Fluid effects on the fracture toughness of gels

Abstract

The fracture of polymeric gels has been of growing interest in the last two decades. Well established continuum theories that couple large deformations and fluid diffusion have been applied to gels to determine crack tip fields and the energy release rate. Some studies have combined experiment and calculations to determine the fracture toughness of gels and have shown that fluid effects make a substantial contribution to the toughness. Here we adopt a micro-mechanical view to estimate the fracture toughness of gels, defined as the critical (total) energy release rate, and show how the initiation toughness can be written as a combination of contributions from fiber scission and of fluid-solid demixing at the crack tip. This estimate is based on knowledge of a critical stretch and an associated volumetric strain when fracture is incipient and reveals dependencies on material properties including the solid volume fraction of gels. There have been no known ways to measure the de-mixing contribution directly from experiments, but the results in this paper provide a methodology. We also show how dissipation due to fluid motion as the crack propagates can contribute to the fracture toughness. Detailed results are presented for fibrin gels, which are the main structural component of blood clots.

1. Introduction

Polymeric and fibrous gels are widely used in household and industrial applications ranging from diapers to plugs for oil wells [1] as well as microfluidic devices [2] and soft robotics [3]. In biology and medicine, fibrin and collagen gels [4] are used as scaffolds for tissue engineering [5]. Fibrin also is the main structural element of blood clots [6]. Gels are soft materials that consist of a fluid filled network of polymer chains or fibers. The large deformation of gels is strongly coupled with the motion of liquid through their pores. There are now well established continuum theories for the analysis of gels which take account of various material properties of the solid network and for the mixing of liquid and solid [7, 8].

The fracture of gels has been of interest for at least the last two decades and there has been much experimental and theoretical work on the topic which has been summarized in review papers [9, 10]. Continuum theories have been applied to analyze crack tip fields and the energy release rate for crack propagation in gels [11–15], where the latter has been defined in terms of a surface-independent integral, commonly labeled $J^{*}$, which depends directly on both the properties of the solid network and those that characterize the mixing of the fluid and solid. Recently, we have coupled experiments with finite element calculations based on an accurate material model fit to experiments in order to calculate the critical energy release rate for fibrin clots of various solid volume fractions for both tensile [16] and shear [17, 18] loading. These results have shown that the liquid makes a substantial contribution to the fracture toughness defined as the critical energy release rate, ${𝒢}_c=J_c^{*}$. The purpose of this paper is to provide a methodology to more simply estimate the fracture toughness of gels, including the de-mixing contribution (see below), as a function of material properties.

We adopt a micro-mechanical view to estimate the fracture toughness of gels without recourse to detailed crack solutions or finite element calculations. There is an extensive literature related to the micro-mechanics of fracture, and several prominent examples are [19–23]. A simple model is adopted in this paper that not only estimates the contribution to the critical energy release rate from the failure of the solid network, but the contributions from the mixing of fluid with the network, and diffusive effects arising from crack growth are also predicted. When fracture occurs under permeable boundary conditions at the crack surface fluid that was initially mixed with the polymer in the gel could become part of the bath, and therefore, no longer be mixed with the polymer. We call this a de–mixing contribution to the fracture toughness, but it cannot be directly measured from experiments. The failure of the gel is assumed to occur at a state corresponding to critical stretch of fibers comprising the network and an associated volume change, which directly correlates with local fluid de-mixing at the crack tip.

This paper is laid out as follows. Section 2 summarizes the continuum theory and the calculation of the energy release rate for a gel in terms of the $J^{*}$-integral. The key result of this section is that the energy release rate can be written as an integral over the surface of a semi-circular crack tip. Section 3 introduces the micro-mechanical picture of the gel at the crack tip. Fracture toughness is then estimated from a simple model that utilizes constitutive parameters and a few experimentally measured quantities including a critical stretch for the scission of fibers and a corresponding volumetric strain. The latter is important in accounting for the fluid contribution to energy release rate in addition to the failure of the network. Section 4 applies the results of Section 3 to predict the fracture toughness of fibrin gels and its variation with material properties including solid volume fraction, fiber diameter and length, elastic moduli, osmotic parameters, etc. Section 5 estimates dissipation due to fluid motion at the crack tip and its contribution to fracture toughness. Section 6 concludes with the main findings of this work. A justification for the simple model is given in Appendix A in terms of the HRR fields of elastic-plastic fracture mechanics. An explanation for the relation between dissipation due to drag vs. diffusion of fluid is given in Appendix B. A table of constitutive parameters for fibrin gels appears in Appendix C.

2. Preliminaries

2.1. Governing equations

We begin with a short review of the continuum mechanics of gels undergoing large deformation. Our interests focus on gels that consist of a solid phase which is a network of fibers, but the formulation can also be applied to molecular networks. The fibers can be a relatively high density of polymer chains, as in synthetic gels, or they could be thick fibers as in fibrin and collagen gels of relatively low solid volume fraction. Let ${\varphi }_s^{ref}$ be the volume fraction of solid in the stress-free state of the gel which is usually in the range ${\varphi }_s^{ref}<0.2$. In a continuum mechanical model a representative volume element (RVE) of the gel is treated as a mixture of solid and liquid. The nominal volume fraction of liquid is $C(\mathbf{X},t)$ while that of the solid is ${\varphi }_s(\mathbf{X},t)$ at reference location $\mathbf{X}$ and time $t$. The Helmholtz free energy density of the gel is given by:

$$\Psi \left(\mathbf{F},C\right)={\Psi }_{net}\left(\mathbf{F}\right)+{\Psi }_{mix}\left(C\right),$$ (1)

where $F_{ij}=\partial x_i/\partial X_j$ is the deformation gradient tensor, ${\Psi }_{net}$ is the free energy density of the solid network and ${\Psi }_{\mathit{\text{mix}}}(C)$ is the free energy density of mixing of the solid and liquid. It is assumed that the components of the gel are both incompressible, so that any local volume changes are due to the motion of the liquid only. In mathematical form, this constraint can be written as:

$$C+{\varphi }_s^{ref}=\text{det}\left(\mathbf{F}\right)=j.$$ (2)

The derivatives of the free energy density give the first Piola stress and the chemical potential of the liquid:

$$\frac{\partial \Psi }{\partial F_{ij}}=P_{ij},\frac{\partial \Psi }{\partial C}=\mu .$$ (3)

It is more convenient to work in terms of the Legendre transform of the free energy density:

$$\hat{\Psi }\left(\mathbf{F},\mu \right)=\Psi \left(\mathbf{F},C\right)-\mu C,P_{ij}=\frac{\partial \hat{\Psi }}{\partial F_{ij}},C=-\frac{\partial \hat{\Psi }}{\partial \mu }.$$ (4)

The first Piola stress $\mathbf{P}(\mathbf{X},t)$ satisfies the equilibrium equation:

$$\frac{\partial P_{ij}}{\partial X_j}=0.$$ (5)

The evolution of nominal liquid volume fraction $C(\mathbf{X},t)$ is governed by mass conservation:

$$\frac{\partial C}{\partial t}+\frac{\partial Q_i}{\partial X_i}=0,$$ (6)

where $Q_i$ is the $i^{\mathit{\text{th}}}$ component of the nominal liquid flux through a reference surface. Darcy’s law is used to quantify the liquid flux $\mathbf{Q}$ in terms of the chemical potential gradient:

$$Q_i=-\frac{k_{ij}}{\eta }\frac{\partial \mu }{\partial X_j},$$ (7)

where $\eta$ is the viscosity of the liquid and $\mathbf{k}$ is the Darcy constant tensor. Equations (5), (6) and (7) together with the constraint (2) are a set of coupled partial differential equations for $\mathbf{P}(\mathbf{X},t)$, $\mu (\mathbf{X},t)$, $\mathbf{Q}(\mathbf{X},t)$ and $C(\mathbf{X},t)$. With appropriate boundary and initial conditions they constitute a poro-elastic initial-boundary value problem that can be solved to give the stress, chemical potential, flux and concentration fields everywhere in the gel. Solutions have been given for cracked specimens with various constitutive laws and boundary conditions in several publications. [11, 12, 14–17, 24]. Traction or displacement boundary conditions are prescribed for the mechanical part of the problem, including the crack faces which are assumed traction free. Flow boundary conditions could be permeable ($\mu ={\mu }_0$, where ${\mu }_0$ is the chemical potential of liquid in a surrounding bath) or impermeable $(\partial \mu /\partial \mathbf{N}=0)$ where $\mathbf{N}$ is the unit outward normal to the boundary in the reference configuration.

2.2. Energy release rate and $J^{*}$-integral

The energy release rate of a cracked specimen is computed by applying the first law of thermodynamics to a system consisting of the specimen immersed in a constant temperature bath containing liquid at fixed chemical potential ${\mu }_0$. Work done on the specimen is either stored as free energy in the fiber network forming the gel, or it is dissipated due to liquid flow through the pores of the gel into or out of the bath. Since the bath is very large in comparison to the specimen it is assumed that the chemical potential of the bath fluid is fixed (no gradients) so that all dissipation occurs in the specimen. Let us denote by $\Phi$ the total energy of the system, then

$$\Phi =\Pi +\Sigma .$$ (8)

In the above, $\Pi$ is the potential energy of the specimen consisting of its free energy, which is an integral of the free energy density over the reference volume of the specimen, and the potential energy of the applied forces, which is a surface integral in the reference configuration. $\Sigma$ is the dissipated energy in the specimen and its rate of change can be expressed as an integral over the reference volume $V_0$ of the specimen as:

$$\frac{\partial \Sigma }{\partial t}=-{\int }_{V_0}{Q}_i\frac{\partial \mu }{\partial X_i}dV.$$ (9)

Conservation of energy requires that $d\Phi /dt=0$. For a cracked specimen $\Phi =\Phi (\mathbf{F},\mu ,a)$ where $a$ is a crack length in the reference configuration and $\mathbf{F}$ and $\mu$ are the deformation gradient and chemical potential fields, respectively.

Consider a specimen with a crack of length $a$ parallel to the $X_1-X_3$ plane with the crack front $s$ parallel to the $X_3$ axis under tensile (mode 1) loading. For an isotropic material (or an orthotropic material aligned with the crack $X_i$ coordinates), crack propagation is assumed to progress in the $X_1$ direction. The energy release rate for uniform propagation along the crack front, is the variation of the total energy $\Phi$ of the system (8) with respect to crack length $a$ under fixed mechanical and chemical boundary conditions:

$$𝒢=-\frac{1}{t}\frac{\partial \Phi }{\partial a},$$ (10)

where $t$ is the thickness along the $X_3$ direction. The energy release rate can be expressed as the sum of two integrals [12, 15, 25]:

$$\frac{1}{t}\frac{\partial \Pi }{\partial a}=-\frac{1}{t}{\int }_{S_0}\left(\Psi N_1-P_{ij}{N}_j\frac{\partial x_i}{\partial X_1}\right)dS,$$ (11)

$$\frac{1}{t}\frac{\partial \Sigma }{\partial a}=\frac{1}{t}{\int }_{V_0}\mu \frac{\partial C}{\partial X_1}dV,$$ (12)

$$𝒢=\frac{1}{t}\left[{\int }_{S_0}\left(\Psi N_1-P_{ij}{N}_j\frac{\partial x_i}{\partial X_1}\right)dS-{\int }_{V_0}\mu \frac{\partial C}{\partial X_1}dV\right]=J_{\mathit{\text{mech}}}+J_{\mathit{\text{flow}}},$$ (13)

where $\Psi (\mathbf{F},C)$ is the free energy of the gel given by (1), $S_0$ is the surface of the specimen with outward normal $\mathbf{N}$ in the reference configuration. The surface integral in (13) is labeled $J_{\mathit{\text{mech}}}$, since it reduces to the classical $J$-integral in the absence of fluid effects, whereas the volume integral, labeled $J_{\mathit{\text{flow}}}$, accounts for dissipation due to fluid flow [15]. The energy release rate can be rewritten in terms of $\hat{\Psi }(\mathbf{F},\mu )$ (4) after using the divergence theorem as:

$$𝒢=\frac{1}{t}{\int }_{S_0}\left[\hat{\Psi }{N}_1-P_{ij}{N}_j\frac{\partial x_i}{\partial X_1}\right]dS+\frac{1}{t}{\int }_{V_0}C\frac{\partial \mu }{\partial X_1}dV.$$ (14)

For an arbitrary surface $S$ enclosing the crack tip, and corresponding volume $V$, one can readily show from energy conservation that the energy release rate $𝒢$ is independent of the surface defined in the integrals in (13) and (14) as shown in [12, 15, 25]. Therefore, the energy release rate $J^{*}$ can be written in terms of an arbitrary surface enclosing the crack tip as:

$$𝒢=J^{*}=\frac{1}{t}{\int }_S\left[\hat{\Psi }{N}_1-P_{ij}{N}_j\frac{\partial x_i}{\partial X_1}\right]dS+\frac{1}{t}{\int }_{V}C\frac{\partial \mu }{\partial X_1}dV,$$ (15)

where $\mathbf{N}$ is the outward normal to any surface $S$ enclosing the crack tip in the reference configuration and $V$ is the volume contained inside that surface. Therefore, (15) is commonly termed a path-independent integral that equals the energy release rate due to crack advance [12].

The $J^{*}$-integral (15) was used to estimate the initiation fracture toughness of fibrin gels in Garyfallogiannis et al. [16] through a combination of experiments and finite element calculations. Experiments were conducted for various solid volume fractions of fibrin gels and force-extension data was collected for edge cracked specimens with different crack lengths. A constitutive law for fibrin gels based on the microscopic physics of deformation of fibrin fibers was developed and predicted, via finite element calculations, force-extension curves for cracked samples that agreed very well with the experimental data over a range of solid volume fractions and crack lengths. Then, $J^{*}$ was computed as the sum of the $J_{\mathit{\text{mech}}}$ and $J_{\mathit{\text{flow}}}$ integrals over the external surface of the specimen and enclosed volume, respectively, at the experimentally determined onset of crack propagation. This calculation captured the effect of fluid in the initiation toughness of fibrin gels. A goal of the current work is to circumvent the need for a finite element calculation and estimate the fracture toughness of gels, including fluid effects, from a knowledge of the gel constitutive law and some experimentally measurable parameters, such as, the critical stretch for fiber (or chain) scission.

In order to estimate $𝒢$ for two-dimensional, plane strain problems (i.e. all fields independent of $X_3$), in what follows we consider a semi-circular crack tip in the reference configuration as depicted in Fig. 1. The radius of the crack tip, $R_0$, is chosen to be the half the height of a Representative Volume Element (RVE) of the fiber network as discussed in the next section. In the analysis below, $R_0$ is not to be associated with an actual notch radius in the reference configuration, e.g. associated with specimen preparation; rather, it is a parameter that enters the calculation of $J^{*}$. Consider the surface and volume $S$ in (15) defined by a cylinder with generators parallel to $X_3$; the contour $\Gamma$ in that figure is the projection of that surface on the $X_1-X_2$ plane. With $dS=tds$, where $s$ is arc length on $\Gamma$ that increases in a counter clockwise sense, and $dV=tdA$, the integrals in (15) reduce to a line integral on $\Gamma$ and an integral over the enclosed area $A$. If $\Gamma$ is shrunk to the notch surface, i.e. $\Gamma \to {\Gamma }_1$ (depicted as the semi-circular notch tip centered at $X_1=X_2=0$ in Fig. 1), then $A\to 0$ and since $N_1=0$ on the entire traction free surfaces of the crack (including the horizontal surfaces parallel to the $X_1$ axis), it follows that

$$𝒢=J^{*}={\int }_{{\Gamma }_1}\hat{\Psi }{N}_{1}ds,$$ (16)

where the integral is evaluated in a counterclockwise sense along the circular notch tip. Remarkably, the energy release rate is defined by an integral only over the blunted crack tip. Under permeable boundary conditions $\mu ={\mu }_0$ on the crack surface. If ${\mu }_0=0$, e.g. when the specimen is immersed in a liquid bath, since $\hat{\Psi }=\Psi -\mu C$ (16) becomes

$$J^{*}={\int }_{{\Gamma }_1}\left[{\Psi }_{net}{N}_1+{\Psi }_{mix}{N}_1\right]ds=J_{net}+J_{mix}.$$ (17)

The first term is the contribution from the fibers forming the network and the second term is the contribution from mixing of the solid and liquid. The breaking of fibers at the crack tip causes the loss of the elastic energy stored in those fibers. When fibers break then all liquid contained in the pore surrounded by those fibers becomes contiguous with the bath liquid (which is at $\mu =0$) so the mixing free energy of that pore liquid is also lost. The latter was referred to as the de-mixing contribution to the fractue toughness in Sec. 1. In the absence of liquid $\hat{\Psi }$ reduces to the free energy of the solid network. Therefore, the fracture toughness of elastomers can be estimated by considering the energy stored in stretched fibers per unit reference area at the crack tip as is done in the commonly used Lake-Thomas model [26]. Note, if the energies in the integral in (17) are constant on ${\Gamma }_1$, then for the semi-circular notch tip, the integral is simply $2R_0\left({\Psi }_{\mathit{\text{net}}}+{\Psi }_{\mathit{\text{mix}}}\right)$ and the ratio $J_{\mathit{\text{mix}}}/J_{\mathit{\text{net}}}={\Psi }_{\mathit{\text{mix}}}/{\Psi }_{\mathit{\text{net}}}$ is constant on ${\Gamma }_1$.

Fig. 1.

A contour around a semi-circular crack tip. The contour is traversed anti-clockwise along $\Gamma$ and ${\Gamma }_1$. The crack faces are all traction free.

Whereas the classical $J$-integral applied to cracks in rate-independent elastic and elastic-plastic materials can readily be evaluated from load deflection measurements on the surface of a specimen, that is not the case for the last integral in (14) which arises due to fluid effects. As we have shown, the energy release rate $𝒢=J^{*}$ can be estimated from (14) by coupling experimental data with full-field finite element calculations [16]. Nevertheless, (16) or (17), which involve an integral over the crack tip only, can be exploited to determine a simple estimate of $𝒢$ from experiments alone as shown in the next section. The decoupling of the network and liquid contributions to the $J^{*}$-integral under permeable boundary conditions is utilized in the next section to estimate the critical energy release rate, or fracture toughness, of gels. We also note that a $J^{*}$ based on a critical stretch ahead of the crack tip shows little difference for tensile dilating gels irrespective of permeable or impermeable boundary conditions as seen in Fig. 16 of Garyfallogiannis et al. [15]. For tensile contracting gels (see [27]), moderate differences are predicted [16,27], with the critical energy release rate up to one-third higher for impermeable boundary conditions. The actual ratio of the fracture toughness for impermeable and permeable boundary conditions depends on many factors such as loading rate, crack length and material properties including the critical stretch. For convective boundary conditions in tensile contracting gels, the critical energy release rate tends to be in between the permeable and impermeable values [28].

3. Estimating the critical energy release rate

A micromechanical view of the gel is adopted with the goal of estimating the fracture toughness, i.e. the critical energy release rate ${𝒢}_c$, in terms of material properties of the gel. From From (16) and (17), ${𝒢}_c$ is evaluated assuming that critical conditions are reached ahead of the crack tip, so that

$${𝒢}_c={𝒢}_c^{net}+{𝒢}_c^{mix},$$ (18)

where ${𝒢}_c^{\mathit{\text{net}}}$ is the contribution due to network failure and ${𝒢}_c^{\mathit{\text{mix}}}$ is the contribution due to de-mixing. For gels, experimental observations indicate that the volume change tends to plateau before failure [16, 29, 30]. This includes both tensile dilating and tensile contracting gels [27]. For mode 1 crack configurations, the critical condition is defined in terms of the critical stretch and an associated volume change over a region ahead of the crack tip that defined by the microstructure. The size scale of the microstructure itself is determined by the average length of fibers between cross-links, $\xi$, and the diameter of those fibers, $d$, both of which can be readily measured in experiments. A cubic unit cell of side $\xi$ is considered in this work. For the purposes of estimating the fracture toughness, the Representative Volume Element (RVE) is chosen to be of rectangular cross-section with dimensions $\xi$, $h$, $t$ in the $X_1$, $X_2$, $X_3$ directions, respectively. Since the mechanical and chemical fields are assumed to be independent of $X_3$ (plane strain conditions), $t\gg \xi$, $h$, $t$ factors out of the calculation of fracture toughness, while $\xi$ is inherited from the unit cell and $h$ needs to be estimated. To estimate the critical energy release rate in what follows, we associate $h$ with $2R_0$, where $R_0$ is the radius of the rounded notch in the reference configuration. With this choice, the $J^{*}$-integral calculation of the previous section, which neglects damage in the bulk and assumes that the free energy does not vary over the notch surface, accurate estimates of the fracture toughness are obtained. Alternatively, if the crack-tip stress and strain fields that include damage are known, which in practice would require detailed finite element calculations, which is not the case for gels, then $h$ could be directly associated with a critical stretch criterion at a microstructural distance ahead of the crack tip, since the latter has been shown to be a good predictor of crack advance in fibrin gels [17, 31]. In Appendix A, an argument is presented that uses a critical stretch criterion at a distance $\xi$ ahead of crack tip together with known expressions for the crack tip stress and strain fields and suggests $h$ must be several times the unit cell size $\xi$ to give the correct magnitude of the energy release rate, e.g. $h/\xi =7$ or 8 for a power-law $J_2$ deformation theory material. Although $J_2$ deformation theory is not directly relevant to gels, this calculation illustrates how a critical stretch criterion may be employed to determine $h$ when stress and strain fields near the crack tip are known. An RVE elongated in the direction perpendicular to the direction of crack advance as suggested above may also be justified for gels due to the height of the large strain region ahead of cracks from finite element calculations that neglect damage [15,16]. For real gels with complex constitutive laws (e.g., including damage), $h$ must be determined through experimental calibration as we discuss in section 4.

For a unit cell with fibers along the cube edges, the so-called 3-chain model of Wang and Guth [32], the solid volume fraction of the gel is:

$${\varphi }_s^{ref}=\frac{\frac{12}{4}\pi \frac{d^2}{4}\xi }{{\xi }^3}=\frac{3\pi }{4}\frac{d^2}{{\xi }^2},$$ (19)

because each fiber in the cube is shared with four neighboring cubes. Most synthetic and biopolymer gels tend to be isotropic with random distribution of fibers. The 8-chain model of Arruda and Boyce [33] has been used to model such networks for which the solid volume fraction is ${\varphi }_s^{\mathit{\text{ref}}}=\sqrt{3}\pi d^2/{\xi }^2$. A combination of the 3-chain and 8-chain model was shown to match uniaxial tensile behavior of istropic elastomer networks very well by Wu and van der Geissen [34]. This motivated the 14-chain model used in Garyfallogiannis et al. [16] for which the solid volume fraction is $2.482\pi d^2/{\xi }^2$. In general, ${\varphi }_s^{ref}=\alpha d^2/{\xi }^2$, where $\alpha$ is a constant determined by the geometry of the network [35]. For fibrin gels this scaling was shown to hold in Garyfallogiannis et al. [16], Fig. 3H. The constant $\alpha$ can be estimated from the slope of the line in that figure.

Fig. 3.

Scaled RVE height $h/\xi$ is plotted as a function of ${\lambda }_{\mathit{\text{crit}}}$, given the known fracture toughness of fibrin gels at various ${\varphi }_s^{ref}$. The measured values of ${𝒢}_c$ and $\xi$ for three different fibrin volume fractions are given in the legend. For ${\lambda }_{\mathit{\text{crit}}}\approx 3$ the height of the RVE is a few hundred microns irrespective of solid volume fraction ${\varphi }_s^{ref}$.

Consider a RVE of dimensions $\xi$, $h$, $t$ in the reference configuration right ahead of the crack tip. The initial crack-tip radius may be on the order of the RVE size, but need not be. Under mode 1 conditions, as the crack-tip RVE fails, the crack advances by an amount $\xi$ along $X_1$, assuming that the unit cells break sequentially along the $X_1$ direction since the stress ${\sigma }_{22}$ and stretch $F_{22}$ increase monotonically as we approach the notched crack tip [15]. Referring to (17), we take ${\Gamma }_1$ to be the face of this RVE exposed to the bath and assume that the free energies ${\Psi }_{\mathit{\text{net}}}$ and ${\Psi }_{\mathit{\text{mix}}}$ are uniform throughout the RVE. Then, computation of the critical energy release rate, $J_c^{*}$, becomes a matter of computing the free energies ${\Psi }_{\mathit{\text{net}}}$ and ${\Psi }_{\mathit{\text{mix}}}$ of the RVE material just before rupture. This simple calculation follows.

When the stretch ${\lambda }_2$ in the RVE right ahead of the crack tip reaches ${\lambda }_c$ then this RVE fails and the reference area of the crack face increases by $t\xi$, where $t$ is the depth of the specimen in the $X_3$ direction. The energy stored in the fiber network comprising the RVE, including the dilational energy, is lost (released), and the liquid contained in that RVE instantaneously merges with the liquid outside (assuming permeable boundary conditions) and is no longer mixed with the solid in the gel. Thus, there is a change in the mixing free energy also when the fibers of the cube break. The released energy (due to fiber breaking and due to liquid exchange) divided by $t\xi$ is an estimate of the critical energy release rate. In experiments, the breaking of fibers occurs over a damage zone near the crack tip and not just in a single unit cell of a microstructural dimension ahead of the crack tip. Therefore, the following analysis is expected to give an estimate of the dependence of toughness on microstructural parameters.

Two pieces of (experimental) information are needed to estimate the fracture toughness of a gel – the critical stretch ${\lambda }_c$ and the corresponding $j_c=\text{det}\left({\mathbf{F}}_c\right)$ at critical conditions. The volume of the RVE in the current configuration is $jht\xi ={\lambda }_1{\lambda }_2{\lambda }_{3}ht\xi$. Under plane strain conditions ${\lambda }_3=1$. It is assumed that the fibers making up the RVE break when ${\lambda }_2={\lambda }_c$ where ${\lambda }_2$ is evaluated at a distance $\xi$ ahead of the notch tip since $\xi$ is typical microstructural length scale. It was shown in Tutwiler et al. [31] and Ramanujam et al. [17] that attaining a critical stretch a distance on the order of a micro-structural length ahead of the crack tip was a good predictor of crack advance irrespective of loading. It remains to determine ${\lambda }_1$ when fiber breaking is imminent. For fibrous gels (fibrin, collagen) under uniaxial tension it is known that increasing stretch results in a decrease in volume until a minimum is reached, after which further stretching causes negligible change in volume. We assume that the minimum $j$ has been reached when fiber breakage occurs. For fibrin gels this minimum corresponds to $j_c=0.1$. We refer to these materials as tensile-contracting materials [27], for which $j_c$ is expected to be bounded below by ${\varphi }_s^{ref}$. For synthetic gels under uniaxial tension, the volume increases with increasing stretch. Since polymer chains are nearly inextensible, it is reasonable to assume that the volume change $j_c$ of these materials is bounded above by $l^3/{\xi }^3$, where $l$ is the contour length of the chains. We refer to these materials as tensile-dilating materials. The reader is referred to Garyfallogiannis et al. [27] for detailed discussions of how cracked specimens of these two types of materials respond to loading. With ${\lambda }_c$ and $j_c$ known from experiments, the fracture toughness is estimated below. The results are expected to apply equally well to both tensile-contracting and tensile-dilating gels. The particular energy densities ${\Psi }_{\mathit{\text{net}}}$ and ${\Psi }_{\mathit{\text{mix}}}$ will be different for each type of gel. For example, ${\Psi }_{\mathit{\text{net}}}$ could be neo-Hookean, Mooney-Rivlin or Gent for an elastomeric (tensile-dilating) gel consisting of a network of polymer chains, while it could be a Fung or Yeoh type expression for a fibrous (tensile-contracting) gel consisting of a network of thick fibers, such as, collagen or fibrin.

Network failure (fibers breaking):

If ${\Psi }_{\mathit{\text{net}}}$ is assumed to be constant on ${\Gamma }_1$, then from (17) and $J_{net}$ evaluated at the critical state, the contribution to the critical energy release rate from the failure of the network is:

$${𝒢}_c^{net}={\Psi }_{net}\left(\frac{j_c}{{\lambda }_c},{\lambda }_c,1\right)h,$$ (20)

which has units of $J/m^2$. Since $h=m\xi$, this contribution to the fracture toughness scales with $\xi$. This result also follows if we simply consider the energy released by failure of the RVE at the crack tip (as given in Appendix (A.2)). The change in network elastic energy due to failure of the RVE is ${\Psi }_{net}(\mathbf{F})ht\xi$ evaluated at principal stretches ${\lambda }_1=j_c/{\lambda }_c$, ${\lambda }_2={\lambda }_c$, ${\lambda }_3=1$. Note that we have ordered the principal stretches as along the $X_1$, $X_2$, $X_3$ directions. Dividing this by the increase in reference area of the crack face $t\xi$ gives (20).

The network part of the free energy density can be partitioned as ${\Psi }_{net}={\Psi }_{fib}(\mathbf{F})+{\Psi }_{\mathit{\text{vol}}}(\text{det}(\mathbf{F}))$, where ${\Psi }_{fib}$ is the free energy density stored in the elastic deformation of the fibers alone and ${\Psi }_{\mathit{\text{vol}}}(\text{det}(\mathbf{F}))$ is volumetric part of the network free energy density [15, 16]. The free energy stored in the fibers can be computed by integrating the $f-\delta$ relation of the fiber where $f$ is the force in the fiber and $\delta$ is its extension. It is easiest to illustrate this for an 8-chain cube in which the stretch of each fiber $\lambda$ is given by $3{\lambda }^2={\lambda }_1^2+{\lambda }_2^2+{\lambda }_3^2$. If the fibers break at $\lambda ={\lambda }_{\mathit{\text{crit}}}$ then ${\lambda }_c$ can be computed in terms of ${\lambda }_{\mathit{\text{crit}}}$ and $j_c$ as given below in (30) for fibrin gels. The length of each fiber is $\sqrt{3}\xi /2$, so assuming that the fibers break at $\delta =\left({\lambda }_{\mathit{\text{crit}}}-1\right)\sqrt{3}\xi /2$ and that the volumetric part of the network free energy is also lost after fiber breakage ${\Psi }_{\mathit{\text{net}}}$ at critical conditions is given by:

$${\Psi }_{net}\left(\frac{j_c}{{\lambda }_c},{\lambda }_c,1\right)=\frac{1}{h{\xi }^2}{\int }_0^{\delta }Mfd\delta +{\Psi }_{vol}\left(j_c\right),$$ (21)

where $M=8h/\xi$ since all 8 fibers in the RVE are stretched by the same amount. ${\Psi }_{\mathit{\text{vol}}}\left(j_c\right)$ is the volumetric part of the network free energy evaluated at the critical value of ${\lambda }_1{\lambda }_2{\lambda }_3=j$. A similar calculation can be done for the 3-chain cube for which energies stored in fibers aligned with various directions are different and must be added together, viz., ${\delta }_1=\left(j_c/{\lambda }_c-1\right)\xi$ and $M_1=1$ for fibers aligned with the $X_1$ direction, ${\delta }_2=\left({\lambda }_c-1\right)\xi$ and $M_2=h/\xi$ for fibers aligned with the $X_2$ direction; fibers aligned with the $X_3$ direction do not contribute because ${\lambda }_3=1$ in plane strain.

Fluid-solid demixing:

The release of the fluid-solid mixing energy as the network fails is computed next. If ${\Psi }_{\mathit{\text{mix}}}$ is assumed to be constant on ${\Gamma }_1$, then from (17), with $J_{\mathit{\text{mix}}}^{*}$ evaluated at the critical state, the contribution to the critical energy release rate from the failure of the network is:

$${𝒢}_c^{mix}=\frac{{\Psi }_{mix}\left(j_c\right)ht\xi }{t\xi }={\Psi }_{mix}\left(j_c\right)h.$$ (22)

Again, since $h=m\xi$ this contribution to the fracture toughness also scales with $\xi$. As with the evaluation of ${𝒢}_c^{\mathit{\text{net}}}$ above, (22) also follows from the release of the fluid-solid mixing energy in the RVE. The RVE of volume $ht\xi$ in the reference configuration has been deformed to a cuboid of sides ${\lambda }_1\xi$, ${\lambda }_{2}h$ and ${\lambda }_{3}t$ in the current configuration. Assuming ${\lambda }_1{\lambda }_2{\lambda }_3=j_c$ when fiber breaking is imminent, fluid of volume $\left(j_c-{\varphi }_s^{ref}\right)ht\xi$ is expelled into the bath when the fibers break at the crack tip, which leads to the same result for ${𝒢}_c^{mix}$ in (22).

4. Application to fibrin gels

4.1. Toughness of networks with ${\varphi }_s^{\mathit{\text{ref}}}\propto d^2/{\xi }^2$

In order to obtain ${𝒢}_c^{\mathit{\text{net}}}$ the force-stretch relation of fibrin fibers must be known. This force-stretch relation also enters the 14-chain model (which combines the 8-chain and 3-chain models [34] in the poro-elastic constitutive laws used in [16, 17]). Rate-dependence enters these constitutive laws only through the diffusion of liquid. Visco-elasticity of fibrin could also give rise to rate effects, but in cross-linked fibrin networks the measured loss modulus is ten times smaller than the storage modulus [36]. Fibrin fibers consist of protofibrils, which in turn are made of fibrin monomers that undergo a change in secondary structure under tensile forces [37]. In fact, there are sevaral force-induced structural changes in fibrin, including an $\alpha$-helix to $\beta$-sheet phase transition followed by unfolidng of a few globular domains. Therefore, the stress-strain relation of fibrin fibers is described in two parts:

Folded phase.

In the folded phase, the stress-strain relation of the fibers is

$$P=\frac{f}{\pi d^2/4}=c_f{E}_p=E{E}_p.$$ (23)

where $d$ is the fiber diameter and $E$ is the Young modulus of fibrin fibers in the folded phase. $f$ is the force in a fiber, $P$ is the axial stress in the fiber and $E_p$ is the axial strain in the fiber. Therefore, the constant $c_f$ appearing in the 14-chain model of Garyfallogiannis et al. [16] is $c_f=E$.

Unfolded phase.

In the unfolded phase, the force-stretch (not strain) relation of the ‘fiber’ is given by a worm-like-chain formula [37]

$$\lambda =E_p+1=\frac{L_u}{L_f}\left(1-\frac{\sqrt{k_{B}TN}}{2\sqrt{l_{p}f}}\right).$$ (24)

In the above, $N=\pi d^2/\left(4a_0\right)$ where $a_0$ is the cross-section area occupied by a single protofibril in the fiber and $l_p$ is the persistence length of the unfolded chain of amino-acids, $L_u/L_f$ is the ratio of contour length in the unfolded state to that in the folded state. The above equation can be solved for $f$ (or $P$) as:

$$P=\frac{f}{\pi d^2/4}=\frac{L_u^2}{L_f^2}\frac{k_{B}TN}{4l_p\pi d^2/4}{\left[\frac{1}{L_u/L_f-1-E_p}\right]}^2.$$ (25)

In the 14-chain model of Garyfallogiannis et al. [16], the stress-strain relation of the fibers in the unfolded phase is:

$$P=\frac{c_u}{{\left({\epsilon }_{10}-E_p\right)}^2}+C_1,$$ (26)

where $c_u$, ${\epsilon }_{10}$ and $C_1$ are constants. By comparing the above expression with (25), it is easy to see that

$${\epsilon }_{10}=\frac{L_u}{L_f}-1,c_u=\frac{L_u^2}{L_f^2}\frac{k_{B}TN}{l_p\pi d^2}.$$ (27)

There is one more parameter ${\epsilon }_{\mathit{\text{transf}}}$ in Garyfallogiannis et al. [16] which sets the strain $E_p$ at which the unfolding transition occurs. This parameter can determine $C_1$ by continuity of force $f$ at $E_p={\epsilon }_{\mathit{\text{trans}}f}$:

$$C_1=c_f{\epsilon }_{\mathit{\text{transf}}}-\frac{c_u}{{\left({\epsilon }_{10}-{\epsilon }_{\mathit{\text{transf}}}\right)}^2}.$$ (28)

If $E$ is independent of $d$ and $\xi$ and $N\propto \pi d^2/4$, then $c_f$, $c_u$ and $C_1$ are all independent of $d$ and $\xi$.

Assuming that the fibers are brittle, so the downward part of the $f-\delta$ law is vertical and that the fibers break at $E_p={\epsilon }_{\mathit{\text{crit}}}$, then following (21) the free energy ${\Psi }_{\mathit{\text{net}}}\left(j_c/{\lambda }_c,{\lambda }_c,1\right)$ at critical conditions can be evaluated as:

$$\begin{gathered} {\Psi }_{net}\left(\frac{j_c}{{\lambda }_c},{\lambda }_c,1\right)=\frac{M}{h{\xi }^2}{\int }_{\xi }^{\left(1+{\epsilon }_{\mathit{\text{crit}}}\right)\xi }fd\delta +{\varphi }_s^{ref}{\alpha }_1\text{log}{j}_c, \\ =\frac{\sqrt{3}M\xi \pi d^2}{2h{\xi }^2}{\int }_0^{{\epsilon }_{\mathit{\text{crit}}}}Pd{E}_p,+{\varphi }_s^{ref}{\alpha }_1\text{log}{j}_c, \end{gathered}$$ (29)

where we have taken fiber strain $E_p=2\delta /\sqrt{3}\xi$ and the volumetric part of the network free energy density as ${\Psi }_{vol}(j)={\alpha }_1{\varphi }_s^{ref}\text{log}j$ (as in Garyfallogiannis et al. [16]). In the above, ${\epsilon }_{\mathit{\text{crit}}}$ is the rupture strain of a single fibrin fiber (e.g. Liu et al. [38] measure $1.5\le {\epsilon }_{\mathit{\text{crit}}}\le 2.3$ using atomic force microscopy) is related to ${\lambda }_c$ and $j_c$ in the 8-chain model under plane strain conditions through:

$$3{\left(1+{\epsilon }_{\mathit{\text{crit}}}\right)}^2={\left(\frac{j_c}{{\lambda }_c}\right)}^2+{\lambda }_c^2+1.$$ (30)

The integral in (29) can be computed using the stress-strain law of the fibers in the two phases:

$$\begin{gathered} {\int }_0^{{ϵ}_{\mathit{\text{crit}}}}Pd{E}_p={\int }_0^{{\epsilon }_{\mathit{\text{transf}}}}{c}_f{E}_{p}d{E}_p+{\int }_{{\epsilon }_{\mathit{\text{transf}}}}^{{\epsilon }_{\mathit{\text{crit}}}}\frac{c_{u}d{E}_p}{{\left({\epsilon }_{10}-E_p\right)}^2} \\ +\left[c_f{\epsilon }_{\mathit{\text{transf}}}-\frac{c_u}{{\left({\epsilon }_{10}-{\epsilon }_{\mathit{\text{transf}}}\right)}^2}\right]{\int }_{{\epsilon }_{\mathit{\text{transf}}}}^{{\epsilon }_{\mathit{\text{crit}}}}d{E}_p \\ =c_f\left({\lambda }_{\mathit{\text{transf}}}-1\right)\left({\lambda }_{\mathit{\text{crit}}}-\frac{{\lambda }_{\mathit{\text{transf}}}+1}{2}\right)+\frac{c_u{\left({\lambda }_{\mathit{\text{crit}}}-{\lambda }_{\mathit{\text{transf}}}\right)}^2}{\left({\lambda }_{10}-{\lambda }_{\mathit{\text{crit}}}\right){\left({\lambda }_{10}-{\lambda }_{\mathit{\text{transf}}}\right)}^2}, \end{gathered}$$

where we have used ${\lambda }_{\mathit{\text{transf}}}=1+{\epsilon }_{\mathit{\text{transf}}}$, ${\lambda }_{\mathit{\text{crit}}}=1+{\epsilon }_{\mathit{\text{crit}}}$ and ${\lambda }_{10}=1+{\epsilon }_{10}$. Therefore, following (20) the final expression for ${𝒢}_c^{\mathit{\text{net}}}$ is:

$${𝒢}_c^{\mathit{\text{net}}}=\frac{\sqrt{3}Mh\pi d^2}{2h\xi }\left[c_f\left({\lambda }_{\mathit{\text{transf}}}-1\right)\left({\lambda }_{\mathit{\text{crit}}}-\frac{{\lambda }_{\mathit{\text{transf}}}+1}{2}\right)+\frac{c_u{\left({\lambda }_{\mathit{\text{crit}}}-{\lambda }_{\mathit{\text{transf}}}\right)}^2}{\left({\lambda }_{10}-{\lambda }_{\mathit{\text{crit}}}\right){\left({\lambda }_{10}-{\lambda }_{\mathit{\text{transf}}}\right)}^2}\right]+h{\alpha }_1{\varphi }_s^{\mathit{\text{ref}}}\text{log}{j}_c,$$ (31)

In the above expression, $M=8h/\xi$ is proportional to the number of unit cells in the RVE ahead of the crack tip. If $c_f$, $c_u$, ${\lambda }_{10}$, ${\lambda }_{\mathit{\text{transf}}}$, ${\lambda }_{\mathit{\text{crit}}}$ are independent of $\xi$ and ${\varphi }_s^{\mathit{\text{ref}}}=\alpha d^2/{\xi }^2$ then the first term in ${𝒢}_c^{\mathit{\text{net}}}$ varies linearly with $\xi {\varphi }_s^{\mathit{\text{ref}}}$. In Garyfallogiannis et al. [16], $d$ increased as the fibrin volume fraction increased, yet $d^2/{\xi }^2\propto {\varphi }_s^{\mathit{\text{ref}}}$ was shown to hold (see their Fig. 3H). An approximate analysis of their data shows that $d\propto {\left({\varphi }_s^{\mathit{\text{ref}}}\right)}^{0.35}$. Therefore, it must be that $\xi \propto {\left({\varphi }_s^{\mathit{\text{ref}}}\right)}^{-0.15}$. Thus, $\xi$ should decrease weakly as ${\varphi }_s^{\mathit{\text{ref}}}$ increases and indeed it does so in the experiments of [16].

Next, the contribution of the (de-)mixing free energy to crack advance will be computed for fibrin gels. The mixing free energy per reference volume in Garyfallogiannis et al. [16] is

$${\Psi }_{mix}(j)=\frac{{\pi }_0}{{\beta }_1-1}\left[\frac{{\left(1-{\varphi }_s^{ref}\right)}^{{\beta }_1}}{{\left(j-{\varphi }_s^{ref}\right)}^{{\beta }_1-1}}-\left(1-{\varphi }_s^{ref}\right)\right]$$ (32)

where, ${\pi }_0$ and ${\beta }_1$ are constants chosen to fit the data. ${\beta }_1=1.02$ was used in Garyfallogiannis et al. and crack advance occurs when ${\lambda }_c\approx 3$ at $100\mu \mathit{\text{m}}$ ahead of the crack tip in fibrin gels [17,31]. At this stretch $j_c=0.1$ [37]. This was measured in uniaxial tension experiments on cylndrical specimens that were a few millimeter in diameter and a few centimeter in length. The appropriate values ${\lambda }_{\mathit{\text{crit}}}$ and $j_c$ in the equations above should be from a plane strain tension experiment, but we are unaware of any such measurements, so we simply use $j_c=0.1$ in our calculations. Following (22), the contribution to the energy release rate of the de-mixing of fluid and solid is:

$${𝒢}_c^{mix}=\frac{h{\pi }_0}{{\beta }_1-1}\left[\frac{{\left(1-{\varphi }_s^{ref}\right)}^{{\beta }_1}}{{\left(j_c-{\varphi }_s^{ref}\right)}^{{\beta }_1-1}}-\left(1-{\varphi }_s^{ref}\right)\right]$$ (33)

The material parameters – $c_f$, $c_u$, ${\lambda }_{\mathit{\text{transf}}}$, ${\lambda }_{\mathit{\text{crit}}}$, ${\lambda }_{10}$ – were independent of $\xi$ and $d$ in Garyfallogiannis et al. [16] and ${\varphi }_s^{\mathit{\text{ref}}}{\alpha }_1=G{\varphi }_s^{\mathit{\text{ref}}}-{\pi }_0$ in their work with $G{\varphi }_s^{\mathit{\text{ref}}}\ll {\pi }_0$, where $G$ was a shear modulus. If ${\pi }_0$, ${\beta }_1$ and $j_c$ are all constants independent of $d$ and $\xi$, then the fluid contribution to the fracture toughness has a more complex dependence on ${\varphi }_s^{\mathit{\text{ref}}}$. Combining the contributions of fiber breaking and fluid-solid de-mixing, we get the fracture toughness as:

$${𝒢}_c=A\xi {\varphi }_s^{ref}+B\xi +C\xi \left[\frac{{\left(1-{\varphi }_s^{ref}\right)}^{{\beta }_1}}{{\left(j_c-{\varphi }_s^{ref}\right)}^{{\beta }_1-1}}-\left(1-{\varphi }_s^{ref}\right)\right],$$ (34)

where $A$, $B$ and $C$ are given by:

$$A=\frac{8\sqrt{3}\pi h}{2\alpha \xi }\left[c_f\left({\lambda }_{\mathit{\text{transf}}}-1\right)\left({\lambda }_{\mathit{\text{crit}}}-\frac{{\lambda }_{\mathit{\text{transf}}}+1}{2}\right)+\frac{c_u{\left({\lambda }_{\mathit{\text{crit}}}-{\lambda }_{\mathit{\text{transf}}}\right)}^2}{\left({\lambda }_{10}-{\lambda }_{\mathit{\text{crit}}}\right){\left({\lambda }_{10}-{\lambda }_{\mathit{\text{transf}}}\right)}^2}\right]+\frac{h}{\xi }G\text{log}{j}_c,$$ (35)

$$B=-\frac{h}{\xi }{\pi }_0\text{log}{j}_c,$$ (36)

$$C=\frac{{\pi }_0}{{\beta }_1-1}\frac{h}{\xi }.$$ (37)

Note that if ${\varphi }_s^{\mathit{\text{ref}}}$ is held fixed but conditions of clot synthesis are varied to allow different $d$ and $\xi$ with $\frac{d^2}{{\xi }^2}$ held fixed, then the above analysis predicts that the initiation toughness varies linearly with $\xi$. This prediction was shown to hold over a range of fiber lengths (or $\xi$) in Ramanujam et al. [39].

Since the initiation fracture toughness of fibrin gels of various ${\varphi }_s^{\mathit{\text{ref}}}$ and structural parameters (fiber diameter, length, etc.) of fibrin fibers are known from experiment and since constitutive parameters $c_f$, $c_u$, etc. are independent of ${\varphi }_s^{ref}$ in [16] the RVE height $h$ can be estimated using (20) and (22). We refer the reader to the Supplementary Information of Garyfallogiannis et al. [16] for values of constitutive parameters $c_f$, $c_u$, ${\lambda }_{\mathit{\text{transf}}}$, ${\lambda }_{10}$ used here and summarized in Appendix C. For illustrative purposes, we consider $j_c=0.1$ [37] which was measured for ${\varphi }_s^{\mathit{\text{ref}}}=0.037$ (corresponding to 10 mg/ml fibrinogen) in all calculations and focus on the ${\lambda }_{\mathit{\text{crit}}}$ dependence. Lower values of $j_c$ are certainly possible since $j_c$ is bounded below by ${\varphi }_s^{ref}$, however we are not aware of any measurements of $j_c$ for fibrin or other tensile contracting gels other than the one mentioned above. We work with the 8-chain model with $\xi$ as the unit cell size.

4.2. Estimating $h$ from coupled experimental and finite element analysis

In our previous work that coupled experimental measurements of the critical value of $J_{\mathit{\text{mech}}}$ with detailed finite element calculations [16], we determined ${𝒢}_c$ as a function of ${\varphi }_s^{\mathit{\text{ref}}}$ for fibrin gels. From those results, we are able to directly estimate $h$ as shown below and, therefore, use (34) – (37) to determine ${𝒢}_c$ from the simplified model of the previous section.

${𝒢}_c$ is plotted as a function of ${\varphi }_s^{\mathit{\text{ref}}}$ in Fig. 2 using (34) together with (35), (36) and (37). Toughness values from Garyfallogiannis et al. [16] are shown as red circles for comparison. In Fig. 2, $\xi =7{\left({\varphi }_s^{ref}\right)}^{-0.15}\mu \mathit{\text{m}}$ (see the discussion below (31)), ${\lambda }_{\mathit{\text{crit}}}=4$ and $h/\xi =28$ with all other material properties taken from the Supplementary Information of Garyfallogiannis et al. [16] and summarized in Appendix C. Equation (34) agrees very well with experimental data with these parameters, in particular with $h/\xi =28$.

Fig. 2.

Fracture toughness ${𝒢}_c$ is plotted as a function of ${\varphi }_s^{\mathit{\text{ref}}}$ for fibrin gels. The curve is based on (34) with ${\lambda }_{\mathit{\text{crit}}}=4$ and $h/\xi =28$. All other material properties and the experimental data points are from Garyfallogiannis et al. [16].

Since the RVE size $h$ and the critical stretch ${\lambda }_{\mathit{\text{crit}}}$ are not known precisely, $h/\xi$ is plotted as a function of ${\lambda }_{\mathit{\text{crit}}}$ in Fig. 3 given the known fracture toughness at various values of ${\varphi }_s^{\mathit{\text{ref}}}$. The height $h$ of the RVE that must be used for computing fracture toughness is estimated to be on the order of a few hundred $\mu \mathit{\text{m}}$ for three different fibrin concentrations corresponding to ${\varphi }_s^{\mathit{\text{ref}}}=0.01,0.0185,0.037$ over a range of ${\lambda }_{\mathit{\text{crit}}}$. This is consistent with the height of the elongated high stretch zone plotted in Fig. 6 of Garyfallogiannis et al. [16].

4.3. Estimating ${𝒢}_c$ and $h$ directly from experimental measurements

The critical energy release rate ${𝒢}_c$ can be estimated from experiment given that $J_{\mathit{\text{mix}}}/J_{\mathit{\text{net}}}$, and therefore ${𝒢}_c^{\mathit{\text{mix}}}/{𝒢}_c^{\mathit{\text{net}}}$, are independent of $h$ which follows from (20) and (22). Furthermore, therefore, $h$ can also be estimated. Indeed, to estimate ${𝒢}_c$ without input from a fracture test, one needs to know $h$ (see previous sub-section). On the other hand, if the critical value of $J_{\mathit{\text{mech}}}$ is known from experiment [16, 17, 31], then ${𝒢}_c$ is approximately that value times a factor that depends only on

$$\omega ={𝒢}_c^{mix}/{𝒢}_c^{net},$$ (38)

which as noted is independent of $h$. This claim results from the fact that the contribution of $\pi$ and $\mu$ on the stress is small [16, 27], so that $J_{\mathit{\text{mech}}}\approx J_{\mathit{\text{net}}}$, and it follows that

$${𝒢}_c\approx \left(1+\omega \right){\widetilde{𝒢}}_c,$$ (39)

where ${\widetilde{𝒢}}_c$ is the critical value of $J_{\mathit{\text{mech}}}$. Therefore, with $\omega$ known from (38) and with (20) and (22), we can estimate $h$ from an experimentally-measured value of ${\widetilde{𝒢}}_c$ [16, 17], which for the fibrin gel turns out to be $h=28\xi$. The fact that the height of the cohesive zone is large (for example compared to the HRR problem introduced in Appendix A to demonstrate such a scaling) is consistent with the vertical extent of the large deformation region ahead of the crack tip seen in Fig. 6 of Garyfallogiannis et al. [15] and Fig. 6 of Garyfallogiannis et al. [16]. In Fig. 4, $\omega ={𝒢}_c^{\mathit{\text{mix}}}/{𝒢}_c^{\mathit{\text{net}}}$ is plotted as a function of ${\lambda }_{\mathit{\text{crit}}}$ using the constitutive parameters in [16] with $j_c=0.1$. Importantly, $\omega$ is independent of $h$. This plot makes clear that ${𝒢}_c^{\mathit{\text{mix}}}$ can be as much as one-half of ${𝒢}_c$ when the critical stretch for fiber rupture ahead of the crack tip is ≈ 1.25. The contribution of ${𝒢}_c^{\mathit{\text{mix}}}$ to the toughness becomes smaller at large ${\lambda }_{\mathit{\text{crit}}}$ because the stretching energy of the fibers dominates at large stretches.

Fig. 4.

The ratio of fluid contribution to the solid contribution of the toughness is plotted at constant $j_c=0.1$ for various solid volume fractions. At low critical stretches the fluid makes a major contribution to the fracture toughness while at high stretches the solid contribution dominates.

4.4. Toughness of other networks

The analysis above assumes that the number of protofibrils in a fiber $N\propto \pi d^2/4$. It has been shown experimentally that this is not true for thick fibrin fibers in which $N\propto d^{1.4}$ [40, 41]. Therefore, we will relax the assumption that ${\varphi }_s^{\mathit{\text{ref}}}\propto d^2/{\xi }^2$. Also, $c_f$ and $c_u$ will no longer be assumed independent of the fiber diameter $d$ (or $N$). We want to determine how the fracture toughness scales with the fiber length $\xi$ under these new assumptions. Recall that $c_f{\epsilon }_{\mathit{\text{transf}}}$ is the stress (in a fiber) at which the folded-to-unfolded transition occurs. In a real fiber, the unfolding force for a given protofibril is known. Therefore, if there are $N$ protofibrils in a cross-section, we expect that $c_f{\epsilon }_{\mathit{\text{transf}}}=c_f\left({\lambda }_{\mathit{\text{trans}}f}-1\right)\propto N/\left(\pi d^2/4\right)$. Note that $c_u\propto N/\left({\xi }^2\pi d^2/4\right)$ from the second equation in (27) with $L_f$ assumed proportional to $\xi$, so $4/\pi d^2$ can be taken out of the square brackets in (31), and we can write

$${𝒢}_c^{\mathit{\text{net}}}=\frac{32h}{{\xi }^2}\left[A_{1}N\left({\lambda }_{\mathit{\text{crit}}}-\frac{{\lambda }_{\mathit{\text{transf}}}+1}{2}\right)+B_1\frac{N}{{\xi }^2}\frac{{\left({\lambda }_{\mathit{\text{crit}}}-{\lambda }_{\mathit{\text{transf}}}\right)}^2}{\left({\lambda }_{10}-{\lambda }_{\mathit{\text{crit}}}\right){\left({\lambda }_{10}-{\lambda }_{\mathit{\text{transf}}}\right)}^2}\right]+h{\alpha }_1{\varphi }_s^{\mathit{\text{ref}}}\text{log}{j}_c,$$ (40)

where $A_1$ and $B_1$ are some constants. The amount of solid in our cubic unit cell of side $\xi$ depends on the number of protofibrils $N$ and their length $\xi$. If we assume ${\varphi }_s^{\mathit{\text{ref}}}\propto N\xi /{\xi }^3=N/{\xi }^2$ in the expression for ${𝒢}_c^{\mathit{\text{net}}}$ and that ${\lambda }_{\mathit{\text{crit}}}$, ${\lambda }_{\mathit{\text{trans}}}$, ${\lambda }_{10}$ do not depend on $\xi$ (or $d$), then we conclude from (40) that the first term in ${𝒢}_c^{\mathit{\text{net}}}$ scales as

$$A_2\xi {\varphi }_s^{ref}+\frac{B_2{\varphi }_s^{ref}}{\xi },$$ (41)

where $A_2$ and $B_2$ are some other constants that can be computed in terms constitutive parameters just like $A$ and $B$ in (35) and (36). The contribution of ${𝒢}_c^{mix}$ when added to the above gives the result:

$${𝒢}_c=A_2\xi {\varphi }_s^{ref}+\frac{B_2{\varphi }_s^{ref}}{\xi }+B\xi +C\xi \left[\frac{{\left(1-{\varphi }_s^{ref}\right)}^{{\beta }_1}}{{\left(j_c-{\varphi }_s^{ref}\right)}^{{\beta }_1-1}}-\left(1-{\varphi }_s^{ref}\right)\right],$$ (42)

which is a more complex dependence on $\xi$ and ${\varphi }_s^{\mathit{\text{ref}}}$ than (34) and arises due to relaxing some assumptions about the geometry of fibers and the constitutive parameters that enter their force-stretch relations.

5. Energy dissipated due to fluid motion around steadily moving crack

There is a third contribution to the fracture toughness which is the energy lost in the flow of liquid due to crack advance. This contribution comes into play when the crack grows. A linear dependence of ${𝒢}_c$ on $\eta v$ for a gel was experimentally obtained by Baumberger et al. [29], where $\eta$ is the liquid viscosity and $v$ is the steady state crack tip speed. They justified this linear dependence by computing the energy dissipated due to a viscous drag force felt by a polymer chain as it moves through fluid during pull-out at the crack tip. That this dissipative contribution is localized to the crack tip was shown by adding a drop of miscible liquid (to change the viscosity) only to the moving tip and not the rest of the gel. Here we arrive at the linear dependence of ${𝒢}_c$ on $\eta v$ via a poro-elastic calculation.

Since we assume plane-strain conditions, $\mathbf{F}$ everywhere in the gel including along the crack tip, $\mu$ is independent of $X_3$ and there are no chemical potential gradients in the $X_3$ direction. Therefore, fluid flows are confined to the $X_1-X_2$ plane. Again consider the RVE ahead of the crack tip. In the unloaded state it contained fluid of volume $\left(1-{\varphi }_s^{ref}\right)ht\xi$, and just before fibers break it contained fluid of volume $\left(j_c-{\varphi }_s^{ref}\right)ht\xi$. Fluid of volume $\left|1-j_c\right|ht\xi$ flows out of or into the reference RVE, depending on whether the material is tensile-contracting or tensile-dilating, respectively, as it deforms to its current state in which fiber breaking is imminent. If the time required for this deformation is $\Delta t$, then the average nominal flux of fluid $Q$ through the top, front and bottom faces of the RVE of areas $t\xi$, $th$ and $t\xi$, respectively, is given by:

$$Q=\frac{\left|1-j_c\right|ht\xi }{t(h+2\xi )\Delta t}=c_Q\frac{\left|1-j_c\right|\xi }{\Delta t},$$ (43)

where $c_Q$ is a constant. With $h/\xi$ constant, $c_Q=h/(h+2\xi )$. Since the back face of the cube (which faces the bath) moves to the right along the $X_1$ axis at speed $v$, $\Delta t\propto \xi /v$. Then, $Q=c_Q\left|1-j_c\right|v$. The nominal fluid flux is related to the chemical potential gradient in the reference configuration through a Darcy law, $Q_i=-(1/\eta )k_{ij}\partial \mu /\partial X_j$. Here $\eta$ is the viscosity of the liquid and $k_{ij}$ is a permeability tensor. We will take $k_{ij}=k{\delta }_{ij}$, where $k$ depends on the fiber diameter and solid volume fraction; $k=d^{2}f\left({\varphi }_s^{ref}\right)$, where $f$ is a functional form available in literature. For example, Wufsus et al. [42] measured the permeability of fibrin gels of various solid volume fractions ${\varphi }_s^{ref}$ and showed that the following forms of $f\left({\varphi }_s^{ref}\right)$ fit the experimental data quite well:

$$f_1\left({\varphi }_s^{ref}\right)=\frac{1}{16{\left({\varphi }_s^{ref}\right)}^{1.5}\left(1+56{\left({\varphi }_s^{ref}\right)}^3\right)},$$ (44)

$$f_2\left({\varphi }_s^{ref}\right)=\frac{3}{20{\varphi }_s^{ref}}\left(-\text{log}{\varphi }_s^{ref}-0.931\right),$$ (45)

$$f_3\left({\varphi }_s^{ref}\right)=0.50941{\left({\left(\frac{\pi }{4{\varphi }_s^{ref}}\right)}^{1/2}-1\right)}^2\text{exp}\left(-1.8042{\varphi }_s^{ref}\right).$$ (46)

Other forms of $f\left({\varphi }_s^{\mathit{\text{ref}}}\right)$ include the Kozeny-Carman equation for flow through granular beds [43]. With $k$ known the chemical potential gradient can be computed by inverting the Darcy law:

$$\frac{\partial \mu }{\partial X_i}=-\frac{Q_i\eta }{k}.$$ (47)

Now the rate of energy dissipated in reference volume $V_0$ is given in terms of fluid flux and chemical potential gradient as

$$\frac{\partial \Sigma }{\partial t}=-{\int }_{V_0}{Q}_i\frac{\partial \mu }{\partial X_i}dV.$$ (48)

Therefore, the energy dissipated by fluid flow into or out of the RVE of reference volume $ht\xi$ in time $\Delta t$ can be estimated as:

$$\Sigma =\frac{3Q^2\eta ht{\xi }^2}{kv}=\frac{3c_Q^2{\left(1-j_c\right)}^{2}v\eta ht{\xi }^2}{k}.$$ (49)

Dividing by change in crack-face area $t\xi$ and recognizing the dependence of $k$ on $d$ and ${\varphi }_s^{\mathit{\text{ref}}}$, $k=d^{2}f\left({\varphi }_s^{\mathit{\text{ref}}}\right)$, one gets

$$G_c^{\mathit{\text{diff}}}=\frac{3c_Q^2{\left(1-j_c\right)}^2\eta h\xi v}{d^{2}f\left({\varphi }_s^{ref}\right)}=3c_Q^2\frac{h}{\xi }\frac{{\left(1-j_c\right)}^2\eta {\xi }^{2}v}{d^{2}f\left({\varphi }_s^{ref}\right)}.$$ (50)

Since, ${\varphi }_s^{ref}=\alpha d^2/{\xi }^2$,

$$G_c^{\mathit{\text{diff}}}=\frac{3c_Q^{2}h}{\alpha \xi }{\left(1-j_c\right)}^2\eta vg\left({\varphi }_s^{ref}\right),$$ (51)

where $g\left({\varphi }_s^{\mathit{\text{ref}}}\right)={\left({\varphi }_s^{\mathit{\text{ref}}}f\left({\varphi }_s^{\mathit{\text{ref}}}\right)\right)}^{-1}$ is a function of solid volume fraction alone.

Finally, the fracture toughness of the gel can be estimated as

$${𝒢}_c={𝒢}_c^{net}+{𝒢}_c^{mix}+{𝒢}_c^{\mathit{\text{diff}}}.$$ (52)

The first two terms on the right hand side of (52) depend on free energies of network stretching and fluid mixing through (20) and (22). The third term depends linearly on the crack speed $v$ through (51). Each term in (52) is linear in the RVE height $h$, so the ratios ${𝒢}_c^{\mathit{\text{mix}}}/{𝒢}_c$ and ${𝒢}_c^{\mathit{\text{diff}}}/{𝒢}_c$ are independent of RVE size.

Note that our estimate of ${𝒢}_c^{\mathit{\text{diff}}}$ is linear in $\eta v$, where $v$ is the crack tip velocity and $\eta$ is the fluid viscosity. This dependence emerged from the dissipation due to fluid diffusion through the pores of the network. In the work of Baumberger et al. [29, 44, 45], linearity came about due to drag on polymer chains being pulled through fluid at the crack tip. Their experiments showed that ${𝒢}_c$ can be chemi-osmotically manipulated by adding ionic solutions to the crack tip alone and that molecular level information (e.g. binding energies of physical cross-links) can be extracted using crack growth experiments. The linear dependence of ${𝒢}_c$ on $\eta v$ obtained by two different methods may seem coincidental, but they are in fact connected. An explanation is given in Appendix B for how the drag felt by a fiber moving through fluid may be connected to diffusion of fluid through the pores of a fibrous network. At the outset, one notes that both these phenomena involve relative motion between the fibers and the fluid molecules. The key idea is that local mass conservation and Darcy’s law combine to give a diffusion equation for the fluid and the diffusion coefficient $D$ entering this equation can be connected to the drag on a fiber through the Stokes-Einstein relation $D=M{k}_{B}T$ where $M$ is the mobility of a microscopic fiber moving through fluid [46].

6. Conclusion

In this paper, we have described several ways in which fluid in a gel contributes to its fracture toughness, which we have defined as the critical energy release rate. We have shown that the energy release rate can be evaluated in terms of an integral of the free energy density defined on the surface of the blunted crack tip in the reference configuration. Then, the fracture toughness of the gel is evaluated at a critical stretch and and an associated volume change of the network from the surface independent integral $J^{*}$ or, equivalently, in terms of the release of network and mixing energy, ${\Psi }_{\mathit{\text{net}}}$ and a mixing component ${\Psi }_{\mathit{\text{mix}}}$ repsectively, in the RVE directly ahead of the crack tip. This leads to contributions ${𝒢}_c^{\mathit{\text{net}}}$ and ${𝒢}_c^{\mathit{\text{mix}}}$ to the critical energy release rate. The latter most often has been neglected in the literature.

From the critical stretch criterion, fibrin fiber material properties, and the fracture toughness of fibrin gels of various solid volume fractions, the appropriate RVE size can be determined (Fig. 3). Nevertheless, knowledge of the fracture toughness required coupling experimental data, which determines the mechanical contribution, with finite element calculations, which determines the demixing contribution, as outlined in our previous work. Then from (18) with (20) and (22) and the micro-mechanical analyses that followed, the simple estimate of the fracture toughness that includes effects of fluid demixing determines the dependency on the many material parameters that control the toughness of gels. We show that RVE can be a few hundred micrometers tall for fibrin gels in the the direction perpendicular to crack advance, which is consistent with damage zone sizes predicted using finite element calculations and with electron microscopy images of fibers ahead of the crack tip.

The aforementioned fracture toughness estimate required knowledge of the RVE size, which required detailed finite element analyses from our previoius work to determine the demixing contribution. In Sec. 4.3, an alternative estimate is proposed that only requires experimental input on the mechanical contribution ${\widetilde{𝒢}}_c=J_{\mathit{\text{mech}}}$, which can be determined from well-known fracture methodology. Then, with the estimate of $\omega ={𝒢}_c^{\mathit{\text{mix}}}/{𝒢}_c^{\mathit{\text{net}}}$ from the micro-mechanical analysis, the fracture toughness is accurately approximated by (38) when the osmotic pressure and chemical potential have a small contribution to stress, which is the case for fibrin gels [31], but may have to be investigated for other gels. This is the key result from this paper.

We show that ${𝒢}_c^{\mathit{\text{mix}}}$ contributes significantly to the initiation toughness for fibrin gels at low critical stretches (see Fig. 4). Since ${𝒢}_c^{\mathit{\text{mix}}}/{𝒢}_c^{\mathit{\text{net}}}$ is independent of the RVE size $h$, a similar plot can be made for other gels (including tensile dilating gels of [47]) whose constitutive behavior ${\Psi }_{\mathit{\text{net}}}$ and ${\Psi }_{\mathit{\text{mix}}}$ are known. We also have shown how fluid flow around a steadily moving crack contributes to dissipation and leads to a contribution to the toughness that is linear in crack speed and fluid viscosity. Our analysis is based on fluid diffusion through the network. A similar experimental result in earlier works also has obtained a linear dependence on crack speed and viscosity experimentally that was explained by considering fluid drag on a fiber during pull-out at the crack tip. We show that these two views may be connected through the fluctuation-dissipation theorem of statistical mechanics at the microscopic scales.

A. Verification of a simple fracture criterion

The following provides perspective to assess the validity of the simple fracture criteria that is applied in the main text and, in particular, to the RVE that is adopted to estimate the energy release with crack advance. It is shown that the simple expressions used to estimate fracture toughness in the main text also holds for a non-linear elastic material that is a widely adopted constitutive law for a deformation-theory elastic-plastic material. As an illustrative example, a classical $J_2$ power-law material is considered, in which case analytical expressions are known for sharp crack-tip stress and strain fields and the $J$-integral (up to a parameter $I_n$ that is known numerically), assuming small deformations. From those results, the height of the RVE to compute the energy release rate can be estimated using a critical stretch criterion for crack advance at a microstructural distance ahead of the crack tip as shown below.

A.1. Cohesive zone model and critical strain fracture criterion

For a Dudgale-Barrenblat cohesive zone of length $L$ straight ahead of a mode 1 crack in a nonlinear elastic material, the well-known result for the classical J-integral that gives the critical energy release rate as a crack propagates with a critical crack tip opening of ${\delta }_c$ [48,49]:

$$J_c=-{\int }_0^L\sigma (x)\frac{d\delta }{dx}dx={\int }_0^{{\delta }_c}\sigma (\delta )d\delta$$ (A-1)

where $\sigma$ is the normal traction in the cohesive zone and $x$ is measured from the crack tip. Consider a power-law, $J_2$ deformation theory which is widely adopted in analyses of elastic-plastic fracture:

$${\epsilon }_{ij}=\frac{3}{2}{\epsilon }_0{\left(\frac{{\sigma }_e}{{\sigma }_0}\right)}^n\frac{s_{ij}}{{\sigma }_0}$$ (A-2)

where $s_{ij}$ is the stress deviator, ${\sigma }_e=\frac{3}{2}{\left(s_{ij}{s}_{ij}\right)}^{1/2}$ is the von Mises effective stress, and ${\epsilon }_0$, ${\sigma }_0$ and $n$ are material parameters. The corresponding strain energy density is:

$$W=\frac{n}{n+1}{\sigma }_0{\epsilon }_0{\left(\frac{{\epsilon }_e}{{\epsilon }_0}\right)}^{\frac{n}{n+1}}$$ (A-3)

where ${\epsilon }_e={\left(\frac{2}{3}{\epsilon }_{ij}{\epsilon }_{ij}\right)}^{1/2}$ denotes the von Mises effective strain. With $\sigma ={\sigma }_{22}$ and $\delta =h{\epsilon }_{22}$, where $h$ denotes the thickness of the cohesive zone

$$J_c=h{\int }_0^{{\epsilon }_c}\sigma (\delta )d\epsilon =h\left[\frac{n{\sigma }_0{\epsilon }_0}{n+1}\right]{\left(\frac{{\epsilon }_c}{{\epsilon }_0}\right)}^{\frac{n+1}{n}}=h{W}_c$$ (A-4)

where $W_c$ denotes the strain energy density (A-3) at the critical strain ${\epsilon }_c$.

In what follows, we consider representative volume elements in the cohesive zone of volume $ht\xi$, where $t$ is the specimen thickness and $\xi$ is the size of a unit cell, e.g. cubic cell of a fiber network.

A.2. A simple estimate of the energy released when the RVE ahead of the crack tip fractures

Next, consider the simple criterion adopted in this paper for estimating the critical energy release rate. For a nonlinear elastic material, the energy released as an RVE directly ahead of the crack of volume $ht\xi$ fractures at a critical strain ${\epsilon }_c$ with an increase in crack surface area of $t\xi$ is

$${𝒢}_c=\frac{W_c(ht\xi )}{t\xi }=h{W}_c=J_c.$$ (A-5)

For the chosen RVE, it is noteworthy that this simple estimate of the critical energy release rate (A-5) is identical to the cohesive zone model result (A-4) for a power-law, $J_2$ deformation theory. It remains to estimate $h$, which is considered next.

A.3. Estimate of $J_c$ from the HRR field for a critical strain fracture criterion

The well-known strain fields ahead of a crack in a power-law, $J_2$ deformation theory material [50, 51] so called HRR fields, can be represented as [51]

$${\epsilon }_{ij}={\epsilon }_0{\left(\frac{J}{{\sigma }_0{\epsilon }_0{I}_{n}r}\right)}^{\frac{n}{n+1}}{\tilde{\epsilon }}_{ij}(\theta ,n)$$ (A-6)

where $r$ and $\theta$ are crack-tip polar coordinates, ${\tilde{\epsilon }}_{ij}(\theta ;n)$ are known nondimensional functions with ${\left({\tilde{\epsilon }}_e\right)}_{\mathit{\text{max}}}=1$ and $I_n$ is a known constant [51, 52]. For these rigorous solutions the amplitude of the crack tip fields is the $J$-integral. If the fracture criterion corresponding to a critical opening strain ${\epsilon }_{22}={\epsilon }_c$ at a distance $r=\xi$ directly ahead of the crack $(\theta =0)$ yields $J_c$, i.e., the critical energy release rate

$$J_c={\sigma }_0{\epsilon }_0{I}_n\xi {\left(\frac{{\epsilon }_c}{{\epsilon }_0{\tilde{\epsilon }}_{22}(0;n)}\right)}^{\frac{n+1}{n}}=\left[\frac{I_n}{{\left({\tilde{\epsilon }}_{22}(0;n)\right)}^{\frac{n+1}{n}}}\right]\xi W_c.$$ (A-7)

Equating (A-5) and (A-7) for a power law material leads to the estimate of $h$ for a simple fracture criteria analogous to the one adopted in this paper for a hydrogel as

$$h=\left[\frac{I_n}{{\left({\tilde{\epsilon }}_{22}(0;n)\right)}^{\frac{n+1}{n}}}\right]\xi .$$ (A-8)

The term in brackets, $h/\xi$, has values of 8.9 for $n=3$ and 5.6 for $n=13$. This result is for a power-law material, $J_2$ deformation theory material (and small strain solutions). The factor $h/\xi$ will depend on the crack-tip fields for a given material model including large strain effects. Therefore, an appropriate RVE for the simple estimates of the critical energy release rate adopted in this paper for gels are expected to be on the order of the fiber network unit-cell dimension $\xi$ in the direction of crack propagation with a height many times $\xi$ perpendicular to the crack plane.

B. Drag and diffusion

In this appendix, we explain how fluid drag on a fiber making up the network in a gel is connected to the diffusion of fluid through the gel. If $C(\mathbf{X},t)$ is the local fluid volume fraction in a gel, then mass conservation requires that

$$\frac{\partial C}{\partial t}+\frac{\partial Q_i}{\partial X_i}=0,$$ (B-1)

where $Q_i$ is the $i^{\mathit{\text{th}}}$ component of the local liquid flux. The flux itself is governed by Darcy’s law:

$$Q_i=-\frac{k_{ij}}{\eta }\frac{\partial \mu }{\partial X_j},$$ (B-2)

Since $\mu =p-\pi$, where $p$ is the local hydrostatic pressure and $\pi$ is the osmotic pressure, let us assume that the fiber volume fraction is in the semi-dilute regime in which interpenetration of polymer molecules occurs so that the osmotic pressure is given by

$$\pi =\beta RT{\left(\frac{{\varphi }_s}{v_s}\right)}^{9/4}=\beta RT{\left(\frac{{\varphi }_s^{ref}}{j{v}_s}\right)}^{9/4}=\beta RT{\left(\frac{j-C}{j{v}_s}\right)}^{9/4},$$ (B-3)

where $\alpha$ is a constant, $v_s$ is the molar volume of the solid and $j=C+{\varphi }_s^{\mathit{\text{ref}}}$ is assumed since all pores are saturated with liquid.¹ This scaling of the osmotic pressure is due to J. des Cloizeuax [53] and it has been verified experimentally in Cohen et al. [54]. Therefore,

$$\mu =p-\beta RT{\left(\frac{j-C}{j{v}_s}\right)}^{9/4}.$$ (B-4)

To keep the mathematics simpler let us assume that we are in a regime in which $j=j_m=j_c$ and it does not change. Then, plugging the above into the Darcy law,

$$Q_i=-\frac{k_{ij}}{\eta }\frac{\partial p}{\partial X_j}-\frac{k_{ij}}{\eta }\frac{9\alpha RT}{4{\left(j_m{v}_s\right)}^{9/4}}{\left(j_m-C\right)}^{5/4}\frac{\partial C}{\partial X_j}.$$ (B-5)

Next, substitute the flux above into the conservation law for liquid in the gel, then

$$\frac{\partial C}{\partial t}=\frac{\partial }{\partial X_i}\left[\frac{k_{ij}}{\eta }\left(\frac{\partial p}{d{X}_j}+\frac{9\beta RT}{4{\left(J_m{v}_s\right)}^{9/4}}{\left(j_m-C\right)}^{5/4}\frac{\partial C}{\partial X_j}\right)\right].$$ (B-6)

If the hydrostatic pressure $p$ is held fixed or if the hydrostatic pressure gradient is held constant (as in experiments to measure permeability), then the above is a diffusion equation for $C(\mathbf{X},t)$. As before we will take $k_{ij}=k{\delta }_{ij}$ for simplicity then the diffusion coefficient is $D=\frac{k}{\eta j_m{v}_s}\frac{9\beta }{4}{\left(\frac{j_m-C}{j_m{v}_s}\right)}^{5/4}RT$. If the volume of one molecule of the solid is $V_m$ then

$$D=\frac{k}{\eta j_m{V}_m}\frac{9\beta }{4}{\left(\frac{j_m-C}{j_m{v}_s}\right)}^{5/4}{k}_{B}T.$$ (B-7)

The form of this equation reminds us of the Stokes-Einstein relation of statistical mechanics $D=M{k}_{B}T$ where $M$ is the mobility of the solid particle in a liquid [46]. In particular, the drag force on the particle is $f_d=v/M$ where $v$ is the velocity of the solid particle relative to the fluid. $M$ is usully computed in the Stokes limit of the Navier-Stokes equations because the particles whose diffusion is studied are very small. Polymer molecules and fibers making up gels are nanometer to micrometer in length, so Stokes expressions for the drag force (or mobility) are utilized here. By comparing (B-7) above with the Stokes-Einstein formula one infers that

$$\frac{k}{\eta j_m{V}_m}\frac{9\beta }{4}{\left(\frac{{\varphi }_s}{v_s}\right)}^{5/4}=M,$$ (B-8)

where ${\varphi }_s$ is used instead of $C$. For a cylindrical molecule of length $L$ and diameter $d$, the longitudinal mobility is given by

$$M=\frac{\text{log}(L/d)+c}{2\pi \eta L},$$ (B-9)

where $c$ is a constant of $O(1)$. The transverse mobility has a factor $4\pi$ in the denominator instead of $2\pi$. Using the longitudinal mobility $k$ can be calculated from the previous equation as

$$k=\frac{V_m[\text{log}(L/d)+c]}{2\pi L}\frac{4j_m}{9\beta }{\left(\frac{v_s}{{\varphi }_s}\right)}^{5/4}=\frac{d^2}{8}[\text{log}(L/d)+c]\frac{4j_m}{9\beta }{\left(\frac{v_s}{{\varphi }_s}\right)}^{5/4},$$ (B-10)

where $V_m=\frac{\pi d^{2}L}{4}$ for a cylindrical molecule stretched taut. Since ${\varphi }_s=\frac{{\varphi }_s^{\mathit{\text{ref}}}}{j_m}$ and ${\varphi }_s^{\mathit{\text{ref}}}=\alpha \frac{d^2}{L^2}$ the above expression for $k$ is indeed of the form $k=d^{2}f\left({\varphi }_s^{\mathit{\text{ref}}}\right)$, exactly as in Wufsus et al. [42]. In particular,

$$k=\frac{d^2{j}_m}{36\alpha }\left[-\text{log}\left({\varphi }_s^{ref}\right)-c_1\right]{\left(\frac{v_s{j}_m}{{\varphi }_s^{ref}}\right)}^{5/4},$$ (B-11)

where $c_1$ is another constant of order $O(1)$. This expression comes close to (6) of Wufsus et al. [42] which was derived by Jackson and James for Stokes flow parallel and normal to a two dimensional array of cylinders:

$$k=d^2\frac{3}{80{\varphi }_s^{ref}}\left[-\text{log}\left({\varphi }_s^{ref}\right)-0.931\right].$$ (B-12)

Their expression did not consider deformations of a network of fibers which leads to the appearance of $j_m$ in (B-11). Even this dependence on $j_m$ assumes that $j_m$ does not change with position $\mathbf{X}$, which is rather simplistic. In reality, a microscopic picture of fluid transport through gels is more complex than diffusion analyzed through Darcy’s law.

The analysis above exposes the microscopic origins of the dissipation due to fluid flow within a poro-elastic material. This can be a major contribution to the fracture toughness of gels (i.e., $G_1(v)/G_0\gg 1$) when crack speeds are high as shown by Baumberger et al. [29,44].

C. Constitutive parameters for fibrin gels

In Sec. 4, we describe a constitutive model for fibrin gels that is based on the force-extension relation of individual fibrin fibers. The numerical values of the parameters entering this model were obtained from Garyfallogiannis et al. [16] and are tabulated in Table 1.

Table 1.

Constitutive parameters for single fibrin fibers and fibrin gels adapted from Garyfallogiannis et al. [16].

$c_f$	70 KPa
$c_u$	1350 KPa
${\epsilon }_{\mathit{\text{transf}}}$	0.4
${\epsilon }_{10}$	5.0
$G$	1 KPa
${\pi }_0$	1 KPa
${\beta }_1$	1.02