What is Stiffness?
Stiffness refers to a material’s resistance to deformation. A more stiff object requires more force to stretch the same distance as a less stiff object of the same dimensions. Hydrogels with high stiffness are harder for cells to pull on, affecting their mobility, lifespans, differentiation behaviors, and more.
What is the Relevant Property?
Shear modulus is a broadly applicable summary parameter for the stiffness of an elastic material, such as a covalently crosslinked hydrogel. While shear modulus originally referred to a material’s resistance to shearing deformations, where two opposing surfaces are pulled in parallel, opposite directions by traction forces, the term has been co-opted for a more general definition in the incompressible Neo-Hookean hyperelastic model, where the shear modulus is used as the single parameter relating applied Cauchy stresses (σ) to the material’s stretching (λ). The equation for uniaxial tensile or compressive deformation of an incompressible Neo-Hookean material is as follows:
\begin{align*} \sigma = G \left( \lambda^2 – \frac{1}{\lambda} \right) \end{align*}Alternative Parameters:
Young’s Modulus (E)
Young’s modulus is the most common and broadly applied descriptor of a material’s stiffness based on tension or compression experiments. Young’s modulus is based directly on linear stress and strain and therefore does not make inherent assumptions about a material’s compressibility, whereas the shear modulus as calculated above assumes an incompressible material. Young’s modulus can be calculated from shear modulus via the following equation, where ν represents Poisson’s ratio (0.5 for incompressible materials).
\begin{align*} E = 2G \left( 1+ \nu \right) \end{align*}In our work, we use the shear modulus (G) because it is closer to a fundamental descriptor of a material’s stiffness, whereas Young’s modulus is best suited for linear deformation experiments and applications. When describing a material’s stiffness, it is important to differentiate between Young’s modulus and shear modulus, since the Young’s modulus is typically three times the shear modulus.
Storage Modulus (G’)
Storage modulus (G’) describes a material’s frequency- and strain-dependent elastic response to twisting-type deformations. It is usually presented alongside the loss modulus (G”), which describes the material’s complementary viscous response or internal flow resulting from the same kind of deformation. The balance of storage modulus and loss modulus within most materials will shift dependent on the frequency and strain applied. Therefore, all measurements of storage modulus and loss modulus must be evaluated within the context of their experimental conditions. Some parallels to shear modulus can be drawn within the Linear viscoelastic range (LVE), or the frequency range (starting from a low frequency) over which the storage modulus does not change significantly for a given strain amplitude. However, even the storage modulus within the linear viscoelastic range typically varies with the strain applied. Ultimately, the storage modulus and loss modulus are critical parameters for viscoelastic materials and characterizing how materials change under changing conditions, but storage modulus is less useful than shear modulus for describing the stiffness of a fully elastic hydrogel. Most hydrogels held together by irreversible covalent bonds are effectively fully elastic, with negligible ratios of loss modulus to storage modulus.
How is it Measured?
Stiffness is measured by applying a force to a sample and measuring the resulting deformation. The type, size, and location of the applied force differs based on the equipment available, resulting in different deformation measurement techniques and mathematical models needed to interpret how the force and deformation are related. For all methods, it is important to set relevant standards for the strain rate and the maximum strain applied since both can greatly affect the measured stiffness. Below, we describe four basic techniques for measuring stiffness, but there are many other ways to make equivalent measurements. One interesting example not listed here is cavitation rheology [Mijailovic et al., 2021], which is especially suited for brain tissue as it can mimic the effects of traumatic brain injuries.
Each method applies forces and creates deformations with specific advantages and disadvantages for measuring stiffness.
Tensile Testing
Tensile testing stretches a sample by applying forces near its ends in opposite directions. The simplest form of tensile testing is uniaxial, where a film is stretched along one axis, although biaxial tensile testing can also be useful for certain applications such as studying the mechanics of skin tissues. In tensile testing, force is usually applied by some kind of actuator, and force is measured by a piezoelectric load cell aligned with the direction of applied force. Displacement can either be measured in bulk by keeping track of the displacement from the motor or locally via digital image correlation. For digital image correlation, a pattern is made on the surface of the sample, and a camera tracks how the pattern is distorted, correlating local stretch to the applied force by matching time with the actuator and load cell. Digital image correlation results in more precise stiffness measurements than bulk displacement because sample clamping methods typically create displacement nonlinearities near the sample edges, and digital image correlation can identify and isolate the homogeneous strain region toward the center of the sample.
For tensile testing, stiffness can be calculated as a shear modulus from the uniaxial Neo-Hookean model shown above or calculated as a Young’s modulus from simple engineering stress and strain.
Compressive Testing
Compressive testing applies forces in the opposite direction of tensile testing. For incompressible samples (including most hydrogels at relatively fast strain rates and short experiment times), unconstrained compression should yield the same stiffness values as tensile testing. Unlike tensile testing though, compressive testing is usually on a relatively short sample, meaning that digital image correlation is usually no viable. Therefore, tensile testing is likely a little more experimentally accurate, but it typically needs larger samples cut in a flat film or dog-bone shape. Unconstrained compression is when there is no lateral forces or constraints placed on the sample, so it is free to expand in those directions as it is compressed along the major axis. Constrained compression effectively boxes in the sample, forcing it to change volume as it is compressed. For hydrogels, constrained compression can be used with a dense sieve to measure how water leaves the network under applied forces.
For unconstrained compressive testing, stiffness can be calculated as a shear modulus from the uniaxial Neo-Hookean model shown above or calculated as a Young’s modulus from simple engineering stress and strain.
Rheology
Rheology using two parallel plates to twist a hydrogel sample slightly squeezed between them yields a storage modulus and a loss modulus. As discussed above, in the “Storage Modulus” section, storage modulus and loss modulus are both dependent on the frequency and amplitude of the applied force. However, within the linear viscoelastic range, at a application-relevant frequency, and at a strain amplitude that does not break the sample, AND if the sample is effectively elastic (which can be claimed if the loss modulus is less than 5% of the storage modulus), then the storage modulus can be used as an approximation of the shear modulus. It is also important to find an appropriate normal force applied by the parallel plates for these studies. This measurement is less direct than tensile, compressive or even indentation studies that yield a sample’s shear modulus, but it is a reasonable alternative for laboratories with limited equipment.
Indentation
Indentation techniques for stiffness measurements differ from tensile and compressive measurements in that they only deform a relatively small part of the sample. For this reason, size and geometry of the indentation probe are especially important when calculating stiffness. Typically, spherical probes are used on a flat surface of the sample, and the resulting shear modulus is calculated from the Hertz model:
\begin{align*} G = \frac{3\left(1-\nu \right) F}{8r^{1/2}d^{3/2}} \end{align*}where ν is Poisson’s Ratio, F is the applied force, r is the probe radius, and d is the distance indented into the sample.
Mostly importantly, indentation can be used at macroscale and microscale, allowing researchers to probe stiffness at length scales relevant for cell-material interactions.
Related Properties
Strength
Strength describes the amount of stress a crack-free material can bear before permanent, plastic deformation or breaking. It can be measured by straining a crack-free material to its breaking point and analyzing the stress and strain profile before, during, and after breaking.
Toughness
Toughness addresses the energy needed to break a material with a crack in it. Toughness can be measured by introducing a crack into a material before stretching it and analyzing the stress and strain profile before, during, and after breaking. Effectively, strength focuses on permanent deformation or breaking of a material, while toughness focuses on the conditions that allow a crack to propagate within the material. Often, strength and toughness are competing properties, and increasing a material’s strength will reduce its toughness.
Viscoelasticity
Viscoelasticity refers to the capability of an object to exhibit both viscous and elastic behavior. This is generally described through the storage and loss moduli which can be measured via rheology techniques (Sakai). While all hydrogels behave viscoelastically at very high frequencies due to the liquid behavior of the contained water, only some hydrogels behave viscoelastically at macroscopically relevant frequencies, typically due to non-covalent interactions controlling their overall network structure.
Our Results on Structural Design of Hydrogel Stiffness
Our published work in Polymer summarizes our results so far on structural design of stiffness for PVA hydrogels. Like with our swelling results, we created PVA hydrogels with three initial polymer volume fractions (φ0) and six degrees of polymerization between junctions (Nj), yielding 18 PVA hydrogel formulations. We then measured the stiffness of each hydrogel formulation using five independent measurement methods: tension, compression, shear rheology, macroindentation, and nanoindentation to cross-evaluate the similarities and differences associated with the different methods. Our major results from the study are summarized below:
We measured the stiffness of PVA hydrogels using five independent methods and compared the results to predictions from the swollen polymer network model.
Cross-evaluation of the different stiffness measurement methods showed that tension and macroindentation have the most accurate relationship, whereas tension and compression have the most precise relationship (based on linear fit R2 values).
Tension experiments with digital image correlation (DIC) allowed measurement of Poisson’s Ratio, and rheology allowed measurement of Viscoelastic Fraction (G”/G’) AKA tan delta, validating incompressibility and elasticity, respectively.
Predictions matched overall trends but consistently overestimated hydrogel stiffnesses and underestimated the deviation from an asymptotic trend associated with lower initial polymer volume fraction.
Swelling and stiffness were highly correlated as predicted by rubberlike elasticity theory.
