

⁠. At large depths

h > R

⁠, the quadratic term dominates, and hence the response is insensitive to the indenter geometry. Additionally, a general power-law-series extension is provided by Fregonese and Bacca for a hemispherical tip geometry, where the reaction force can be expressed as a summation of power law terms [194]

\frac{F}{G R^{2}} = \sum_{j = 1}^{\infty} B_{j} {(\frac{h}{R})}^{β_{j}}

(52)

where $B_{j}$ and $β_{j}$ as the power-law-series fitting parameters.

At the point of rupture F

= F_{c}

⁠,

h = h_{c} ≫ R

⁠. The fracture nucleation energy

Γ_{0}

can be estimated from the critical force

F_{C}

⁠, using cohesive zone theory as [25,193]

Γ_{0} \approx \frac{F_{c}}{R}

(53)

It's noted that the critical energy release rate (⁠ $J_{C}$ ⁠) measured by other fracture experiments is typically smaller than the fracture nucleation energy, attributed to the higher energy required for the crack nucleation compared to the propagation of the crack in soft materials.

Once the critical point is reached a crack is formed, and further indentation leads to the post–rupture crack propagation regime. Crack propagation typically requires external penetration work (⁠

W_{external}

⁠) to balance the energy losses in the form of material compression (⁠

W_{stiffness}

⁠), fracture surface formation (⁠

W_{fracture}

⁠) and friction/adhesion related dissipation (⁠

W_{friction}

⁠) [195–197]. This leads to the following component-wise distribution of the penetration force:

δ W_{external} = δ W_{stiffness} + δ W_{fracture} + δ W_{friction}

(54)

F_{external} = F_{stiffness} + F_{fracture} + F_{friction}

(55)

This energy balance equation is from integrating the force–displacement relation. Closed-form force–displacement expressions only exist for simple cases. For general cases, numerical simulations are conducted using cohesive elements with approximate material parameters tuned to match the experimental force–displacement response, which is then used to estimate the fracture energy [184,198,199]. A few closed-form force–displacement relations are discussed here.

The mode of crack propagation depends on the indenter shape. A mode-

II

ring crack is formed from a flat-bottomed punch [200], while a mode-

I

planar crack is formed from a sharp-tipped punch, including a hemispherical punch [194,200]. The hyperelastic response is commonly described using an incompressible single-term Ogden strain energy function

ϕ

[201]

ϕ = \frac{2 G}{α} (λ_{1}^{α} + λ_{2}^{α} + λ_{3}^{α} - 3)

(56)

where

G

is the small strain shear modulus,

α

is the strain-hardening exponent and

λ_{i}

(⁠

i = 1, 2, 3

⁠) denotes the principal stretches. For the penetration of an Ogden hyperelastic material using a flat-bottomed punch of radius

R

without friction, the relation between penetration force (⁠

F_{F}

⁠) and mode-

II

crack propagation toughness (⁠

J_{IIC})

is given by (Fig. 10(a))

\frac{F_{F}}{π G R^{2}} = {[1 - {(\frac{b}{R})}^{2}]}^{- 1} [\frac{2 b}{R} \frac{J_{IIC}}{G R} + f_{1} (\frac{b}{R})]

(57)

f_{1} (\frac{b}{R}) = \frac{2}{α^{2}} {[2 {(\frac{b}{R})}^{2 - α} + {(\frac{b}{R})}^{2 α + 2} - 3 {(\frac{b}{R})}^{2}] + \int_{1}^{\infty} f_{2} (η, \frac{b}{R}) d η}

(58)

f_{2} (η, \frac{b}{R}) = {(1 - \frac{1 - {(b / R)}^{2}}{η})}^{α / 2} + {(1 - \frac{1 - {(b / R)}^{2}}{η})}^{- α / 2} - 2

(59)

where $α$ is the Ogden strain-hardening exponent given in Eq. (56), $b$ is the radius of the material column below the flat-punch in the undeformed configuration (Fig. 10(b)), $f_{1} (b / R)$ and $f_{2} (η, b / R)$ are dimensionless functions. For estimating $J_{IIC}$ ⁠, the undeformed ring crack radius $b$ needs to be measured experimentally.

Fig. 10

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Indentation penetration schematic with mode- $II$ crack propagation. (a) Flat-punch penetration in postrupture regime. The mode- $II$ ring crack propagates ahead of the indentation depth. The material column below the punch is deformed to a radius $R$ ⁠. (b) After the flat-punch is removed, the column shrinks back to its undeformed radius $b$ ⁠. (Reproduced with permission from [200]. Copyright The Royal Society 2004).

Similarly, for a sharp-tipped punch of radius

R

and crack propagation without friction, the relation between penetration force (⁠

F_{S}

⁠) and mode-

I

crack propagation toughness (⁠

J_{IC})

is given by (Fig. 11)

\frac{F_{S}}{π G R^{2}} = \frac{2}{π} \frac{l}{R} \frac{J_{IC}}{G R} + \frac{1}{π} g (\frac{l}{R})

(60)

where $l$ is the half-length of the crack (Fig. 11(c)) and $g (l / R)$ is the normalized strain energy function, $g (l / R) = W_{stiffness} / (G R^{2} h)$ ⁠, $W_{stiffness}$ is the total strain energy stored in the sample. To calculate $J_{IC}$ ⁠, the crack half-length $l$ needs to be experimentally measured. Additionally, since the dimensionless function $g (l / R)$ cannot be expressed analytically, numerical method must be applied to calculate the total strain energy $W_{stiffness}$ for a known $l / R$ [194,200].

Fig. 11

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Indentation penetration schematic with mode- $I$ crack propagation. (a) Represents the prerupture regime, (b)represents the postrupture regime, and (c) shows the width of the planar mode- $I$ crack. (Reproduced with permission from [194]. Copyright Elsevier Ltd. 2021).

It is worth noting that for frictionless penetration in both above-discussed indenter geometries, the penetration force is independent of the indentation depth. Whereas for penetration with friction, the indentation force typically increases linearly with indentation depth. For mode-

I

crack propagation with friction, Fregonese and Bacca proposed the following extension to Eq. to describe the effect of interfacial friction and adhesion [202]:

F = 2 J_{IC} l + G R^{2} g (l / R) + 2 π R h 〈τ_{c}〉

(61)

where $〈τ_{c}〉$ is the average contact shear stress. This formulation results in a linear force–displacement response in the post–rupture regime as shown in Fig. 9. The slope of this response can be used to fit $〈τ_{c}〉$ according to Eq. (61). The study of frictional force response on the indenter in the postrupture regime is of great importance for accurate needle steering [184]. Various friction models have been provided, such as the Karnopp friction model [203], the Stribeck effect [204], and the Dahl model [205].

Alternatively, to directly measure

J_{IC}

⁠, Azar and Hayward performed a second penetration test at the same spot after a sufficient waiting time for recovery [196]. It is assumed that during the second indentation test, all energy interactions are nearly identical except for the fracture surface formation energy. The difference between the two forces can be related to

J_{IC}

as follows:

(F - F^{'}) d h = J_{IC} 2 l d h

(62)

J_{IC} = \frac{1}{2 l (h - h_{c})} \int_{h_{c}}^{h} (F - F^{'}) d h

(63)

where $F^{'}$ is the force response during the second test, $h_{c}$ is the critical depth, $h > h_{c}$ is the final depth such that the half-space assumption is still valid and $2 l$ is the crack length. This method of repeated penetration tests at the same location has been employed to study fracture properties [26,196,206] and friction–adhesion response [207].

5.2 Application of Indentation–Penetration on Soft Hydrated Materials.

Indentation–penetration tests have been performed on various biological tissues and soft hydrated materials. Using the penetration force–displacement relations discussed previously, the fracture initiation and propagation energies can be measured, and these results are summarized in Table 7. Typically, a mode- $I$ crack is expected for sharp indenters and a mode- $II$ crack is expected for flat punches. However, for soft materials, mode- $I$ is more prevalent as the fracture energy for mode- $II$ is larger [194], and hence mode- $I$ fracture propagation energy is more widely measured compared to mode- $II$ ⁠, as listed in Table 7. It is also reported that the fracture energies depend on various factors, such as indenter radius [26], indentation rate [196,209,210], and load cell compliance [211]. For example, a linear reduction in the measured fracture propagation toughness was seen with increasing indenter radius for the bovine liver sample [26]. Additionally, the critical penetration force and depth decrease with an increase in indentation velocity due to the time-dependent material behavior [210]. In addition to measuring the fracture energies, determining the strain field and rupture near the indenter presents a challenge in penetration tests for soft materials. Techniques like digital image correlation [212,213], autofluorescence with confocal imaging [208], birefringence [214], and mechanoluminiscence [215] have been utilized to address this issue.

Table 7

Fracture properties of soft-hydrated materials measured by penetration method

Material	References	Value	Punch geometry
Bovine liver	Gokgol et al. [26]	$J_{1 C} = 164 \pm 6 J / m^{2}$	Sharp (conical)
Porcine liver	Azar and Hayward [196]	$J_{1 C} = 95 J / m^{2}$	Bevel tip and Franseen tip
Porcine tissue mimic gel	Misra et al. [185]	$J_{1 C} = 114 kJ / m^{2}$	Flexible needle with bevel tip
Chicken breast tissue	Misra et al. [185]	$J_{1 C} = 24.2 kJ / m^{2}$	Flexible needle with bevel tip
Gelatin	Oldfield et al. [198]	$J_{1 C} = 17.43 J / m^{2}$	Sharp (conical, prismatic, double-edged)
Scleral tissue	Park et al. [206]	$J_{1 C} = 570 \pm 40 J / m^{2}$	Hemispherical (Sphere-ended (⁠ $R = 9$ – $50 μ m$ ⁠)
RLP-PEG hydrogel	Lau et al. [208]	RLP rich: $Γ_{0} = 730 \pm 50 J / m^{2}$ PEG rich: $Γ_{0} = 2844 \pm 140 J / m^{2}$	Sphere-ended (⁠ $R = 0.8$ – $35 μ m$ ⁠)
PAAm hydrogel	Fakhouri et al. [193]	$Γ_{0} = 142 \pm 40 J / m^{2}$	Flat punch, sphere-ended

Material	References	Value	Punch geometry
Bovine liver	Gokgol et al. [26]	$J_{1 C} = 164 \pm 6 J / m^{2}$	Sharp (conical)
Porcine liver	Azar and Hayward [196]	$J_{1 C} = 95 J / m^{2}$	Bevel tip and Franseen tip
Porcine tissue mimic gel	Misra et al. [185]	$J_{1 C} = 114 kJ / m^{2}$	Flexible needle with bevel tip
Chicken breast tissue	Misra et al. [185]	$J_{1 C} = 24.2 kJ / m^{2}$	Flexible needle with bevel tip
Gelatin	Oldfield et al. [198]	$J_{1 C} = 17.43 J / m^{2}$	Sharp (conical, prismatic, double-edged)
Scleral tissue	Park et al. [206]	$J_{1 C} = 570 \pm 40 J / m^{2}$	Hemispherical (Sphere-ended (⁠ $R = 9$ – $50 μ m$ ⁠)
RLP-PEG hydrogel	Lau et al. [208]	RLP rich: $Γ_{0} = 730 \pm 50 J / m^{2}$ PEG rich: $Γ_{0} = 2844 \pm 140 J / m^{2}$	Sphere-ended (⁠ $R = 0.8$ – $35 μ m$ ⁠)
PAAm hydrogel	Fakhouri et al. [193]	$Γ_{0} = 142 \pm 40 J / m^{2}$	Flat punch, sphere-ended

Indentation–penetration tests are crucial in various applications. For example, indentation penetration tests have been an important tool for medical diagnostics. Yu et al. designed a probe for needle-based biopsy procedures for in vivo tumor diagnosis, in which penetration force response of cancerous lesions can be used for rapid characterization of tissues [216]. Bao et al. performed penetration tests on tracheal tissues and determined minimal invasive penetration angle and punch radius to reduce diagnostic and surgical risks during puncture examination of tracheal tissue [217]. Indentation–penetration tests also reveal the relation between the structural properties and mechanical behaviors of soft hydrogels and tissues. Jiang et al. studied the penetration of porcine liver tissues. The penetration force–displacement response indicates the tissue type and the existence of vital vessels in the path of insertion [218]. Matthews et al. performed indentation penetration tests on cornea and sclera tissues for the purpose of surgery and drug delivery of eyes [219]. Greater critical penetration force was reported at the midline than at the central cornea [219]. Lau et al. performed penetration tests on RLP–PEG (resilin-like polypeptide poly(ethylene glycol)) hydrogels and showed that PEG-rich hydrogels have higher fracture initiation energy than the RLP-rich hydrogels [208].

6 Summary and Outlook

In this review, we discussed the methodologies and applications of the indentation method on soft hydrated materials. Indentation measurement has been an effective tool for measuring soft hydrated materials in various aspects of mechanical behaviors. The elasticity of the material can be determined by analyzing the force–displacement response of the indenter. Analytical expressions have been given for various indenter types, and the solutions have been extended for special geometries and large deformation situations. For time-dependent elasticity, including viscoelasticity and poroelasticity, either time domain measurement or frequency domain measurement is required. Notably, indentation responses for viscoelastic materials and poroelastic materials exhibit distinct characteristics due to disparate dissipation mechanisms. While time-dependent and frequency-dependent indentation responses for viscoelastic materials are length-independent, those for poroelastic materials depend on the size of the contact. As a result, for poroelastic materials, relaxation indentation test is usually preferred over creep test, as the size of contact maintains constant during the relaxation process. Analytical solutions have been presented for indentation relaxation following fast loading and indentation oscillation test to determine the intrinsic material properties for pure viscoelastic materials and pure poroelastic materials. For coupled viscoporoelastic materials, the accurate force response needs to be calculated via numerical methods. To ensure the uniqueness of the obtained material parameters, additional measurement is usually required to independently determine certain intrinsic parameters. The applications of the indentation method to obtain the elastic and time-dependent elastic properties of various soft hydrated materials are summarized in this review. Through these systematic studies, the structure–property relations of the soft hydrated materials can be explored. Particularly in the context of cells and biological tissues, the elastic and time-dependent elastic properties hold significant implications for understanding the physiological activity and discerning pathological conditions.

In the case where the adhesion between the indenter and the material is not negligible, the anticipated force–displacement relations during indentation may deviate from the theoretical predictions for nonadhesive counterparts. For soft hydrated materials, traditional methods, such as JKR and Maugius models, are usually applied for analyzing the force–response and obtaining the adhesion properties. These models, however, may not suffice in certain situations. For instance, for hydrogels with nonspecific bonds, adhesion hysteresis can be pronounced, and the pull-off force may exhibit length-dependent behavior, necessitating consideration of the transition from Hertz contact to adhesive contact. Moreover, for soft hydrated materials, the surface tension can exert a significant influence on indentation force response, particularly when the capillary length scale is comparable to the size of contact. The review provides a comprehensive overview of extending adhesion models to incorporate surface tension effects, thus enriching our understanding of adhesive behavior in soft hydrated materials.

The application of indentation methods extends to the measurement of fracture properties in soft hydrated materials. Analysis of the force–displacement response prior to rupture enables the determination of fracture nucleation energy. Subsequently, as crack propagation ensues, fracture toughness is ascertained through the relationship between reaction force and crack length. Theoretical models elucidating the utilization of indentation penetration methods for measuring fracture properties in soft hydrated materials are comprehensively outlined in this review. Furthermore, the review delves into the phenomenon of indentation penetration across various applications. Notably, it highlights that factors such as the indenter's geometry, tip spacing, and loading rate can significantly alter the reaction force observed during the indentation penetration process.

Despite the significant advancements in indentation methods over recent decades for probing various mechanical behaviors across diverse mechanical systems, several challenges persist, necessitating further research endeavors. Foremost among these challenges is the refinement of theoretical models tailored for indenting anisotropic materials, a common occurrence in soft hydrated materials. Examples abound in biological tissues, such as cornea [220], cartilage [221,222], brain tissue [223,224], and blood vessels [225], which often exhibit transversely isotropic behavior owing to the presence of embedded fibers. Moreover, even ostensibly isotropic materials can demonstrate anisotropic responses under the influence of internal stresses leading to substantial deformations, as observed in constraint swollen gels [226] and cells adhering to substrates [227]. In such scenarios, conventional isotropic indentation predictions fall short of capturing the intrinsic anisotropic mechanical properties of the material. To address this challenge, various attempts have been made to devise methodologies for characterizing anisotropic materials using indentation techniques. For instance, Nia et al. utilized indentation oscillation tests on cartilage to explore its poroelastic behavior, revealing the significant influence of embedded fibers on frequency-dependent phase lag [140]. Namani et al. proposed a combined dynamic shear and asymmetric indentation approach to measure transversely isotropic fibrin gels, wherein mechanical information along different directions is obtained, albeit necessitating numerical simulations for force response calculation in a three-dimensional context [228]. Yue et al. analytically solved the indentation problem of constrained swollen gels, enabling the calculation of reaction force and contact region size for a given constrained swollen state [226]. Moghaddam et al. advocated for using indentation methods to determine the anisotropic properties of biological tissues, relying on both reaction force and contact region size measurements, alongside numerical simulations for property determination [229]. In general, characterizing anisotropic tissues via indentation necessitates probing responses along different directions, with comprehensive data interpretation often mandating three-dimensional numerical simulations. However, the computational complexity associated with such simulations poses a barrier to the widespread application of indentation methods on soft hydrated anisotropic materials in practical settings. Therefore, there is a pressing need for the development of simplified, straightforward methodologies to facilitate the characterization of soft hydrated anisotropic materials in future studies.

Another significant challenge in real-world applications involving soft hydrated materials is the coupling of various mechanical behaviors with large deformations, whereas existing analytical methods for indentation tests typically concentrate on the small deformation regime. Despite efforts to address this challenge, such as exploring elasticity under large deformation [58] and analyzing large deformation during the prerupture regime [193], tackling nonlinear time-dependent responses under large deformation remains daunting and often necessitates resorting to numerical methods. For instance, Meloni et al. investigated the poroelastic response of cartilage under large deformation, modeling the cartilage matrix as a neo-Hookean material and fitting the relaxation behavior using biphasic finite element simulations [230]. Similarly, Greiner et al. conducted a range of mechanical tests to elucidate the nonlinear viscoporoelasticity of brain tissues, conducting parameter studies to derive nonlinear viscous and porous material parameters [145]. Basilio et al. employed large deformation indentation tests in conjunction with inverse finite element simulations to measure the nonlinear viscoelastic properties of brain tissue across different regions [231]. However, in such endeavors, the sheer number of material parameters identified from finite element simulations often proves extensive, and validating the uniqueness of these parameters poses a significant challenge. Hence, there is a pressing need for more robust theoretical models capable of describing the indentation of nonlinear time-dependent materials, particularly for measuring soft hydrated materials under large deformation. Future studies are anticipated to focus on refining and developing theoretical frameworks to better capture the complexities inherent in such materials and their behavior under substantial deformations.

Summarily, indentation techniques have proven to be highly effective and broadly utilized in determining the mechanical properties of soft hydrated materials. These methods, supported by diverse theoretical models, have shed light on various aspects of mechanical behaviors across different material systems. Despite their widespread application, challenges remain in refining the indentation methodology specifically for soft hydrated materials. Future research is anticipated to enhance the precision and applicability of indentation techniques for these materials. Notably, while indentation methods have been developed to assess a range of mechanical behaviors, for soft hydrated materials, the primary analytical framework employed for interpreting results from soft hydrated materials often remains the basic elastic contact model. This review aims to serve as a practical guide for employing indentation techniques to ascertain mechanical properties of soft hydrated materials, advocating for the exploration of more complex mechanical behaviors across various material systems. Through such comprehensive investigations, a deep understanding of structural and biological properties inherent to soft hydrated material systems can be achieved.

Funding Data

National Science Foundation (NSF) (Grant No. 2019783; Funder ID: 10.13039/501100001809).

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