Figure 1

(Color online) Multiscale mechanisms of materials failure. (a) Multiscale view of failure of glass, from macro to nano. (b) Fracture can be envisioned as dissipation of elastic (reversible) energy. This basic view of fracture holds for a very broad range of failure phenomena, from failure of the Earth during earthquakes, failure of engineering materials, to failure of proteins in cells, tissues, or organisms (32).Reuse & Permissions

Figure 2

(Color online) Basic energy balance during crack extension, the basic mechanism of brittle failure (e.g., of a material such as glass) in a cracked solid under remotely applied load $\sigma$. The energy stored in element (1) of width $\tilde{a}$ ahead of the crack is given by $W_P^{\left(1\right)}=\frac{1}{2}\xi \tilde{a}B{\sigma }^2∕E$, where $B$ is the out-of-plane thickness of the specimen, $\xi$ is the width of the thin strip, $\sigma$ is the applied stress, and $E$ is the Young’s modulus of the material (see right panel, where $\sigma =E\epsilon$). The energy stored in element (2) is $W_P^{\left(2\right)}=0$ since it is completely relaxed. During crack propagation by a distance $\tilde{a}$, the energy of $W_P^{\left(1\right)}-W_P^{\left(2\right)}=\frac{1}{2}\xi \tilde{a}B{\sigma }^2∕E$ is dissipated. This energy is used to create two new material surfaces; thus $\frac{1}{2}\xi \tilde{a}B{\sigma }^2∕E=2\gamma \tilde{a}B$, leading to the critical fracture condition $\frac{1}{2}\xi {\sigma }^2∕E=2\gamma$, or in terms of the applied stress $\sigma =\sqrt{4{\gamma }_{s}E∕\xi }$.Reuse & Permissions

Figure 3

(Color online) Physical mechanisms of failure in ductile (a) and brittle (b) materials, representing two fundamental failure modes. In both cases, dissipation of stored elastic energy drives the failure process. However, the mechanism of energy dissipation is different in the two cases. In (a), dislocations are the key dissipative mechanism (shown by the ⊥ symbol). In (b) the extension of the crack through the creation of new material surfaces is the governing energy dissipation mechanism. The existence of a stress concentration around a cracklike defect shown in (c) leads to locally much higher stresses than those found further away from the crack (stress is denoted by $\sigma$, and the distance from the crack tip denoted by $r$). These local stresses can cause cracks to extend as the large interatomic forces induce bond breaking [(b), as in brittle materials] or bond shearing [(a), as in ductile materials] (note that the stress concentration appears at cracks or flaws in any solid, regardless if it is brittle or ductile).Reuse & Permissions

Figure 4

(Color online) Schematic views of the hierarchical structure of spider silk and its fundamental beta-sheet protein building blocks (84). A spider web, a spider silk fiber, the microstructure of a spider silk fiber, and a detailed view of beta-sheet crystals, as well as individual H-bonded beta-strands that make up beta-sheet crystals are shown. The particular hierarchical structure of biological and natural materials makes it challenging to develop fracture models; mechanisms of failure and energy dissipation may occur at multiple scales and cannot be identified easily. Spider web image courtesy Nicolas Demars.Reuse & Permissions

Figure 5

(Color online) Basic approach of the molecular dynamics simulation method. (a) Atomistic structure (neutrons, electrons, and protons) replaced by a point representation in the molecular dynamics approach. (b) Illustration of the energy decomposition in classical molecular dynamics force fields, along with a representation of a simple potential function between pairs of atoms.Reuse & Permissions

Figure 6

(Color online) Multiparadigm molecular dynamics simulation of dynamic fracture of silicon ([25]; 26; 30), carried out based on a pure Tersoff model and a hybrid ReaxFF-Tersoff model. The hybrid model schematically shown in (a) describes the fracture mechanics of silicon by a combination of a simple Tersoff force field in regions far away from bond rupture events, with the ReaxFF reactive force field used to more accurately describe the rupture processes at the crack tip. (b) Comparison between (left) the pure Tersoff model and (right) the hybrid ReaxFF-Tersoff model. In the pure Tersoff model the crack does not extend, in contrast with experimental results. In the hybrid model where an accurate representation of the chemistry of bond breaking is provided at the crack tip, the crack extends under application of load. This comparison illustrates the significance of providing an accurate representation of chemical bond breaking events for modeling fracture.Reuse & Permissions

Figure 7

(Color online) Breaking a single protein molecule by pulling at the ends of a small protein ($\alpha$ conotoxin PnIB from conus pennaceus; PDB identification code 1AKG) (21). The study shown here reveals the differences between a nonreactive (CHARMM) force field and the ReaxFF reactive force field. (a) Snapshots as the molecule is being stretched, modeled using the reactive model (ReaxFF). As the molecule is being pulled, the covalent cross-links (disulfide bonds) within the molecule break. These breaking points correspond to the peaks in the force-extension plot shown in (c); and the force drops significantly after each breaking point as the elastic energy stored in the protein is released. In the case of the CHARMM model (b), bond breaking cannot be described, and the force continues to rise once the covalent cross-link within the protein is being stretched (c).Reuse & Permissions

Figure 8

(Color online) Illustration of coarse-graining approach for a simple one-dimensional fibrillar protein filament (collagen). This schematic shows how a full atomistic representation is coarse-grained and used in a mesoscale model formulation. This mesoscale model formulation enables one to reach much larger time and length scales. The systematic parametrization from the bottom up provides a rigorous link between the chemical structure of proteins (for example, through their amino acid sequence) and the overall functional material properties. This computational approach is a key component in the advancement of materiomics as it provides us with the ability to reach microsecond and micrometer length scales.Reuse & Permissions

Figure 9

(Color online) Experimental, theoretical, and computational tools for the characterization and modeling of deformation and failure of materials, plotted over their respective time and length scale domain of applicability. Experimental methods include x-ray diffraction, TEM (transmission electron microscopy), AFM (atomic force microscopy), OT/MT (optical/magnetic tweezers), and MEMS (mechano-electro-mechanical system) testing, as well as nanoindentation. Frequently used theoretical and simulation tools include quantum mechanics (DFT), molecular dynamics, coarse-grained models, mesoscale atomistically informed continuum theories, and continuum models. The lower part indicates respective classes and scales of materials that can be studied with these types of techniques. Adapted from 31.Reuse & Permissions

Figure 10

(Color online) Basic fracture process associated with an earthquake, here exemplified for the case of the San Andreas fault. Two tectonic plates are sheared against each other, while elastic energy is stored in the system, which is released in a catastrophic event once the earthquake occurs. Energy dissipation mechanisms include friction between along the fault line, where local slip occurs. Maps and photographs of the San Andreas fault courtesy of the U.S. Geological Survey (the detailed aerial view of the San Andreas fault is taken near Carrizo Plain, Central California).Reuse & Permissions

Figure 11

Earthquake dynamics of the 1999 Kocaeli, Turkey earthquake (136). The plot shows the path of the earthquake along a weak plane in the Earth’s crust. From 136.Reuse & Permissions

Figure 12

(Color online) Analysis of the dynamics (and speed) of the earthquake’s rupture front. The location (distance) of the front of the earthquake is shown as a function of time. It propagates at a speed of approximately $3\mathrm{km}∕\sec$ initially (subsonic), then followed by a sudden jump to a much higher propagation velocities of around $5.8\mathrm{km}∕\sec$. This propagation is faster than the speed of shear wave (thus referred to as “supershear” or “intersonic”). From 136.Reuse & Permissions

Figure 13

(Color online) Laboratory earthquakes reveal a similar phenomenon of intersonic rupture along a shear fault, mimicking the phenomena observed in earthquakes. The Mach cones in (c) show intersonic crack propagation, similar as observed in earthquakes (see Fig. 11). From 157.Reuse & Permissions

Figure 14

(Color online) Molecular dynamics simulations of interfacial failure between two dissimilar materials (22). A rupture propagates along a weak plane between two elastically dissimilar materials, moving at an intersonic speed [see shock fronts in (d)]. Intersonic fracture is possible through so-called mother-daughter mechanism where a secondary crack is born ahead of the primary (mother) crack. This mechanism is illustrated in the blowup in (c) (the two arrows indicate the mother and daughter crack, respectively).Reuse & Permissions

Figure 15

Example of fracture of bone, a common form of body injury. (a) A macroscopic view of bone failure. (b) SEM micrographs of bone fracture experiments as reported by 104. From 104.Reuse & Permissions

Figure 16

(Color online) Hierarchical structure of bone, showing seven hierarchies, and their relation to the mechanisms of mechanical properties (bottom three inlays from 155; figure adapted from [126]). Collagen protein molecules (“tropocollagen”), formed by three chains of amino acids, provide the structural basis mineralized collagen fibrils, the smallest building block of bone. Several mineralized collagen fibrils form fibril arrays, linked by an organic phase. Several fibril arrays (or collagen fibers) form geometric fibril array patterns, which provide structure to cellular components of bone; this is also known as the lamellar structure of bone. The boundaries between the packets of fibers comprise the lamellar interfaces. This microstructure of bone forms distinct mesoscale structural arrangements, such as spongy/compact bone and eventually macroscopic bone. Distinct toughening mechanisms occur at each level of hierarchy. Molecular uncoiling and intermolecular sliding of molecules are observed at the smallest level of tropocollagen molecules and mineralized collagen fibrils. Microcracking and fibrillar sliding are observed at the level of fibril arrays. At larger levels, the breaking of sacrificial bonds contributes to increasing the energy dissipation capacity of bone at the interface of fibril arrays. Crack bridging by collagen fibrils is observed at scales of several hundred micrometers, as well as microcracking and crack branching at scales of centimeters and beyond. Cortical (compact) bone is the dense bone that is found at the surface of all bones; trabecular (cancellous) bone is spongy with struts of order $100–200\mu \mathrm{m}$ in diameter and holes of approximately $1\mathrm{mm}$ in diameter. Notice the similarity of the structural makeup in multiple hierarchies to the structure of spider silk shown in Fig. 4.Reuse & Permissions

Figure 17

(Color online) Example of a protein undergoing unfolding and the schematics for the theoretical model that represents a key resisting H-bond cluster acting as a mechanical clamp (81). (a) A large beta-sheet protein consisting of H-bonded strands as shown in (b). (c) The schematic for the theoretical model presented in Sec. 4C. Per unit rupture advancement $d\lambda$, an energy balance criterion between entropic elasticity and H-bond dissociation energy can be written to obtain the failure strength. Elasticity of the chain is modeled by the wormlike chain theory, and has one parameter, the persistence length ${\xi }_P$, that is linked to the bending rigidity of the polypeptide, $EI$. Note that the beta-strand also represents a fundamental constituent in spider silk, one of the toughest materials known (see Fig. 4).Reuse & Permissions

Figure 18

(Color online) Strength model of H-bond clusters in beta-sheets, including a direct comparison between experiment and theory. (a) The force-extension behavior displayed before and after rupture, illustrating the dissipated energy (81). The continuum WLC (cWLC) expression used in the derivation could be replaced by discrete WLC (dWLC) formula proposed in 128 to see if there is any variation in the predicted strength. Both forms of the WLC expression predict similar elastic behavior and rupture force. (b) Summary of studies on rupture force of beta-domains as a function of pulling rate. We summarize findings [adapted from 144] of the strength of beta-sheets rich proteins (focus on fibronectin, immunoglobin domains in ECM and titin). The overall behavior suggests that the rupture force asymptotically approaches a limiting value for vanishing pulling rates [continuous line is a power law fit to data for Ig27 (140)]. The inset shows a close-up of the strength limit and asymptotic behavior. The shaded region shows the range of values admissible by the theoretical prediction. The theoretical prediction range and experimental scattering of data show close agreement.Reuse & Permissions

Figure 19

(Color online) Turning weakness into strength: size dependence of hydrogen bond strength. Size effects of the shear strength of beta sheets [the geometry of a beta sheet is shown in the inlay of (a)] composed of two polypeptide chains connected through $N$ H bonds; this number of H bonds has a significant effect on the shear strength of the structure (82). (a) The shear strength as a function of the number of H bonds of a beta-sheet structure (inset shows an example with four H bonds). (b) The physical significance of this size effect. In the upper plot, only H bonds at the boundary participate in the rupture process and provide resistance. In the lower plot, all H bonds throughout the entire structure contribute to the strength, making the overall structure three times stronger. The shear strength if defined as the strength of the beta strand divided by the sheared area.Reuse & Permissions

Figure 20

(Color online) Characteristic dimensions of the size of H-bond clusters in common protein structures (82). We compare the characteristic dimensions of alpha-helices, beta-sheets, and beta-helices to the dimensions associated with the optimal strength presented in Fig. 19 (see left bar, suggesting an optimal strength at three to four H bonds). Since the theoretical derivation reviewed here considers uniform deformation of hydrogen bonds with no particular specificity to geometry, it may also apply to other protein structures where nature utilizes geometric confinement to achieve higher mechanical stability. The fact that three to four H bonds per convolution exists on alpha-helices and beta-sheets on the sides of helices occur in clusters of approximately four may be indicative of such a universal biological concept that may be based on the evolutionary driving force to provide maximum strength.Reuse & Permissions

Figure 21

(Color online) Hierarchical flaw tolerance mechanism in a cellular alpha-helix protein network (3). (a) Seven levels of hierarchies are considered, from intrabackbone hydrogen bond (H0), alpha-helical turns (H1), filaments of alpha-helices (H2), to the representative unit cell (H3) of protein networks (H4) that form the cell nucleus (defects in the network highlighted) (H5) of eukaryotic cells (H6). The structure at each level is adapted to provide an optimal mechanical response and plays a key role in the overall mechanical behavior. Unfolding of alpha-helix turns (H1) proceeds via breaking of strong clusters of three to four H bonds (H0). The large deformation of alpha-helix filaments (with maximum strains of 100–200 %) (H2) is enabled by the serial arrangements of many alpha-helical turns (H1). The severe stiffening of the filaments is enabled by alpha-to-beta-sheet transitions and backbone stretching, followed by interprotein sliding at the filament level (H2), is a direct consequence of the structure of coiled alpha-helical proteins. The lattice structure (H3) is the key to facilitate large strain gradients in the protein network, enabling gigantic strain gradients at virtually no energetic cost at the network level (H4). This behavior is crucial for the flaw-tolerant behavior of the nuclear envelope level (H5), which is relevant to provide robust structural support to cells under large deformation (H6). (b) Protein network deformation with marked strain gradient, illustrating the change of the crack orientation from a horizontal to a vertical one. (c) Schematic of crack geometry transition, plotting the crack corner stress concentration ${\sigma }_{\text{tip}}$ and the applied stress far away from the crack ${\sigma }_0$. The schematic shows that the initial horizontal crack orientation features a large stress concentration at the crack tip. In contrast, the transformed vertical crack orientation features only marginal stresses at its corners.Reuse & Permissions

Figure 22

(Color online) Role of changes in biological protein material properties in diseases, here exemplified for the rapid aging disease progeria. This shows how changes in the lamin microstructure due to the genetic mutation and effect on deformation mechanism. (a) A structural analysis of a healthy nucleus compared with a disease nucleus; showing that the progeria nucleus is more brittle and shows the formation of several small fractures. The lower part shows micropipetting experiments, also suggesting a more brittle behavior of the progeria nucleus. (b) A possible schematic of the mechanism behind these observations, suggesting a structural change of the protein filament microstructure due to the mutation. It is noted that the number of filaments is not altered significantly, but their structural molecular makeup leads to changes in the microstructure. Adapted from 43.Reuse & Permissions

Figure 23

(Color online) Mutations in osteogenesis imperfecta disease and impact on mechanics properties. Geometry of point mutations in osteogenesis imperfecta (brittle bone disease, often abbreviated as OI) in (a) the tropocollagen molecule and (b) effect on mechanical properties of tendon. The analysis shows that the maximum strain and maximum stress of tendon is severely reduced under mutations (lower part, data plotted from 101). The graphs shown in (c) illustrate that the relative strength of the effect of different mutations on the molecular stiffness and similarly, the intermolecular adhesion, correlates directly with the clinically measured severity of various types of mutations (63).Reuse & Permissions

Figure 24

(Color online) Influence of osteogenesis imperfecta mutations on the mechanical properties of a collagen fibril, leading to a significant reduction of mechanical strength and yield strain (for most severe mutation, located at the end of the molecule) (63). (a) The decrease of the strength for a mutation (in a cross-link deficient fibril and highly cross-linked fibril), including a qualitative comparison with experimental results (101) in osteogenesis imperfecta tendon (note the difference in scale; the MD studies are focused on single collagen fibrils; the experiments have been carried out at much larger tendon tissue scales). (b) The stress-strain curve for the reference case and a mutated case, showing that the presence of mutations can severely influence the overall mechanical signature (highly cross-linked fibril). Intermolecular gliding sets in at 30% vs 42% in the mutated fibril, and the maximum stress is significantly reduced (to about 50% of its reference value). The small-deformation elastic modulus is reduced by approximately 15% under the presence of the osteogenesis imperfecta mutation, due the formation of internal stress concentrations between collagen molecules at the sites of the mutations. (c) A schematic of the stress concentrations caused at the mutation sites. The formation of stress concentrations within collagen fibrils suggests that the breakdown of the homogeneous stress state as achieved in the healthy state (19) results in a severe reduction of the strength and toughness of the material.Reuse & Permissions