## 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.

## 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 $\nu $. The shear modulus $G$ represents the material's ability to resist shape change, while the Poisson's ratio $\nu $ 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 $\nu $ 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 $E=2G(1+\nu )$. For incompressible materials, $E=3G$. In this review, we use shear modulus $G$ and Poisson's ratio $\nu $ to represent the linear elastic property of the material for consistency, and one can easily calculate $E$ from the value of $G$ and $\nu $.

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 $G/(1\u2212\nu )$ for all the geometries while the dependency on the indentation depth $h$ is geometry specific. For cylindrical flat punch, it is $F\u221dh$; for spherical indenter, $F\u221dh1.5$; and for conical indenter, $F\u221dh2$. 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, $F\u221dh2$, 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.

Geometry | Reaction force, $F$ | Contact radius, $a$ |
---|---|---|

Spherical indenter, radius $R$ [35–37] | $F=8GhRh3(1\u2212\nu )$ | $a=Rh$ |

Conical indenter, half opening angle $\theta $ [35,38,39] | $F=4Gh2\pi (1\u2212\nu )tan\theta $ | $a=2\pi htan\theta $ |

Cylindrical flat punch, radius $R$ [35,40] | $F=4GhR1\u2212\nu $ | $a=R$ |

Plane-strain cylindrical indenter, radius $R$ [35,41] | $F=\pi Ga22R(1\u2212\nu )$ | $h=a24R[2ln(4Ra)\u22121]$ |

Geometry | Reaction force, $F$ | Contact radius, $a$ |
---|---|---|

Spherical indenter, radius $R$ [35–37] | $F=8GhRh3(1\u2212\nu )$ | $a=Rh$ |

Conical indenter, half opening angle $\theta $ [35,38,39] | $F=4Gh2\pi (1\u2212\nu )tan\theta $ | $a=2\pi htan\theta $ |

Cylindrical flat punch, radius $R$ [35,40] | $F=4GhR1\u2212\nu $ | $a=R$ |

Plane-strain cylindrical indenter, radius $R$ [35,41] | $F=\pi Ga22R(1\u2212\nu )$ | $h=a24R[2ln(4Ra)\u22121]$ |

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

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, $R*=R1R2/(R1+R2)$, where $R1$ is the radius of the indenter and $R2$ 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.

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.

By combining Eqs. (10) and (11), the force–displacement curve can be expressed in terms of the neo-Hookean parameter $\mu 0$, 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].

Material | References and comment | Shear modulus, $G$ | Indentation geometry |
---|---|---|---|

Cartilage | Hori and Mockros [11], human articular cartilage | 0.4–3.5 MPa | Sphere, $R=10\u2009mm,20\u2009mm,\u2009and\u200930.4\u2009mm$ |

Korhonen et al. [59], bovine humeral, patellar, and femoral cartilages | 0.16–0.42 MPa | Sphere, $R=0.5\u2009mm\u2009and\u20091.5\u2009mm$ | |

Park et al. [32], bovine humeral cartilage | 15.3±6.26 kPa | Sphere, $R=2.5\u2009\mu m$ | |

Darling et al. [60], articular cartilage | Pericellular matrix: 6.3–98 kPa; extracellular matrix: 22–275 kPa | Sphere, $R=2.5\u2009\mu m$ | |

Li et al. [61], murine meniscus | 3.0–4.6 MPa | Sphere, $R=5\u2009\mu m$ | |

Brain | Miller et al. [62], human brain, in vivo | 1.08 kPa | Cylindrical flat punch, $R=5\u2009mm$ |

Dommelen et al. [63], porcine brain | 0.67–1.2 kPa | Sphere, $R=1\u2009mm$ | |

Budday et al. [14], bovine brain | White matter: 0.631±0.197 kPa; gray matter: 0.456±0.096 kPa | Cylindrical flat punch, $R=0.375\u22120.75\u2009mm$ | |

Antonovaite et al. [64], mouse hippocampus | 0.5–0.8 kPa | Sphere, $R=60\u2212105\u2009\mu m$ | |

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, $R=1\u2009\mu m$ |

Last et al. [65], human cornea | Bowman's layer: 36.6±4.4 kPa; anterior stroma: 11.0±2.0 kPa | Sphere, $R=1\u2009\mu m$ | |

Last et al. [66], human Trabecular Meshwork (HTM) | Normal 0.56–2.93 kPa, glaucoma: 26–83 kPa | Sphere, $R=1\u2009\mu m$ | |

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, $R=2.25\u2009mm$ |

Shkumatov et al. [69], lung airway tissue | 0.7–16 kPa | Sphere, $R=2.5\u2009\mu m$ | |

Liver | Nava et al. [70], human liver in vivo | Normal: 0.09 MPa; cholestatic: 0.25 MPa | Sphere, $R=2.25\u2009mm$ |

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, $R=4.5\u2009\mu m$ |

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, $R=2.5\u2009\mu m$ | |

Li et al. [77], breast cancer cells | 0.07–0.47 kPa | Sphere, $R=2.25\u2009\mu m$ | |

Maloney et al. [78], human mesenchymal stem cells (hMSCs) | 0.6–3 kPa | Sphere, $R=25\u2009nm$ | |

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, $R=50\u2009\mu m$ |

Nowatzki et al. [50], extracellular matrix (aECM) protein thin film | 0.1–0.3 Mpa | Sphere, $R=300\u2009nm$ | |

Soofi et al. [83], matrigel | 440±250 Pa | Sphere, $R=0.5\u2009\mu m$ | |

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, $R=6\u2009\mu m$ | |

Dingle et al. [18], cortical cells self-assembled into three-dimensional spheroid | 25–100 Pa | Sphere, $R=2.5\u2009\mu m$ | |

Welsch et al. [86], fibrin gel | 460±260 Pa | Sphere, $R=1.75\u2009\mu m$ | |

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, $R=0.5\u2009\mu m$ | |

Henderson et al. [87], ionic cross-linking PMMA hydrogel | 1.5 kPa–7 MPa | Cylindrical flat punch, $R=220\u2009\mu m$ | |

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, $R=1\u2009mm$ |

Material | References and comment | Shear modulus, $G$ | Indentation geometry |
---|---|---|---|

Cartilage | Hori and Mockros [11], human articular cartilage | 0.4–3.5 MPa | Sphere, $R=10\u2009mm,20\u2009mm,\u2009and\u200930.4\u2009mm$ |

Korhonen et al. [59], bovine humeral, patellar, and femoral cartilages | 0.16–0.42 MPa | Sphere, $R=0.5\u2009mm\u2009and\u20091.5\u2009mm$ | |

Park et al. [32], bovine humeral cartilage | 15.3±6.26 kPa | Sphere, $R=2.5\u2009\mu m$ | |

Darling et al. [60], articular cartilage | Pericellular matrix: 6.3–98 kPa; extracellular matrix: 22–275 kPa | Sphere, $R=2.5\u2009\mu m$ | |

Li et al. [61], murine meniscus | 3.0–4.6 MPa | Sphere, $R=5\u2009\mu m$ | |

Brain | Miller et al. [62], human brain, in vivo | 1.08 kPa | Cylindrical flat punch, $R=5\u2009mm$ |

Dommelen et al. [63], porcine brain | 0.67–1.2 kPa | Sphere, $R=1\u2009mm$ | |

Budday et al. [14], bovine brain | White matter: 0.631±0.197 kPa; gray matter: 0.456±0.096 kPa | Cylindrical flat punch, $R=0.375\u22120.75\u2009mm$ | |

Antonovaite et al. [64], mouse hippocampus | 0.5–0.8 kPa | Sphere, $R=60\u2212105\u2009\mu m$ | |

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, $R=1\u2009\mu m$ |

Last et al. [65], human cornea | Bowman's layer: 36.6±4.4 kPa; anterior stroma: 11.0±2.0 kPa | Sphere, $R=1\u2009\mu m$ | |

Last et al. [66], human Trabecular Meshwork (HTM) | Normal 0.56–2.93 kPa, glaucoma: 26–83 kPa | Sphere, $R=1\u2009\mu m$ | |

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, $R=2.25\u2009mm$ |

Shkumatov et al. [69], lung airway tissue | 0.7–16 kPa | Sphere, $R=2.5\u2009\mu m$ | |

Liver | Nava et al. [70], human liver in vivo | Normal: 0.09 MPa; cholestatic: 0.25 MPa | Sphere, $R=2.25\u2009mm$ |

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, $R=4.5\u2009\mu m$ |

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, $R=2.5\u2009\mu m$ | |

Li et al. [77], breast cancer cells | 0.07–0.47 kPa | Sphere, $R=2.25\u2009\mu m$ | |

Maloney et al. [78], human mesenchymal stem cells (hMSCs) | 0.6–3 kPa | Sphere, $R=25\u2009nm$ | |

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, $R=50\u2009\mu m$ |

Nowatzki et al. [50], extracellular matrix (aECM) protein thin film | 0.1–0.3 Mpa | Sphere, $R=300\u2009nm$ | |

Soofi et al. [83], matrigel | 440±250 Pa | Sphere, $R=0.5\u2009\mu m$ | |

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, $R=6\u2009\mu m$ | |

Dingle et al. [18], cortical cells self-assembled into three-dimensional spheroid | 25–100 Pa | Sphere, $R=2.5\u2009\mu m$ | |

Welsch et al. [86], fibrin gel | 460±260 Pa | Sphere, $R=1.75\u2009\mu m$ | |

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, $R=0.5\u2009\mu m$ | |

Henderson et al. [87], ionic cross-linking PMMA hydrogel | 1.5 kPa–7 MPa | Cylindrical flat punch, $R=220\u2009\mu m$ | |

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, $R=1\u2009mm$ |

## 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 $F(\u221e)$ 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 $h(\u221e)$ (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 $\delta $ can be obtained, which can be used to determine the viscoelastic properties of the material.

#### 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.

Model | Relaxation modulus $G(t)$ | Storage modulus $G\u2032(\omega )$ | Loss modulus $G\u2033(\omega )$ |
---|---|---|---|

Maxwell model [103,104] | $G(t)=G0e\u2212t/\tau $, $\tau =\eta /G0$, $G\u221e=0$. | $G\u2032(\omega )=\eta 2\omega 2G0G02+\eta 2\omega 2$ | $G\u2033(\omega )=\eta \omega G02G02+\eta 2\omega 2$ |

Kelvin-Voigt model [103,104] | $G(t)=G\u221e+\eta \delta (t)$, $G0=+\u221e$, $\delta (t):$ Dirac delta function | $G\u2032(\omega )=G\u221e$ | $G\u2033(\omega )=\eta \omega $ |

Standard linear solid model [104] | $G(t)=G\u221e+G1e\u2212t/\tau 1$, $\tau 1=\eta 1/G1$, $G0=G\u221e+G1$ | $G\u2032(\omega )=G\u221e+\eta 12\omega 2G1G12+\eta 12\omega 2$ | $G\u2033(\omega )=\eta 1\omega G12G12+\eta 12\omega 2$ |

Generalized Maxwell model (Prony series) [104,105] | $G(t)=G\u221e+\u2211i=1NGie\u2212t/\tau i$, $\tau i=\eta i/Gi$, $G0=G\u221e+\u2211i=1NGi$ | $G\u2032(\omega )=G\u221e+\u2211i=1N\eta i2\omega 2GiGi2+\eta i2\omega 2$ | $G\u2033(\omega )=\u2211i=1N\eta i\omega Gi2Gi2+\eta i2\omega 2$ |

Model | Relaxation modulus $G(t)$ | Storage modulus $G\u2032(\omega )$ | Loss modulus $G\u2033(\omega )$ |
---|---|---|---|

Maxwell model [103,104] | $G(t)=G0e\u2212t/\tau $, $\tau =\eta /G0$, $G\u221e=0$. | $G\u2032(\omega )=\eta 2\omega 2G0G02+\eta 2\omega 2$ | $G\u2033(\omega )=\eta \omega G02G02+\eta 2\omega 2$ |

Kelvin-Voigt model [103,104] | $G(t)=G\u221e+\eta \delta (t)$, $G0=+\u221e$, $\delta (t):$ Dirac delta function | $G\u2032(\omega )=G\u221e$ | $G\u2033(\omega )=\eta \omega $ |

Standard linear solid model [104] | $G(t)=G\u221e+G1e\u2212t/\tau 1$, $\tau 1=\eta 1/G1$, $G0=G\u221e+G1$ | $G\u2032(\omega )=G\u221e+\eta 12\omega 2G1G12+\eta 12\omega 2$ | $G\u2033(\omega )=\eta 1\omega G12G12+\eta 12\omega 2$ |

Generalized Maxwell model (Prony series) [104,105] | $G(t)=G\u221e+\u2211i=1NGie\u2212t/\tau i$, $\tau i=\eta i/Gi$, $G0=G\u221e+\u2211i=1NGi$ | $G\u2032(\omega )=G\u221e+\u2211i=1N\eta i2\omega 2GiGi2+\eta i2\omega 2$ | $G\u2033(\omega )=\u2211i=1N\eta i\omega Gi2Gi2+\eta i2\omega 2$ |

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

where $G\u221e$ is the equilibrium shear modulus, $N$ is the number of dashpots in the generalized Maxwell segments, $Gi$ is the shear modulus of the spring connected to the $ith$ dashpot, and $\tau i$ is the $ith$ time scale with $\tau i=\eta i/Gi$, where $\eta i$ is the viscosity of the $ith$ dashpot. The instantaneous modulus can also be calculated, $G0=G\u221e+G1+G2+\cdots +GN$. 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.

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].

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.

where $\omega $ is the angular frequency of the cyclic loading and $j$ is the imaginary unit. Since $G*(\omega )$ is a complex function, it can be written in the form of $G*(\omega )=G\u2032(\omega )+jG\u2033(\omega )$, where $G\u2032(\omega )$ is the storage modulus and $G\u2033(\omega )$ 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 $G\u2032(\omega )$ is associated with the solid-like elastic behavior during the periodic deformation of the angular frequency $\omega $, while the loss modulus $G\u2033(\omega )$, on the other hand, describes the fluid-like viscous dissipation of the energy during the deformation [103]. The analytical expressions of $G\u2032(\omega )$ and $G\u2033(\omega )$ 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.

where $Fa$ is the amplitude of the oscillational force (Fig. 1(c)), $and\u2009R$ 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, $G\u221e/G0$. 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].

Material | References | Geometry | Rheological model | Initial shear modulus $G0$ | Ratio of relaxation $G\u221e/G0$ | Time scale/frequency |
---|---|---|---|---|---|---|

Porcine brain | Gefen et al. [109] | Sphere, $R=2\u2009mm$ | 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, $R=0.75\u2009mm$ | 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, $R=60\u2212105\u2009\mu m$ | — | 0.8±0.1 kPa | ∼0.63 | 1–10 Hz |

Porcine kidney | Mattice et al. [98] | Sphere, $R=2.78\u2009mm$ | Prony series, 3 time scales | 5–12 kPa | 0.21±0.03 | 0.04–40 s |

Meniscus extracellular matrix | Li et al. [113] | Sphere, $R=5\u2009\mu m$ | 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, $R=1\u22123\u2009mm$ | 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, $R=3\u2009mm$ | 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, $R=1.58\u2009mm$ | 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, $R=1\u2009mm$ | Prony series, 2 time scales | 1.72–2.27 kPa | 0.15–0.30 | 1.5–60 s |

Fibroblast | Mahaffy et al. [97] | Sphere, $R=0.5\u22126\u2009\mu m$ | — | 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, $R=2.5\u2009\mu m$ | Standard linear viscoelastic model | 0.11–0.20 kPa | ∼0.57 | ∼2.1 s |

Chondrosarcoma cell | Darling et al. [118] | Sphere, $R=2.5\u2009\mu m$ | 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, $R=2.5\u2009\mu m$ | Standard linear viscoelastic model | 0.83–1.07 kPa | 0.14–0.7 | 7.3–10.1 s |

Osteoblast | Darling et al. [119] | Sphere, $R=2.5\u2009\mu m$ | Standard linear viscoelastic model | 0.87–2.2 kPa | 0.23–0.69 | 6.9–15.4 s |

Adipocyte | Darling et al. [119] | Sphere, $R=2.5\u2009\mu m$ | 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, $R=5\u2009\mu m$ | Standard linear viscoelastic model | 0.10–0.15 kPa | 0.72–0.86 | 1.1–1.4 s |

Fibroblasts | Sousa et al. [121] | Sphere, $R=3\u2009\mu m$ | Power law | ∼10 kPa | ∼0.5 | 3 ms–41 ms |

MDCK cell | Guan et al. [122] | Sphere, $R=3.5\u22129\u2009\mu m$ | Combined exponential and power law | 0.25–4.8 kPa | <0.02 | 2.61–6.80 ms |

HeLa cell | Guan et al. [122] | Sphere, $R=3.5\u22129\u2009\mu m$ | 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 $G0$ | Ratio of relaxation $G\u221e/G0$ | Time scale/frequency |
---|---|---|---|---|---|---|

Porcine brain | Gefen et al. [109] | Sphere, $R=2\u2009mm$ | 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, $R=0.75\u2009mm$ | 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, $R=60\u2212105\u2009\mu m$ | — | 0.8±0.1 kPa | ∼0.63 | 1–10 Hz |

Porcine kidney | Mattice et al. [98] | Sphere, $R=2.78\u2009mm$ | Prony series, 3 time scales | 5–12 kPa | 0.21±0.03 | 0.04–40 s |

Meniscus extracellular matrix | Li et al. [113] | Sphere, $R=5\u2009\mu m$ | 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, $R=1\u22123\u2009mm$ | 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, $R=3\u2009mm$ | 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, $R=1.58\u2009mm$ | 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, $R=1\u2009mm$ | Prony series, 2 time scales | 1.72–2.27 kPa | 0.15–0.30 | 1.5–60 s |

Fibroblast | Mahaffy et al. [97] | Sphere, $R=0.5\u22126\u2009\mu m$ | — | 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, $R=2.5\u2009\mu m$ | Standard linear viscoelastic model | 0.11–0.20 kPa | ∼0.57 | ∼2.1 s |

Chondrosarcoma cell | Darling et al. [118] | Sphere, $R=2.5\u2009\mu m$ | 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, $R=2.5\u2009\mu m$ | Standard linear viscoelastic model | 0.83–1.07 kPa | 0.14–0.7 | 7.3–10.1 s |

Osteoblast | Darling et al. [119] | Sphere, $R=2.5\u2009\mu m$ | Standard linear viscoelastic model | 0.87–2.2 kPa | 0.23–0.69 | 6.9–15.4 s |

Adipocyte | Darling et al. [119] | Sphere, $R=2.5\u2009\mu m$ | 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, $R=5\u2009\mu m$ | Standard linear viscoelastic model | 0.10–0.15 kPa | 0.72–0.86 | 1.1–1.4 s |

Fibroblasts | Sousa et al. [121] | Sphere, $R=3\u2009\mu m$ | Power law | ∼10 kPa | ∼0.5 | 3 ms–41 ms |

MDCK cell | Guan et al. [122] | Sphere, $R=3.5\u22129\u2009\mu m$ | Combined exponential and power law | 0.25–4.8 kPa | <0.02 | 2.61–6.80 ms |

HeLa cell | Guan et al. [122] | Sphere, $R=3.5\u22129\u2009\mu m$ | 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].

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 $\nu $. Moreover, the kinetics of the solvent migration is governed by a diffusion equation with the diffusivity $D$ (SI unit: $m2/s$) as a material parameter. Alternatively, the kinetics of solvent migration can also be characterized by the permeability $k$ (SI unit: $m2$), also known as hydraulic permeability or intrinsic permeability, which can be related to the poroelastic properties [5], $k=1\u22122\nu 2(1\u2212\nu )D\eta sG$, where $\eta s$ is the viscosity of the solvent. Another alternative measure of the solvent migration kinetic is the hydraulic conductivity $K$, defined as $K=k/\eta s$ (SI unit: $m4/(N\xb7s)$), 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.

Geometry . | Initial reaction force $F(0)$ and size of contact $a$ . | Relaxation function $g(\tau )$ . |
---|---|---|

Spherical indenter, radius $R$ | $F(0)=16GhRh3a=Rh$ | $g(\tau )=0.491exp(\u22120.908\tau )+0.509exp(\u22121.679\tau )$ |

Conical indenter, half opening angle $\theta $ | $F(0)=8Gh2\pi tan\theta a=2h\pi tan\theta $ | $g(\tau )=0.493exp(\u22120.822\tau )+0.507exp(\u22121.348\tau )$ |

Cylindrical flat punch, radius $R$ | $F(0)=8GhRa=R$ | $g(\tau )=1.304exp(\u2212\tau )\u22120.304exp(\u22120.254\tau )$ |

Plane-strain cylindrical indenter, radius $R$ | $F(0)=\pi Ga2Rh=a24R[2ln(4Ra)\u22121]$ | $g(\tau )=0.791exp(\u22120.213\tau )+0.209exp(\u22120.95\tau )$ |

Geometry . | Initial reaction force $F(0)$ and size of contact $a$ . | Relaxation function $g(\tau )$ . |
---|---|---|

Spherical indenter, radius $R$ | $F(0)=16GhRh3a=Rh$ | $g(\tau )=0.491exp(\u22120.908\tau )+0.509exp(\u22121.679\tau )$ |

Conical indenter, half opening angle $\theta $ | $F(0)=8Gh2\pi tan\theta a=2h\pi tan\theta $ | $g(\tau )=0.493exp(\u22120.822\tau )+0.507exp(\u22121.348\tau )$ |

Cylindrical flat punch, radius $R$ | $F(0)=8GhRa=R$ | $g(\tau )=1.304exp(\u2212\tau )\u22120.304exp(\u22120.254\tau )$ |

Plane-strain cylindrical indenter, radius $R$ | $F(0)=\pi Ga2Rh=a24R[2ln(4Ra)\u22121]$ | $g(\tau )=0.791exp(\u22120.213\tau )+0.209exp(\u22120.95\tau )$ |

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

Here, $\nu $ 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.

The functional form of $g(\tau )$ 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 $ah$, 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 $a2$, 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 $[F(t)\u2212F(\u221e)]/[F(0)\u2212F(\u221e)]$ versus $Dt/a2$ can fit with the theoretical curve very well.

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

#### 3.2.3 Poroelastic Oscillation Indentation.

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 $\delta c$ and the critical frequency $\omega c$ can be determined from the plot of phase lag $\delta $ versus normalized angular frequency $a2\omega $, as shown in Fig. 5.

Now, with a known Poisson's ratio, the shear modulus $G$ of the material can be calculated using the equilibrium reaction force $F(\u221e)$ 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 *μ*m 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 $\omega a2$, the three curves overlap (Fig. 5(c)). Taking the peak phase lag value $\delta c$ and the corresponding critical normalized angular frequency $a2\omega c$, 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 $F(\u221e)$ 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 $k=1\u22122\nu 2(1\u2212\nu )D\eta sG$, 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 $\xi $ can be approximated by the diameter of these cylindrical tubes, which scales with the square root of the permeability, $\xi \u223ck1/2\u2009$ [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].

Material | References | Geometry | Shear modulus, $G$ | Drained Poisson's ratio, $\nu $ | Diffusivity, $D$ | Permeability, $k$ |
---|---|---|---|---|---|---|

Osteoblast | Shin and Athanasiou [137] | Cylindrical flat punch, $R=2.5\u2009\mu m$ | 0.51±0.17 kPa | 0.37±0.03 | (1.58±0.87)×10^{−7} m^{2}/s | (1.18±0.65) × 10^{−13} m^{2} |

MDCK cell | Moeendarbary et al. [138] | Sphere, $R=5\u2009\mu m$ | 0.9±0.4 kPa | ∼0.3 | (6.1±1.0) ×10^{−11} m^{2}/s | (1.9±0.3) × 10^{−17} m^{2} |

HeLa cell | Moeendarbary et al. [138] | Sphere, $R=5\u2009\mu m$ | 0.4±0.1 kPa | ∼0.3 | (4.1±1.1) × 10^{−11} m^{2}/s | (2.9±0.8) × 10^{−17} m^{2} |

HT1080 cell | Moeendarbary et al. [138] | Sphere, $R=5\u2009\mu m$ | 0.4±0.2 kPa | ∼0.3 | (4.0±1.0) ×10^{−11} m^{2}/s | (2.9±0.7) × 10^{−17}m^{2} |

Alginate hydrogel | Hu et al. [5] | Conical indenter | 27.9 kPa | 0.28 | 3.24 × 10^{−8} m^{2}/s | 3.55 × 10^{−16} m^{2} |

Polyacrylamide gel | Lai and Hu [6] | Sphere, $R=12.5\u2009\mu m$ | 16.0±1.0 kPa | 0.32±0.02 | (6.7±0.8) × 10^{−11} m^{2}/s | (1.1±0.1) × 10^{−18} m^{2} |

Polyacrylamide particle | Berry et al. [136] | Sphere, $R=6.35\u2009\mu m$ | 6.0±2.2 kPa | 0.37±0.03 | (3.6±1.6) × 10^{−11} m^{2}/s | (1.4±1.1) × 10^{−18} m^{2} |

Mouse articular cartilage | Cao et al. [139] | Cylindrical flat punch, $R=55\u2009\mu m$ | 2.0±0.3 MPa | 0.20±0.03 | (5.41±2.98) × 10^{−10} m^{2}/s | (1.1±0.4) × 10^{−19} m^{2} |

Bovine femoral condylar cartilage | Mow et al. [12] | Cylindrical flat punch, $R=0.75\u2009mm$ | 0.16 MPa | 0.39 | 3.9 × 10^{−10} m^{2}/s | 4.40 × 10^{−19} m^{2} |

Bovine patellar groove cartilage | Mow et al. [12] | Cylindrical flat punch, $R=0.75\u2009mm$ | 0.31 MPa | 0.24 | 1.29 × 10^{−9} m^{2}/s | 1.42 × 10^{−18} m^{2} |

Bovine femoropatellar groove cartilage | Nia et al. [140] | Sphere, $R=12.5\u2009\mu m$ | 0.19±0.04 MPa | 0.1 (prescribed) | (4.28±0.49) × 10^{−9} m^{2}/s | (1.00±0.11) × 10^{−17} m^{2} |

Human coagulation clot | He et al. [110] | Sphere, $R=250\u2009\mu m$ | 0.059 kPa | 0.12±0.05 | 4.8 × 10^{−12} m^{2}/s | (3.5±0.9) × 10^{−14} m^{2} |

Material | References | Geometry | Shear modulus, $G$ | Drained Poisson's ratio, $\nu $ | Diffusivity, $D$ | Permeability, $k$ |
---|---|---|---|---|---|---|

Osteoblast | Shin and Athanasiou [137] | Cylindrical flat punch, $R=2.5\u2009\mu m$ | 0.51±0.17 kPa | 0.37±0.03 | (1.58±0.87)×10^{−7} m^{2}/s | (1.18±0.65) × 10^{−13} m^{2} |

MDCK cell | Moeendarbary et al. [138] | Sphere, $R=5\u2009\mu m$ | 0.9±0.4 kPa | ∼0.3 | (6.1±1.0) ×10^{−11} m^{2}/s | (1.9±0.3) × 10^{−17} m^{2} |

HeLa cell | Moeendarbary et al. [138] | Sphere, $R=5\u2009\mu m$ | 0.4±0.1 kPa | ∼0.3 | (4.1±1.1) × 10^{−11} m^{2}/s | (2.9±0.8) × 10^{−17} m^{2} |

HT1080 cell | Moeendarbary et al. [138] | Sphere, $R=5\u2009\mu m$ | 0.4±0.2 kPa | ∼0.3 | (4.0±1.0) ×10^{−11} m^{2}/s | (2.9±0.7) × 10^{−17}m^{2} |

Alginate hydrogel | Hu et al. [5] | Conical indenter | 27.9 kPa | 0.28 | 3.24 × 10^{−8} m^{2}/s | 3.55 × 10^{−16} m^{2} |

Polyacrylamide gel | Lai and Hu [6] | Sphere, $R=12.5\u2009\mu m$ | 16.0±1.0 kPa | 0.32±0.02 | (6.7±0.8) × 10^{−11} m^{2}/s | (1.1±0.1) × 10^{−18} m^{2} |

Polyacrylamide particle | Berry et al. [136] | Sphere, $R=6.35\u2009\mu m$ | 6.0±2.2 kPa | 0.37±0.03 | (3.6±1.6) × 10^{−11} m^{2}/s | (1.4±1.1) × 10^{−18} m^{2} |

Mouse articular cartilage | Cao et al. [139] | Cylindrical flat punch, $R=55\u2009\mu m$ | 2.0±0.3 MPa | 0.20±0.03 | (5.41±2.98) × 10^{−10} m^{2}/s | (1.1±0.4) × 10^{−19} m^{2} |

Bovine femoral condylar cartilage | Mow et al. [12] | Cylindrical flat punch, $R=0.75\u2009mm$ | 0.16 MPa | 0.39 | 3.9 × 10^{−10} m^{2}/s | 4.40 × 10^{−19} m^{2} |

Bovine patellar groove cartilage | Mow et al. [12] | Cylindrical flat punch, $R=0.75\u2009mm$ | 0.31 MPa | 0.24 | 1.29 × 10^{−9} m^{2}/s | 1.42 × 10^{−18} m^{2} |

Bovine femoropatellar groove cartilage | Nia et al. [140] | Sphere, $R=12.5\u2009\mu m$ | 0.19±0.04 MPa | 0.1 (prescribed) | (4.28±0.49) × 10^{−9} m^{2}/s | (1.00±0.11) × 10^{−17} m^{2} |

Human coagulation clot | He et al. [110] | Sphere, $R=250\u2009\mu m$ | 0.059 kPa | 0.12±0.05 | 4.8 × 10^{−12} m^{2}/s | (3.5±0.9) × 10^{−14} m^{2} |

### 3.3 Viscoporoelastic Indentation.

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 $\tau v$, $D$ and $Fi$ [148].

where $G\u221e$ is the long-time equilibrium shear modulus, $Gi$ and $\tau i$ are the viscoelastic Prony series parameters, $\nu $ is the Poisson's ratio, $cp$ is the consolidation coefficient, and $Ai$, $\beta i$, and $\tau iPE$ 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].

## 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 $\gamma $, 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].

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 $\gamma $ can be easily determined from the pull-off force and by substituting the adhesion energy $\gamma $ 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].

where $h$ is the indentation depth, $K=8G/[3(1\u2212\nu )]$ is the reduced modulus, $c$ is the apparent contact radius including the cohesive region, $R$ is the radius of the sphere, and $\lambda $ is the Maugis parameter that derives from normalization process. Here, $F\xafM\u2212D=FM\u2212D\pi \gamma R$ is the normalized reaction force, $h\xafM\u2212D=h/(\pi 2\gamma 2RK2)13$ is the normalized indentation depth, $a\xafM\u2212D=aM\u2212D/(\pi \gamma R2K)13$ is the normalized contact radius, and $\lambda =2\sigma 0/(\pi \gamma K2R)13$ 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 $\gamma $ 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 $\sigma 0$, the adhesion energy $\gamma $, and the separation distance $l0=\gamma /\sigma 0$.

It has been shown that the JKR and DMT models are two extreme cases of the Maugis' solution. When $\lambda \u2192\u221e$, the Maugis' solution is close to the JKR approximation, and when $\lambda \u21920$, 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 $\sigma 0$ and the separation distance $l0$ of the different cohesive relations are comparable [165]. Conversely, $\lambda $ can be used to define the applicable range of the JKR and DMT models. It has been shown that if $\lambda >5$, 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 $c0$ 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 $\gamma $, and the value of $\lambda $ 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, $\gamma $ and $\lambda $ can be obtained, and the cohesive strength $\sigma 0$ and the separation distance $l0$ 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 $0.5\u22122.5\u2009mJ/m2$, and the cohesive strength is in the range of $0.25\u22121.5\u2009kPa$ [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 $\sigma $ 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 $\sigma (GR)\u22122/3\gamma \u22121/3$, where $\sigma $ is the surface tension, $G$ is the shear modulus of the material, $R$ is the radius of spherical indenter, and $\gamma $ is the adhesion energy [174,175]. For $\sigma (GR)\u22122/3\gamma \u22121/3\u226a1$, 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 $\sigma (GR)\u22122/3\gamma \u22121/3\u226b1$, 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].

Combining Eqs. (45)–(50), the force–displacement relation can be determined. Here, for $\beta \u226a1,$ 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 $G/[2(1\u2212\nu )]$ [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.

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, $\Gamma 0$, and the fracture propagation energy also known as the critical energy release rate, $JIC$ and $JIIC$, 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 $\Gamma 0$ and $JC$ are energy per area ($J/m2$). In this section, we review the theoretical formulations and applications of the penetration method for experimentally estimating $JC$ and $\Gamma 0$. 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.

where $Bj$ and $\beta j$ as the power-law-series fitting parameters.

It's noted that the critical energy release rate ($JC$) 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.

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.

where $\alpha $ 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)), $f1(b/R)$ and $f2(\eta ,b/R)$ are dimensionless functions. For estimating $JIIC$, the undeformed ring crack radius $b$ needs to be measured experimentally.

where $l$ is the half-length of the crack (Fig. 11(c)) and $g(l/R)$ is the normalized strain energy function, $g(l/R)=Wstiffness/(GR2h)$, $Wstiffness$ is the total strain energy stored in the sample. To calculate $JIC$, 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 $Wstiffness$ for a known $l/R$ [194,200].

where $\tau c$ 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 $\tau c$ 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].

where $F\u2032$ is the force response during the second test, $hc$ is the critical depth, $h>hc$ is the final depth such that the half-space assumption is still valid and $2l$ 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.

Material | References | Value | Punch geometry |
---|---|---|---|

Bovine liver | Gokgol et al. [26] | $J1C=164\xb16\u2009\u2009J/m2$ | Sharp (conical) |

Porcine liver | Azar and Hayward [196] | $J1C=95\u2009J/m2$ | Bevel tip and Franseen tip |

Porcine tissue mimic gel | Misra et al. [185] | $J1C=114\u2009kJ/m2$ | Flexible needle with bevel tip |

Chicken breast tissue | Misra et al. [185] | $J1C=24.2\u2009kJ/m2$ | Flexible needle with bevel tip |

Gelatin | Oldfield et al. [198] | $J1C=17.43\u2009J/m2$ | Sharp (conical, prismatic, double-edged) |

Scleral tissue | Park et al. [206] | $J1C=570\xb140\u2009J/m2$ | Hemispherical (Sphere-ended ($R=9$–$50\u2009\mu m$) |

RLP-PEG hydrogel | Lau et al. [208] | RLP rich: $\Gamma 0=730\xb150\u2009J/m2$ PEG rich: $\Gamma 0=2844\xb1140\u2009J/m2$ | Sphere-ended ($R=0.8$–$35\u2009\mu m$) |

PAAm hydrogel | Fakhouri et al. [193] | $\Gamma 0=142\xb140\u2009J/m2$ | Flat punch, sphere-ended |

Material | References | Value | Punch geometry |
---|---|---|---|

Bovine liver | Gokgol et al. [26] | $J1C=164\xb16\u2009\u2009J/m2$ | Sharp (conical) |

Porcine liver | Azar and Hayward [196] | $J1C=95\u2009J/m2$ | Bevel tip and Franseen tip |

Porcine tissue mimic gel | Misra et al. [185] | $J1C=114\u2009kJ/m2$ | Flexible needle with bevel tip |

Chicken breast tissue | Misra et al. [185] | $J1C=24.2\u2009kJ/m2$ | Flexible needle with bevel tip |

Gelatin | Oldfield et al. [198] | $J1C=17.43\u2009J/m2$ | Sharp (conical, prismatic, double-edged) |

Scleral tissue | Park et al. [206] | $J1C=570\xb140\u2009J/m2$ | Hemispherical (Sphere-ended ($R=9$–$50\u2009\mu m$) |

RLP-PEG hydrogel | Lau et al. [208] | RLP rich: $\Gamma 0=730\xb150\u2009J/m2$ PEG rich: $\Gamma 0=2844\xb1140\u2009J/m2$ | Sphere-ended ($R=0.8$–$35\u2009\mu m$) |

PAAm hydrogel | Fakhouri et al. [193] | $\Gamma 0=142\xb140\u2009J/m2$ | 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).

## References

**7**(1), p. 45575.10.1038/srep45575

^{TM}as Determined by Atomic Force Microscopy