Tear resistance of soft collagenous tissues

Fracture toughness characterizes the ability of a material to maintain a certain level of strength despite the presence of a macroscopic crack. Understanding this tolerance for defects in soft collagenous tissues (SCT) has high relevance for assessing the risks of fracture after cutting, perforation or suturing. Here we investigate the peculiar toughening mechanisms of SCT through dedicated experiments and multi-scale simulations, showing that classical concepts of fracture mechanics are inadequate to quantify and explain the high defect tolerance of these materials. Our results demonstrate that SCT strength is only modestly reduced by defects as large as several millimeters. This defect tolerance is achieved despite a very narrow process zone at the crack tip and even for a network of brittle fibrils. The fracture mechanics concept of tearing energy fails in predicting failure at such defects, and its magnitude is shown to depend on the chemical potential of the liquid environment.


Analysis of failure behavior
Crack analysis based on a continuum model representation inevitably leads to a problem of singularity of deformation measures at the crack tip. This is obviated in fracture mechanics analysis through the quantification of energy release rate and J-integral [7][8][9] , or through the introduction of a characteristic length scale for the assessment of the field quantities at a specific distance from the crack tip (e.g. ref 10 ).
Such workarounds are not needed for the hybrid modeling approach, which inherently incorporates a failure criterion and a characteristic distance associated with the network properties. General agreement with experimental observations supports the reliability of the model, which is thus qualified as a useful tool for investigation of factors (i.e. fiber network properties) potentially affecting the defect tolerance of  Fig. 2c) affects the critical stretch λ F and the tearing energy Γ only for DNM radii less than 100µm ( Supplementary Fig. 2d). This justifies the selection of a DNM radius of 150µm in all simulations (if not specified otherwise). The influence of the mean fiber length L c in the DNM on Γ and λ F indicates a significant effect of this parameter, with a tougher response for larger L c (Supplementary Figs. 2e,f). Note that the crosslink density was adapted with varying L c so that the macroscopic mechanical response remains relatively unaffected. The relative reduction in critical stretch associated with the presence of a notch seems less sensitive to variations of L c . A similar analysis was performed for the influence of fiber behavior, i.e. ε c and ε s (Supplementary Figs. 2g,h). Γ as well as λ F of notched and intact samples change significantly when the critical fiber strain is increased for the same slackness. Likewise, for the same critical fiber strain ε c =0.35, a reduction of slackness results in higher Γ and lower λ F values. Interestingly, the notch sensitivity, observed as the difference between notched and un-notched samples, increases for larger differences between critical and slackness strain ( Supplementary Fig. 2h).
All variations in the DNM parameters (L c , ε c and ε s ) as well as the failure criterion (i.e. number of failed fibers) were shown to influence the absolute values of Γ in mode I fracture simulations. On the other hand, the characteristic dependence of the apparent tearing energy on initial sample length L 0 is comparable ( Supplementary Fig. 2i) with a pronounced reduction of Γ a for lower values of L 0 . Correspondingly, the reduction of critical elongation in samples with small central defects (normalized with respect to intact samples) is shown to be only modestly dependent of the parameter selection ( Supplementary Fig. 2k).
These results indicate that substantial weakening is expected in SCT for crack sizes in the range of mm, while typical defects in soft collagenous membranes have dimensions of hundreds of microns, as reported in ref 11 for human amnion and shown for a representative defect in a GC sample in Supplementary  Supplementary Fig. 3a shows that the critical elongation is constant for c>5 mm, while it increases for smaller values of c. Obviously, for c=0 mm the critical elongation of an intact sample is measured. Similarly, a constant value of Γ a is computed for c>5 mm (Supplementary Fig. 3b) and values are larger for smaller notch sizes, in line with an increasing value of the critical elongation. The validity of the results obtained with c>10 mm in the present investigations is thus confirmed. In fact, for c<5 mm false values of λ F and Γ a would be measured, as the geometrical criteria indicated for a mode I fracture tests are not satisfied, cf. ref 12 . Nearfield size is determined at under-critical loading states, i.e. just before crack propagation, by splitting up the x-axis in intervals of 10 µm, calculating maximum fiber strains for each interval and characterizing the nearfield as the region where the maximum fiber strain of an interval is larger than 105% of the maximum fiber strain in the stabilized farfield. The results show that for a notch size c<5 mm both peak fiber strain as well as nearfield size increase, while for c>5 mm they are practically independent of notch size (Supplementary Fig. 3c). On the other hand, in line with the results reported in Fig. 1, the nearfield size depends on L 0 , and, similarly to Γ a , it increases with a larger initial sample length.
The fiber compaction mechanism at the notch tip is associated with the network kinematics in a uniaxial tension state, as resulting from the very compliant response of the fibers in compression (governed through the fiber model parameter k 0 ) and very stiff in tension (k 1 ). We hypothesized that the nearfield behavior of the network would change significantly if the fiber response would display the same stiffness in tension as in compression, i.e. if k 0 =k 1 . Corresponding hybrid model simulations of mode I fracture tests were performed, and included a continuum model with adapted parameters so to provide a mechanical response similar to that of the modified DNM. Results are reported in Supplementary Fig. 4.
Comparison with Supplementary Fig. 1e and with Fig. 2d indicates that the nearfield size is about 4 times larger than in the reference case and that the characteristic sample length for transition to flaw insensitive response is significantly smaller. The linear fiber response leads to a nearfield size (0.4-0.8 mm) that is comparable to the critical flaw dimensions (∼0.3 mm), as is commonly the case for rubber-like materials 13 . Interestingly, the corresponding ratio between Γ and the work to rupture W * (see Fig. 3g) leads to a transition length scale of 300 µm, i.e. more than 1 order of magnitude smaller than for the SCT models and thus closer to that of Sylgard 184. Computations used the parameters from Methods, and are shown with respect to the initial configuration. e Peak fiber elongation (hybrid model) and maximal principal stretches (continuum model) in mode I fracture simulations. Decay is shown along the loading direction (y) and perpendicular to it (x) for a stretch λ of 1.4. f Maximal principal stretches decay in x-and y direction from the notch for continuum models representative of elastomers (Neo-Hooke and Ogden for Sylgard184) and GC evaluated at λ=1.4. g Notch shape predicted by the hybrid approach for different initial sample lengths (L 0 =1 mm, 10 mm and 40 mm) at λ=1.2. h increase of osmotic pressure as computed with the biphasic 3D fiber network model at the notch tip (black) and in the farfield (red). Fig. 2 Influence of model parameters on the results of fracture computations. a The influence of the failure criterion (number of failed fibers) on the nominal failure stretch λ F in mode I fracture simulations with the hybrid approach, shown for n=3 different DNM realizations. b Peak DNM fiber strain decay along the direction perpendicular to loading for n=5 realizations at a nominal stretch λ=1.5, without applying a fiber failure limit. Data are represented as mean and standard deviation. c,d Influence of the DNM region size (i.e. its radius "r", as indicated) on the critical elongation λ F and the tearing energy Γ. Note that a new model was generated for every DNM size. e,f Tearing energy Γ and λ F for mean DNM fiber lengths of L c =10 µm and L c =20 µm. λ F in notched samples is compared to that of intact samples. g,h Effect of fiber slackness and critical strain on Γ and λ F . For the latter, the results are compared with those for samples without a notch. i,k For all parameter variations (ε s , ε c , L c and number of failed fibers) the evolution of Γ a (normalized with respect to Γ) vs. initial sample length is shown (i). Likewise, for simulations of samples with small central defects (cf. Figure 3), the influence of model parameters on the reduction of the critical tissue elongation is quantified (k). l MPM image of GC in the reference state with a typical native defect. Collagen is shown in green from second harmonic generation, fluorescence stained nuclei in blue and the defect is colored in yellow. Scale bar: 100 µm. Results in (a,e-h) are presented as mean± standard deviation. Supplementary Fig. 3 Influence of the notch depth in mode I fracture computations. a Critical elongation λ F calculated for different notch depths c, for sample lengths L 0 =5, 10 and 20 mm. 25 failed fibers were selected as failure criterion, for fibers with fiber slackness and critical elongation of ε s =0.21 and ε c =0.35. b Corresponding values of Γ a . c For the corresponding under-critical loading states, the nearfield size along x-direction is determined. Note that the x-axis is split up in intervals of 10 µm and for each interval, the maximum fiber strain is determined. Then, the nearfield was defined as the region where the maximum fiber strains were larger than 105% of the maximum farfield fiber strain. Supplementary Fig. 4 Influence of the fiber behavior in mode I fracture computations. a,b Peak fiber elongation (hybrid model) and maximum principal stretches (continuum model) in mode I fracture simulations for a linear force law of the fibers in the DNM and correspondingly adapted parameters of the continuum model. The decay is shown in in y-direction (a) and x-direction (b). These results are to be compared with those reported in Supplementary Fig. 1e. The dashed line represents 105% of the farfield stretch, and intersection with the continuum model indicates the nearfield size. c Dependence of λ F on initial sample length L 0 , for a failure criterion of 25 failed fibers (solid green), together with the fracture mechanics based prediction of λ F (Γ) (dashed) and λ F of intact samples (dotted). Note that the fracture mechanics based prediction of λ F (Γ) and λ F of intact samples of the reference model with the bi-linear force law is also shown (cf. Fig. 2d). For both models the transition length is indicated as the intersection of λ F (Γ) and λ F of intact samples.