{'fraction_non_alphanumeric': 0.037495786990225814, 'fraction_numerical': 0.019211324570273004, 'mean_word_length': 4.1474627728784155, 'pattern_counts': {'":': 0, '<': 6, '<?xml version=': 0, '>': 5, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 6, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Rupture of blood clots: Mechanics and pathophysiology

We investigate the biomechanical and structural origins of blood clot rupture with implications for thrombotic embolization.

(e) (f) (g) Figure S1: Characterization of fibrin gels. (a) Fibrin gels made from pooled plasma from 25 donors were clotted by the addition of 25 mM calcium chloride and tissue factor (thromboplastin, 1:80 vol:vol dilution). The diameter of fibrin fibers was quantified from scanning electron microscopy; the width of the fiber was determined for > 100 fibers in uncracked, unstretched samples. Average fiber diameter was determined to be 127 ± 35 nm. (b) Pore size was determined in uncracked, unstretched samples by measuring the length of two distances for > 20 spaces between fibrin fibers from confocal microscopy images. Average pore size was determined to be 3.2 ± 0.18 µm. (c) Pore shape ahead of the crack tip as a function of strain. Pore shape was characterized as a ratio of pore height (tensile direction)/pore width (transverse direction) from confocal microscopy images of gels stretched at strains from 0-40%. * shows significance relative to unstrained (0%) sample and # shows significance relative to 10% strain (3mm). Significance was determined using a one-way ANOVA where * p < 0.05, * * p < 0.01, * * * p < 0.001, * * * * p < 0.0001. (d) The pore height does not change from strains 0-40% whereas (e) the pore width decreases with increasing strain. (f) Water expulsion in fibrin gels under tension. Uncracked gels were stretched to varying degrees of extension (% of the original length 30 mm), excised from the tensile tester, and weighed. Final weight of the specimen is plotted as function of the extension. The decrease in the weight is due to water loss. (g) Exercise identical to (d) is repeated for cracked specimens. Here the specimens are stretched until the crack becomes unstable leading to rupture. The ruptured specimens are then weighed. The figure shows the final weight of the specimens as a function of crack length (which is plotted as % of original dimension 30 mm).

Constitutive modeling of fibrin gels
Fibrin hydrogel is a biphasic material comprising a solid mass-a fibrous network-filled with liquid. Much like rubber or other polymeric materials, fibrin gels can be stretched to more than three times their original length with little hysteresis in quasistatic experiments. However, there are certain aspects of fibrin mechanics that are different from rubber. While rubber is a dense network of cross-linked polymer chains, fibrin gels are a porous network of relatively stiff branched fibers. As a result, fibrin fibers buckle [36] under compression. Buckling (in the direction transverse to tension) and subsequent bundling of fibers results in expulsion of fluid from fibrous gels [35], hence they cannot be viewed as incompressible like rubber [17,5].
Finally, fibrin fibers are built from protofibrils that can undergo an 'unfolding transition' at large forces which results in high extensibility of fibrin gels [17,9,10] and also contributes to bundling and water expulsion. For these reasons, we use a modified neo-Hookean model as the constitutive description of the fibrin network in our gels. We propose a network model based on chemoelasticity [28,29,35] which accounts for the three key microstructural details mentioned above and the interplay between the solid and liquid components. Our model is a nonlinear elastic model with an additive decomposition for the energy density functional Ψ: where, φ S ≤ 0.01 is the fraction of the solid mass, Ψ s and Ψ l are the energy density functionals for the solid and liquid phases, respectively. Next, we discuss the modeling of the solid and liquid phases separately.
Solid mass Ψ s : The solid mass consists of an isotropic network of fibrin fibers (diameters in the range of 100-200nm [17]). These fibers consist of protofibrils which are comprised of linearly arranged fibrin monomers. The fibrin monomers have several regions -a triple helical part which can undergo an α-helix to β-sheet transition at large forces, a globular γ-nodule that can unfold at forces smaller than the triple-helical coiled-coil and disordered αC regions which can straighten or stretch under forces [10,9]. To take account of these phenomena in the force-extension response of fibrin protofibrils we assume that they are capable of undergoing a transition from a 'folded' state to an 'unfolded' state. The 'folded' state of the protofibril is characteristic of low forces in which it has a linear force-extension response corresponding to reversible elongation. The 'unfolded' state is characterized by forces large enough to stretch the αC regions from their disordered blob like state or even unfold the globular γ nodule so that it is best described by a non-linear force-extension relation for polymers. The protofibrils comprising a fiber are covalently connected by factor XIIIa so that there is no slippage of protofibrils. The isotropic network is generated by the branching of individual fibrin fibers.
Due to the reasons explained above we decompose the stress response of the solid phase into two components: a nearly incompressible neo-Hookean energy functional Ψ h and a free energy functional accounting for unfolding and buckling Ψ f , For the isotropic fiber network in the fibrin gel [17], we consider an N − fiber model [31,35].
Respecting isotropy, we assume that the unit vectors along the fibers are distributed uniformly on the surface a unit sphere whose directions are registered using M p , where 1 ≤ p ≤ N . In the interest of continuity, we postpone the choice of N until later. The generalized structure tensors H p in the reference configuration are, where 0 ≤ κ ≤ 1/3, characterizes the von-Mises distribution of the fibers (for details see sec. [31] ). The strain-energy contribution from these N fibers is given by,

in Gasser et al
Here, E p = H p ij C ij − 1 is the strain in p th fiber, α(E p )-the stored energy function-is identical for each fiber. The resultant Cauchy stress σ f is: where, The protofibrils in each of the N fibers undergoes an unfolding transition upon stretching [17].
We distinguish between two states, viz., 'folded' and 'unfolded' (see above), using the magnitude of strain E p : E p < 0.1 implies folded state, while E p ≥ 0.1 implies unfolded. This is a simple way of accounting for the two states of the protofibrils, but it results in good fits to the forceextension curve of the gels as shown later. We have different constitutive laws for the folded and unfolded states of the protofibrils and these are inherited by the constitutive laws for each fiber. In the folded state, g(E p ) = g f (E p ) is, where, x = x if x > 0, and 0 otherwise. Thus, the fibers in compression are assumed to not contribute to the energy since they buckle. For the unfolded state, g(E k ) = g u (E k ), which is based on the worm-like chain model for polymers [17]. We will fit the parameters c f , c u and E 10 in the following section using the experimental data from a uniaxial tension test. Note that the parameters c f , c u and E 10 are identical for all N fibers.
The strain energy for the neo-Hookean component is, where, I 1 = tr (FF T ), J = det F, and c, m and a are the material parameters. The corresponding Cauchy stress σ is, where B = FF T , and I is the identity matrix. We need the neo-Hookean contribution for the stability of our finite element calculations presented later.
Fluid mass: The solid mass is merely φ S = 0.01 of the entire volume. The remaining volume is occupied by fluid. We assume the fluid to be incompressible and inviscid, but it can flow in response to stresses and changes in the chemical potential µ. The expulsion of this fluid is a crucial factor impacting various aspects of constitutive modeling of the gel. The chemical potential itself depends on the fluid (pore) pressure p and the osmotic pressure π through µ = p − π [35]. We assume that the fluid (or solvent) chemical potential in the gel is equal to the fluid chemical potential outside because our specimen is thin, the pores are large and k = 0.7µm 2 is obtained using Davie's formula given in Wufsus et al. [37] who measured fibrin gel permeabilities and fitted experimental data to various formulae. We used φ s = 0.005 and hydrodynamic radius of fiber =60 nm in Davie's formula (the actual φ s in our gels could be smaller, implying higher k and larger fluid velocity v). Thus, the fluid will move through the thickness of 6 mm in about 40 seconds. This is fast compared to the time of 10-20 minutes over which the maximum tensile stress of 1.11 × 10 3 Pa is reached, so our assumption that the fluid pressure is the same in the gel and outside is justified. Another way to show that poroelastic effects can be neglected is to use the length scale R p = √ Dt e from Long and Hui [38]. Here D is the effective diffusion coefficient of the solvent molecules in the gel and t e is the loading time to initiate crack propagation. Assuming the solvent is water, D ≈ 1670 µm 2 /sec (using the Stokes-Einstein formula D = k B T 6πµr where r is radius of solvent molecule) and t e = 600 sec, which gives R p ≈ 1mm. This is small compared to the 6.5mm thickness of our samples and much smaller than the 30mm length and width. On the basis of the above two estimates we neglect poroelastic effects in our computations of the fracture toughness of fibrin gels. If osmotic pressure is present then v = M ∇µ where M is a mobility [16]. If we assume that the chemical potential of the fluid in the gel quickly becomes equal to the chemical potential of the fluid outside then ∇µ is zero in the gel. Thus, the fluid (pore) pressure in the gel is equal to the osmotic pressure π everywhere. Thus, the fluid pore pressure which enters the expression for the Cauchy stress can be replaced by the osmotic pressure in the expression. The osmotic activity in the fibrin gel is accounted for by the energy functional term Ψ l = Ψ l (J) resulting in an isotropic stress Following Ehret et al. [35], who performed experiments on collagen gels (which are isotropic networks like fibrin networks, but do not have protein unfolding) which show reduction in volume upon stretching, we use the following expression for the osmotic pressure π(J): The constants β 0 , β 1 will be chosen to fit the experimental data on fibrin gels. Based upon these contributions, the final expression for Cauchy stress in our gel is, Note that our expression for the Cauchy stress depends only on current values of the deformation gradient. The last term in the expression above is the nothing but the pore pressure of the fluid.
Thus, our material constitutive response is purely elastic.
We use N = 14 in the computations, henceforth. The 14 fiber cubic unit cell is shown in fig. S2 (a). Each edge of the cube is aligned with the principal directions of the right Cauchy-Green strain tensor. Note that if we did not use the 8 body diagonal fibers then our cube will be analogous to the 3-chain model for isotropic polymers; if we did not use the 6 fibers from the center of the cube to the centers of the faces then our cube will be analogous to the 8-chain model for isotropic polymers [32]. Wu and van der Geissen [33] showed that the uniaxial tension response of a full isotropic network model of polymers integrating over all directions on the unit sphere lies in between the force-extension response predicted by the 3-chain and Symbol Description φ S The volume fraction of solid mass. φ S = 0.01 t Thickness of the specimen a = 6.5 mm a Crack length Osmotic pressure c, ca neo-Hookean moduli g, g f , g u Moduli of the fibers κ Von-Mises distribution parameter H p , h p Structure tensors (see eqn. 3 and 6 ) β 0 , β 1 Parameters in the osmotic pressure model (eqn. 12)

Analytical solution for a long cylindrical specimen
In this section, we consider a uniaxial tension test on a cylindrical specimen of fibrin gel. The aim of this section is to demonstrate how the above formulation accounts for three crucial features: i) the unfolding transition of the protofibrils in fibers, ii) the volume decrease while stretching, and iii) the alignment of fibers in the tensile direction. For uniaxial tension the deformation gradient F is a diagonal 3 × 3 matrix such that, Here, the strain in the fibers E p is The elastic energies stored in solid phase Ψ s = Ψ f + Ψ h where, The osmotic pressure contribution is π(J) = α 0 , the Cauchy stress stress σ, which is also diagonal is, Let us assume the uniaxial tension is along e 1 direction and the boundaries are traction free.
This entails that for a fixed stretch λ 1 , where the expressions for σ k , k = 2, 3 can be written down from eqn. (17). We use MATLAB to solve these equations for λ 2 and λ 3 . Thus, we can gain insight into the effects of various model parameters before performing finite element calculations for complex geometries, such as, cracked specimens. We define the order parameter χ to quantify the alignment of the fibers in terms of λ max = max(λ 1 , λ 2 , λ 3 ) and λ min = min(λ 1 , λ 2 , λ 3 ) as follows [17]: Note that in the reference state when λ 1 = λ 2 = λ 3 = 1, then χ = 0, indicating no alignment.
As the the specimen is stretched, λ min → 0 and χ → 1, beckoning perfect alignment. We explore the effect of various parameters on the σ 11 − λ 1 , J − λ 1 ,and χ − λ 1 curves in fig.S3. The parameters used here are in the neighborhood of those that give the best fit to the experimental data in fig. 3A of the main text. It is apparent that the results are not very sensitive to c f and c u , but they depend strongly on E 10 and κ. E 10 is linearly related to the ratio of the contour length of a monomer in the unfolded state to that in the folded state [17]. κ is a parameter that connects the individual fiber response to the network response -0 ≤ κ ≤ 1 3 , as discussed earlier in the context of the structure tensor (see Gasser et al. [31] for details). Recall that the 14 fiber symmetric unit cell sums up the contributions from these fibers to yield an overall isotropic response for the fibrin gel. We will show later that force-extension response also depends strongly on the fibrin volume fraction φ S .

Boundary value problem
Here, we discuss the boundary value problem solved in ABAQUS finite element software (see fig.S2(b)). We assume that the tensile experiments on our fibrin gels are quasi-static; this implies that any dependence on time drops out of the governing equations. We pull our specimens at 3 mm/min equivalent to a strain rate 3 30×60 = 1.67 × 10 −3 /s. Brown et al [17] use a pulling rate of 10 mm/min on 60 mm long cylindrical specimens, and compare the resulting force-extension curve to a quasi-static force extension curve obtained by incrementally straining an identical specimen. The comparison between the two force-extension curves reveals that the strain rate of (a) Figure S3: Results for uniaxial test from the MATLAB calculations. Fig. (a) a is computed such that σ 11 = 0 when λ 1 = 0. These values were obtained from a fit of the model to the uniaxial tension data of an uncracked specimen as discussed in section 6. 60×60 = 2.7 × 10 −3 /s is almost quasi-static. Our strain rate 1.67 × 10 −3 /s is smaller. To further justify our assumption of quasistatic response (hence, no rate effects), we conduct experiments at two different pulling rates and report the force-extension curves in fig.S2(c). The curves lie almost on top of each other, which precludes any rate-dependence. The conservation of linear momentum assuming that the distributed body force is zero is: where P is the First Piola-Kirchoff stress. The associated boundary conditions (see fig. S2(b)) are, where u i is the displacement in e i direction, i = 1, 2, 3, and N is the normal to the boundary in the reference configuration. We assume that the chemical potential is constant throughout the specimen. This together with the slow quasi-static pulling rate ensures that the remaining governing equations are trivially satisfied.

Computation of the tangent stiffness
The expression for the Cauchy stress derived in sec.2 is, The tangent stiffness C ijkl needed for Abaqus UMAT is computed as, where τ = Jσ is the Kirchhoff stress. The following derivatives are needed, Using these, we get, Finally, 6 Results for a uniaxial tension test We use a 20-node brick element (C3D20) in ABAQUS to perform the FEM calculations.
There are various parameters in our model that need to be fit to the experimental data: c, c f , c u , E 10 , β 0 . We use the experimental results for the force-extension curve ( fig. S4(a)) to fit these parameters. A uniaxial tension test is performed on an uncracked specimen pulling it at 3 mm/min, equivalent to strain rate of 3 30×60 = 1.67 × 10 −3 /s. c f = 4.4 × 10 4 N/m 2 , c u = 2.6 × 10 6 N/m 2 , E 10 = 5.0, β 0 = 1.1 × 10 3 give the best fit (r 2 = 0.997) to the experimental data (see fig.3A in the main text). c is chosen such that the reference configuration is stress-free. We use our model to record the change in the volume of the specimen, and we find that it indeed decreases as the specimen is stretched, in qualitative agreement with the experimental data points (inset in fig.3A in the main text); the reason for the mismatch between the experimental and numerical values will be given shortly. Fig.S4(a) shows the fraction of fibrin monomers in the 'unfolded' state in the deformed specimen. We note the excessive lateral contraction due to water release by comparing the deformed shape obtained from our model to the one obtained using a neo-Hookean solid with no fibers ( fig.S4(b)). This figure shows a prominent effect of water drainage on the shape of our deformed specimens. Fig.S4(c) shows the surface plot of fiber alignment χ. Next, we use the above parameters to compute the force-extension curve for the specimen without the fibers by setting c u = c f = 0 ( fig.S4(d)).
A visual comparison reveals that a majority of load-bearing capacity emerges from the elastic fibers, not the neo-Hookean solid.
The water loss measured in the experiments is greater than our predictions (see inset in Fig.3A of the main text). This is because to measure the water loss, we took the initial weight of the specimen minus the final weight. Water is lost (a) while we transfer the sample from the mold to the tensile testing machine, (b) while we set up and perform the experiment, and (c) when we transfer the specimen to the weighing machine to measure the initial and final weights.
On the other hand, our FEM model only accounts for the water pushed out due to changes in hydrostatic and osmotic pressures (see eqn.12) as the specimen is stretched. More water is inevitably lost in handling the experimental samples.
To establish the usefulness of fitted parameters, we perform three tests i) compare predictions from the model to previously published uniaxial tension test data at a higher fibrin volume fraction, ii) predict the force-extension response of whole blood clot specimens, and iii) predict the force-extension response of cracked specimens with cracks of various lengths. For i) the experimental data for a cylindrical specimen of height 50 mm and diameter 4 mm is available in Brown et al. [17]. We performed FEM simulations for this specimen and were able to capture experimental data by choosing φ S = 0.075 (> 0.01 for plasma samples in this work) and κ = 0.05, while keeping all the other constants unchanged (see Fig.S4(e)). Note that these parameters give a relatively good prediction (r 2 = 0.9938) for the force-extension response even though the geometry of the specimens here and in Brown et al. are different. The low value of κ is expected because most of the load is carried by fibers aligned with the axis of the cylinder.
For ii) we refer the reader to Fig. S4f. The force-extension curve of the plasma clots is predicted by the same parameters as in Fig. 3A of the main text (in which φ S = 0.01) as shown by the black curve in Fig.S4f. Since the stored energy function of the gel, Ψ, is linear in the solid volume fraction φ S we expect that the nominal (first Piola) stress will also be linear in φ S .
Then, since the geometry of the plasma and whole blood clot specimen is the same the force should be linear in φ S for all values of strain (and extension). Thus, if the same values of c u , c f , E 10 etc., apply to both plasma and whole blood clots then we should be able to scale down the plasma clot's forces and obtain the whole blood clot curve. This is indeed what happens, as shown by the blue curve in Fig.S4f which is obtained by scaling down the plasma force by a factor of 3.6 so that it sits right on top (r 2 = 0.9983) of the red curve for the whole blood clot upto the initiation of failure. Thus, all the same parameters c f , c u , E 10 , etc., can predict the force-extension response of whole blood clots, except φ S = 0.01/3.6 = 0.0028. Lastly, for iii) we show in the next section that the same parameters can predict the force-extension response of specimens with cracks of four different lengths with r 2 > 0.99.
The sensitivity of the force-extension curve to changes in the fitted parameters c f , c u , E 10 and κ for identical cuboidal specimen is documented in fig.S4 (g-j). Here, the superscript 0 denotes the fitted values in fig. 3A of the main text. Just as we showed with MATLAB calculations in fig.S3 the primary implication here is that the force-extension curve varies in a smooth fashion as the values of parameters are varied. The parameter E 10 seems to have the most effect. The effect of parameters c, β 0 , β 1 is not shown here because (a) most of the Figure S4: Results for uniaxial tension test from finite element calculations. (a) Deformed shape and the unfolded fraction. (b) Comparison between the deformed shapes for an incompressible neo-Hookean and our model for identical boundary conditions. (c) The fiber alignment in the loading direction computed using eqn.19. (d) The force-extension curves for the tensile test specimen with and without fibers. This figure shows that a majority of load-bearing capacity comes from the elastic fibers. (e) FEM simulations for blood clots with higher solid volume fraction and cylindrical geometry. The experimental data is taken from Brown et al. [17]. To fit the data we changed the volume fraction φ S from 0.01 to 0.075 and κ from 0.225 to 0.05 while all other parameters were kept the same. (f) The blue force-extension curve is obtained by scaling down the black curve for plasma clot forces by a factor of 3.6 and it sits right on top of the red (experimental) curve for whole blood clots. (g-j) Sensitivity of the force extension curve to the fitted parameters using FEM simulations. The symbols with superscript 0 denote the fitted values from fig. 3A in the main text. load is carried by the fibers represented by g(E p ) in the eqn. (13) for Cauchy stress σ, not the neo-Hookean matrix whose stiffness is proportional to c (see Fig.S4d), and (b) the osmotic pressure, which is quantified by β 0 and β 1 , is too small to play a major role (see Fig.S7).

Critical energy release rate calculation using an edge crack specimen of a fibrin gel
Here, we focus on computing the critical energy release rate G c for the fibrin gel. We will repeat our computations for G c using various specimen geometries and crack lengths to establish certainty.
Our first specimen geometry consists of a traction free edge crack (see fig.S5(c)). The cracked specimen is held fixed on one end and pulled at a quasi-static rate on the other-for a formal definition of the boundary conditions see sec. 4. Our objectives in this section are two-fold: i) we want to validate the model developed in the previous section, and ii) compute the critical energy release rate for the fibrin gel. Since, our model relies on fitting various parameters, we need to validate it. We do so by computing the force-extension curve for cracked specimens using the parameters obtained by fitting to the uniaxial tension test and comparing it to the force-extension curve extracted from the experiments. A good agreement here validates our fitted values.
Our second focus is the computation of critical energy release rate G c , defined as, where E is the elastic energy of the specimen, t is the thickness of the sample, a is the crack length, a * is the crack length at which G c needs to be evaluated and ∆ * c is the displacement at which the crack with length a * starts propagating. To compute G c for a sample with cracklength a * , we evaluate the above derivative numerically, where δ = 0.3 mm is a small change in the crack length. ∆ * c , the displacement at which the crack of length a * starts propagating, is obtained from the experimental force-extension curves by looking at the displacement corresponding to the maximum load. E(∆ * c , a * ± δ) are the elastic energies of the specimen with crack-length a * ± δ, respectively, stretched to a displacement ∆ * c . We use our model to compute E(∆ * c , a * ± δ). We start by discussing the results for the edge crack specimen with crack length a * = 3.3 mm.
First, we compute the force-extension curve for the cracked specimen using the parameters fitted from the uniaxial test, and get excellent agreement (r 2 = 0.9992) with the experimental data (see fig.S5(a)). In fact, the force-extension curves for specimens with other crack lengths also agree reasonably well with FEM calculations with the same constitutive parameters. This ensures that our fitted paramters are reliable. Next, we proceed on to compute the critical energy release rate. The elastic energies at three crack lengths: a * − δ, a * , a * + δ are plotted, then, eqn.28 is used to compute G c (see fig.S5(b)). We find that G c = 4.44 J/m 2 . Fig.S5(c) and (d) show the unfolding fraction and the fiber alignment for the entire deformed specimen, respectively. The unfolding begins at the crack tip, and spreads to other regions of the specimen, as the specimen is stretched. Our model predicts a strong fiber alignment and unfolding near the crack tip (see fig.S5(e)). We show the magnified 3D view of unfolding and fiber alignment in fig.S5(g) and (h) and observe that they are both concentrated around the crack tip. Note that the traction free surfaces of the crack still have unaligned fibers and the unfolding fraction there is near zero. In fig.S5(i), the areal density of the fibers is computed using Nanson's formula: nda = JF −T NdA, where n, da and N, dA are the normal vector and area element in deformed and reference configurations, respectively. Assuming that the fibers are uniformly distributed in the reference configuration, the number of fibers per unit area ξ ∝ 1 |da/dA| . Fig. S5(i) shows that ξ exhibits a sharp increase near the crack tip which is consistent with the experimental observations using confocal microscopy (see main text).
We also present the results for crack lengths a * = 5.4, 7.2 and 10.5 mm in fig. S5 (j-q).
Excellent agreement is observed between the force-extension curves from experiments and FEM simulations (r 2 values 0.9987, 0.9907, 0.9916, respectively) done using the same set of parameters that were originally fitted to the uniaxial tension test. This ensures that our model is capable 8 Fracture mechanism-critical stretch in the fibrin fibers In our FEM simulations, we hold one end fixed and apply a displacement u 2 on the other end of our cracked specimen. Recall that, u 0 is the displacement at which the crack starts to propagate. Here, we claim that when u 2 = u 0 , the strain close to the crack tip reaches a critical value leading to rupture. Contrary to the displacement u 0 which is different for different crack lengths, this critical strain should be independent of the crack length. We use the logarithmic strain tensor [30] defined as, where λ i are the principal stretches, and n i are the eigen-stretch directions in the current con- To test our claim, we record E 2 = ln λ 2 values at a distance 100 µm away from the crack tip. The results for different crack lengths appear in fig. 5 of the main text. Note that for different crack lengths E 2 is nearly constant 100µm ahead of the crack tip. The average of the E 2 data = 1.04 is used as the critical rupture strain. Furthermore, the strain profiles ahead of the crack tip just before crack propagation starts seem to be independent of the crack length. This is consistent with the nearly constant crack opening angle observed in experiments just before crack propagation begins. Crack tip opening angle was determined using the ImageJ angle tool for measurement.
The angle was approximated at the center of the crack tip right before the onset of rupture (critical strain) as in fig. 5 in the main text.
9 Critical energy release rate calculation using a center crack specimen We concluded in the previous section, that the crack begins to propagate when the strain E 2 at the crack tip reaches a critical value of 1.04. Here, we use this fact to compute the energy release rate for a center crack specimen shown in fig. S6(a) and compare it to the values obtained using the edge crack specimens in sec. 8. The specimen is 30 mm ×30 mm ×6.5 mm, with a horizontal 10 mm crack symmetric about the center. We stretch the specimen until E 2 at 100 µm from the crack tip reaches the critical value of 1.04. For symmetry, we use the upper left half of the specimen. The unfolded fraction plotted on the deformed shape of the specimen is shown in fig.S6 (a). Here too, the unfolding originates from the crack tips. We use eqn.28 to compute the critical energy release rate and find G c = 5.7 J/m 2 which is similar to the values obtained using the edge crack specimens in sec. 8.

Critical energy release rate without osmotic contribution
We compute the critical energy release rate without the stress-induced water release. To eliminate the osmotic pressure contribution, we set β 0 = 0 which gives π = 0. We do not change any other parameters and use the fitted values in sec. 6. We find that force-extension curve for a uniaxial tension test on an uncracked specimen resembles closely the experimental values even for β 0 = 0, however, the change in the volume of the specimen is smaller as shown in fig.S7 (a,b). Using β 0 = 0, we find that the FEM force-extension curve for the cracked specimen with crack-length a * = 3.3 mm is also quite close to the experimental values. The critical energy release rate for no water release-G c = 4.90 J/m 2 from fig. S7(d)-is comparable to its value with water release-G c = 4.44 J/m 2 from fig. S5(b). This shows that the value of the critical energy release rate is largely independent of the effects of stress-induced water release.
11 Effect of the fibrin volume fraction φ S on G c The fibrin volume fraction φ S can vary over a wide range-from less than 0.01 in in vitro blood clots to larger than 0.4 in in vivo thrombi and thrombotic emboli. In fibrin sealants φ S ∼ 0.2 [6,7]. Consider the force-extension curves of fibrin gel prepared from whole blood (red) compared to platelet-free plasma (blue) shown in fig. 6E of the main text. The latter exhibits a stiffer response because it has a higher fibrin volume fraction (φ S ). Notably, both the gels rupture at a similar critical displacement meaning that fibrin is the main determinant of rupture resistance of blood clots. In this section we examine the effect of fibrin volume fraction φ S on the critical energy release rate G c of the hydrogel. The constitutive model developed in sec.2 is used to compute the force-extension curves for various values of φ S as shown in fig. 6A of the main text. Unsurprisingly, an increase in φ S leads to stiffer fibrin hydrogels. To compute the critical energy release rate, we use the fracture criterion found in the previous section-the sample is stretched until the strain 100µm ahead of the crack tip reaches the critical value E 2 = 1.04 found in sec. 8. The resulting strain profiles are depicted in fig. 6B of the main text. The self-similarity of the strain profile ahead of the crack tip (just before crack propagation begins) is evident. The critical energy release rate is then computed using eqn.28. It is observed that G c increases linearly with φ S as shown in fig. 6C of the main text. This is in contrast to G c ∝ φ 2/3 S mentioned in [14] which is traced back to the work of Lake and Thomas [40]. In our fibrin hydrogels the fibers that rupture at the crack tip are aligned in the direction of the tensile force (in experiments, as seen in Figure 1 of the main text, and the finite element model, as seen in Figure 4 of the main text), not randomly oriented in space as the polymer chains of a swellable elastomer. This may partly explain why G c ∝ φ S ; a thorough study of this dependence must take account of the variation in solid volume fraction in the current configuration of the gel (since J varies as a function of position in the loaded sample). The fact that G c increases with increasing fibrin volume fraction may partly explain why reduced contraction of blood clots resulting in a lower fibrin density correlates with the increased propensity of venous thrombi to embolize [26].

Relevance of fracture results to thrombi in vivo
Our experiments and finite element calculations indicate that in 30 × 30 × 6.5mm specimens an edge crack of length 10 mm begins to propagate when the applied tensile force reaches 0.2N.
This corresponds to a nominal tensile stress around 0.2 30×6.5×10 −6 ≈ 1000Pa. We wish to estimate how constricted a blood vessel should be in order that shear stress τ acting on a thrombus attached to the wall reaches 1000 Pa. Recall that the maximum tensile stress (largest principal stress) in the thrombus, which is responsible for crack growth in mode I, is equal to τ . Consider the carotid artery with diameter D = 5mm. The shear rate (of blood flow) in this vessel is aroundγ = 400 s −1 . Assuming that the viscosity of blood is µ bl = 4 × 10 −3 Pa.s the shear stress τ = µ blγ = 1.6P a. On a specimen of width 30 mm and thickness 6.5mm this corresponds to a tensile force about 0.0003 N. Thus, a nominal stress of 1.6 Pa is too low to grow a 10 mm crack in our fibrin gel specimens. If the wall shear stress was about 650 times larger then it would be sufficient to grow a 10 mm crack in our specimens. To get larger shear stresses τ we need higher shear rates, which can be achieved in a constricted blood vessel with diameter D b < D. For a cylindrical pipe of radius R with fully developed laminar flow, the flow rate Q = vmaxπR 2 2 and τ = µ bl vmax R where v max is the fluid velocity in the center of the pipe. Hence, τ = 2µ bl Q πR 3 . Since τ ∝γ, it is clear thatγ ∝ R −3 if Q is held fixed. Thus, if D b D ≈ 0.11 then a 600 times higher shear rate of 240,000 sec −1 will be achieved in the carotid artery which would be sufficient to cause crack growth. This shear rate is in the region of 80,000 sec −1 measured by Colace et al. [39] in microfluidic devices to cause embolization of platelet-fibrin deposits. Their channel was 90-95% occluded under constant flow rate conditions when embolization occurred, which is consistent with D b D ≈ 0.11. This calculation shows that our measurements and modeling of the toughness of fibrin gels are medically relevant.
(a) Figure S8: Scanning electron microscopy image of the surface of an arterial thrombus representing the non uniformity of thrombi that form in vivo.