Effect of connectivity on the elasticity of athermal network materials.

Network materials with stochastic structure are ubiquitous in biology and engineering, which drives the current interest in establishing relations between their structure and mechanical behavior. In this work we focus on the effect of connectivity defined by the number of fibers emerging from a crosslink, z, and compare networks with identical (z-homogeneous) and distinct (z-heterogeneous) z at the crosslinks. We observe that the functional form of strain stiffening is z-independent, and that the central z-dependent parameter is the small strain stiffness, E0. We confirm previous results indicating that the functional form of E0(z) is a power function with 3 regimes and observe that this applies to a broad range of z. However, the scaling exponents are different in the z-homogeneous and z-heterogeneous cases. We confirm that increasing z across the Maxwell's central force isostatic point leads to a transition from bending to axial energy storage. However, we observe that this does not necessarily imply that deformation becomes affine in the large z limit. In fact, networks of fibers with low bending stiffness retain a relaxation mode based on the rotational degree of freedom of the crosslinks which allows E0 in the large z limit to be smaller than the affine model prediction. We also conclude that in the z-heterogeneous case, the mean connectivity z̄ is sufficient to evaluate the effect of connectivity on E0 and that higher moments of the distribution of z are less important.

Recyclability of Vitrimer Materials: Impact of Catalyst and Processing Conditions.

With sustainability at the forefront of material research, recyclable polymers, such as vitrimers, have garnered increasing attention since their introduction in 2011. In addition to a traditional glass-transition temperature (T g), vitrimers have a second topology freezing temperature (T v) above which dynamic covalent bonds allow for rapid stress relaxation, self-healing, and shape reprogramming. Herein, we demonstrate the self-healing, shape memory, and shape reconfigurability properties as a function of experimental conditions, aiming toward recyclability and increased useful lifetime of the material. Of interest, we report the influence of processing conditions, which makes the material vulnerable to degradation. We report a decreased crosslink density with increased thermal cycling and compressive stress. Furthermore, we demonstrate that shape reconfigurability and self-healing are enhanced with increasing compressive stress and catalyst concentration, while their performance as a shape memory material remains unchanged. Though increasing the catalyst concentration, temperature, and compressive stress clearly enhances the recovery performance of vitrimers, we must emphasize its trade-off when considering the material degradation reported here. While vitrimers hold great promise as structural materials, it is vital to understand how experimental parameters impact their properties, stability, and reprocessability before vitrimers reach their true potential.

Mechano-chemical regulation of bat wing bones for flight.

Distal phalanges in bat wings have been hypothesized to be cartilaginous to allow for flight. We provide new evidence on how bat wing development might facilitate flight though protein-based regulation of bone mineralization and lead to more deflection at phalanx than humerus. Between Pteropus poliocephalus and Pteropus hypomelanus, two large bat species, we detected 112 proteins including 11 associated with mineralization and analyzed their distribution between the wing bones. Here, in contrast to previous reports, we found no cartilage-specific proteins and demonstrate that distal phalanges in bat wings are in fact low density bone that contain collagen I (the main constituent of bone's organic matrix) and proteins associated with mineralization in bone such as osteomodulin, bone-specific protein osteocalcin. The functional relevance of these changes was explored by measuring changes in mineral (crystal sizes, packing and density), material (Young's modulus and hardness) and structural characteristics. Consistent with changes in proteins associated with mineralization, mineral crystal thickness and alignment decreased from humerus to phalanges, and the mineral platelets were less densely packed along the wing length. Crystal thickness was negatively correlated with proteins associated with inhibition of mineralization as well as with two types of small leucine-rich proteoglycans, indicating the mineral growth and maturity is down regulated by these proteins independent of mineral quantity. The Young's modulus decreased across the wing and was significantly correlated with bone mineral density. Thus, the results from two bat species, studied here, demonstrate progressive alterations in bone mineralization occur in concert with the changes in secretion of bone regulatory proteins along the wing length. This altered mineralization together with structural changes serve to lighten the limb bone and optimize biomechanical properties conducive to flight.

Image-based multi-scale mechanical analysis of strain amplification in neurons embedded in collagen gel.

A general multi-scale strategy is presented for modeling the mechanical environment of a group of neurons that were embedded within a collagenous matrix. The results of the multi-scale simulation are used to estimate the local strains that arise in neurons when the extracellular matrix is deformed. The distribution of local strains was found to depend strongly on the configuration of the embedded neurons relative to the loading direction, reflecting the anisotropic mechanical behavior of the neurons. More importantly, the applied strain on the surrounding extracellular matrix is amplified in the neurons for all loading configurations that are considered. In the most severe case, the applied strain is amplified by at least a factor of 2 in 10% of the neurons' volume. The approach presented in this paper provides an extension to the capability of past methods by enabling the realistic representation of complex cell geometry into a multi-scale framework. The simulation results for the embedded neurons provide local strain information that is not accessible by current experimental techniques.

Stochasticity in materials structure, properties, and processing-A review.

We review the concept of stochasticity-i.e., unpredictable or uncontrolled fluctuations in structure, chemistry, or kinetic processes-in materials. We first define six broad classes of stochasticity: equilibrium (thermodynamic) fluctuations; structural/compositional fluctuations; kinetic fluctuations; frustration and degeneracy; imprecision in measurements; and stochasticity in modeling and simulation. In this review, we focus on the first four classes that are inherent to materials phenomena. We next develop a mathematical framework for describing materials stochasticity and then show how it can be broadly applied to these four materials-related stochastic classes. In subsequent sections, we describe structural and compositional fluctuations at small length scales that modify material properties and behavior at larger length scales; systems with engineered fluctuations, concentrating primarily on composite materials; systems in which stochasticity is developed through nucleation and kinetic phenomena; and configurations in which constraints in a given system prevent it from attaining its ground state and cause it to attain several, equally likely (degenerate) states. We next describe how stochasticity in these processes results in variations in physical properties and how these variations are then accentuated by-or amplify-stochasticity in processing and manufacturing procedures. In summary, the origins of materials stochasticity, the degree to which it can be predicted and/or controlled, and the possibility of using stochastic descriptions of materials structure, properties, and processing as a new degree of freedom in materials design are described.

Stiffness Percolation in Stochastically Fragmented Continua.

We study the mechanical behavior of three-dimensional, randomly microcracked continua for crack densities up to and above the transport percolation threshold. We show the existence of a fully fragmented material state in which stiffness is preserved due to topological interlocking of fragments. In this regime, the mechanical behavior is controlled by the contacts between fragments and becomes nonlinear. The upper limit of crack densities for which this behavior is observed, the stiffness percolation threshold, is identified. The variation of the effective material stiffness for crack densities ranging from 0 to the stiffness percolation threshold is reported.

Collagen Organization in Facet Capsular Ligaments Varies With Spinal Region and With Ligament Deformation.

The spinal facet capsular ligament (FCL) is primarily comprised of heterogeneous arrangements of collagen fibers. This complex fibrous structure and its evolution under loading play a critical role in determining the mechanical behavior of the FCL. A lack of analytical tools to characterize the spatial anisotropy and heterogeneity of the FCL's microstructure has limited the current understanding of its structure-function relationships. Here, the collagen organization was characterized using spatial correlation analysis of the FCL's optically obtained fiber orientation field. FCLs from the cervical and lumbar spinal regions were characterized in terms of their structure, as was the reorganization of collagen in stretched cervical FCLs. Higher degrees of intra- and intersample heterogeneity were found in cervical FCLs than in lumbar specimens. In the cervical FCLs, heterogeneity was manifested in the form of curvy patterns formed by collections of collagen fibers or fiber bundles. Tensile stretch, a common injury mechanism for the cervical FCL, significantly increased the spatial correlation length in the stretch direction, indicating an elongation of the observed structural features. Finally, an affine estimation for the change of correlation length under loading was performed which gave predictions very similar to the actual values. These findings provide structural insights for multiscale mechanical analyses of the FCLs from various spinal regions and also suggest methods for quantitative characterization of complex tissue patterns.

Effect of Fiber Crimp on the Elasticity of Random Fiber Networks With and Without Embedding Matrices.

Fiber networks are assemblies of one-dimensional elements representative of materials with fibrous microstructures such as collagen networks and synthetic nonwovens. The mechanics of random fiber networks has been the focus of numerous studies. However, fiber crimp has been explicitly represented only in few cases. In the present work, the mechanics of cross-linked networks with crimped athermal fibers, with and without an embedding elastic matrix, is studied. The dependence of the effective network stiffness on the fraction of nonstraight fibers and the relative crimp amplitude (or tortuosity) is studied using finite element simulations of networks with sinusoidally curved fibers. A semi-analytic model is developed to predict the dependence of network modulus on the crimp amplitude and the bounds of the stiffness reduction associated with the presence of crimp. The transition from the linear to the nonlinear elastic response of the network is rendered more gradual by the presence of crimp, and the effect of crimp on the network tangent stiffness decreases as strain increases. If the network is embedded in an elastic matrix, the effect of crimp becomes negligible even for very small, biologically relevant matrix stiffness values. However, the distribution of the maximum principal stress in the matrix becomes broader in the presence of crimp relative to the similar system with straight fibers, which indicates an increased probability of matrix failure.

Investigating Orientational Defects in Energetic Material RDX Using First-Principles Calculations.

Orientational defects are molecular-scale point defects consisting of misaligned sterically trapped molecules. Such defects have been predicted in α-RDX using empirical force fields. These calculations indicate that their concentration should be higher than that of vacancies. In this study we confirm the stability of a family of four orientational defects in α-RDX using first-principles calculations and evaluate their formation energies and annealing barrier heights. The charge density distribution in the defective molecules is evaluated and it is shown that all four orientational defects exhibit some level of charge reduction at the midpoint of the N-N bond, which has been previously related to the sensitivity to initiation of the material. We also evaluate the vibrational spectrum of the crystal containing orientational defects and observe band splitting relative to the perfect crystal case. This may assist the experimental identification of such defects by Raman spectroscopy.

Softening in Random Networks of Non-Identical Beams.

Random fiber networks are assemblies of elastic elements connected in random configurations. They are used as models for a broad range of fibrous materials including biopolymer gels and synthetic nonwovens. Although the mechanics of networks made from the same type of fibers has been studied extensively, the behavior of composite systems of fibers with different properties has received less attention. In this work we numerically and theoretically study random networks of beams and springs of different mechanical properties. We observe that the overall network stiffness decreases on average as the variability of fiber stiffness increases, at constant mean fiber stiffness. Numerical results and analytical arguments show that for small variabilities in fiber stiffness the amount of network softening scales linearly with the variance of the fiber stiffness distribution. This result holds for any beam structure and is expected to apply to a broad range of materials including cellular solids.

A discrete network model to represent the deformation behavior of human amnion.

A discrete network model (DNM) to represent the mechanical behavior of the human amnion is proposed. The amnion is modeled as randomly distributed points interconnected with connector elements representing collagen crosslinks and fiber segments, respectively. This DNM is computationally efficient and allows simulations with large domains. A representative set of parameters has been selected to reproduce the uniaxial tension-stretch and kinematic responses of the amnion. Good agreement is found between the predicted and measured equibiaxial tension-stretch curves. Although the model represents the amnion phenomenologically, model parameters are physically motivated and their effect on the tension-stretch and in-plane kinematic responses is discussed. The model is used to investigate the local response in the near field of a circular hole, revealing that the kinematic response at the circular free boundaries leads to compaction and strong alignment of the network at the border of the defect.

Effect of defects on the intrinsic strength and stiffness of graphene.

It is important from a fundamental standpoint and for practical applications to understand how the mechanical properties of graphene are influenced by defects. Here we report that the two-dimensional elastic modulus of graphene is maintained even at a high density of sp(3)-type defects. Moreover, the breaking strength of defective graphene is only ~14% smaller than its pristine counterpart in the sp(3)-defect regime. By contrast, we report a significant drop in the mechanical properties of graphene in the vacancy-defect regime. We also provide a mapping between the Raman spectra of defective graphene and its mechanical properties. This provides a simple, yet non-destructive methodology to identify graphene samples that are still mechanically functional. By establishing a relationship between the type and density of defects and the mechanical properties of graphene, this work provides important basic information for the rational design of composites and other systems utilizing the high modulus and strength of graphene.

Strength of DNA sticky end links.

Sticky ends are unpaired nucleotides at the ends of DNA molecules that can associate to link DNA segments. Self-assembly of DNA molecules via sticky ends is currently used to grow DNA structures with desired architectures. The sticky end links are the weakest parts of such structures. In this work, the strength of sticky end links is studied by computational means. The number of basepairs in the sticky end and the sequence are varied, and the response to tension along the axis of the molecule is evaluated using a full atomistic model. It is observed that, generally, increasing the number of basepairs in the sticky end increases the strength, but the central factor controlling this parameter is the basepair sequence. The sticky ends are divided into two classes of low and high strength. The second class has strength comparable with that of a double stranded molecule with one nick in one of the strands. The strength of the first class is roughly half that of the strong sticky ends. For all strong sticky ends tested, the enhanced stability is associated with the formation of an unusually stable complex composed from two basepairs and two flanking bases of certain sequence. This complex rotates and aligns with the direction of the force allowing significant deformation and providing enhanced strength. This is similar to a mechanism recently suggested to enhance the mechanical stability of an RNA kissing loop from the Moloney murine leukemia virus. The model is tested against experimental structural data for sticky ends and against published simulation results for the stretch of double stranded DNA. The results provide guidance for the design of DNA self-assembled structures and indicate the types of sticky ends desirable if maximizing the strength and stability of these structures is targeted.

Peierls stress of dislocations in molecular crystal cyclotrimethylene trinitramine.

Dislocation mediated plasticity in the α phase of the energetic molecular crystal cyclotrimethylene trinitramine (RDX) was investigated using a combination of atomistic simulations and the Peierls-Nabarro (PN) model. A detailed investigation of core structures and dislocation Peierls stress was conducted using athermal atomistic simulations at atmospheric pressure to determine the active slip systems. Generalized stacking fault energy surfaces calculated using atomistic simulations were used in the PN model to also estimate the critical shear stress for dislocation motion. The primary slip plane is found to be (010) in agreement with experimental observations, with the (010)[100] slip systems having the lowest Peierls stress. In addition, atomistic simulations predict the (021)[01[overline]2], (021)[100], (011)[100], (001)[100], and (001)[010] slip systems to have Peierls stress values small enough to allow plastic activity. However, there are less than five independent slip systems in this material in all situations. The ranking of slip systems based on the Peierls stress values is provided, and implications are discussed in relation to experimental data from nanoindentation and shock-induced plastic deformation.

A coupled fiber-matrix model demonstrates highly inhomogeneous microstructural interactions in soft tissues under tensile load.

A soft tissue's macroscopic behavior is largely determined by its microstructural components (often a collagen fiber network surrounded by a nonfibrillar matrix (NFM)). In the present study, a coupled fiber-matrix model was developed to fully quantify the internal stress field within such a tissue and to explore interactions between the collagen fiber network and nonfibrillar matrix (NFM). Voronoi tessellations (representing collagen networks) were embedded in a continuous three-dimensional NFM. Fibers were represented as one-dimensional nonlinear springs and the NFM, meshed via tetrahedra, was modeled as a compressible neo-Hookean solid. Multidimensional finite element modeling was employed in order to couple the two tissue components and uniaxial tension was applied to the composite representative volume element (RVE). In terms of the overall RVE response (average stress, fiber orientation, and Poisson's ratio), the coupled fiber-matrix model yielded results consistent with those obtained using a previously developed parallel model based upon superposition. The detailed stress field in the composite RVE demonstrated the high degree of inhomogeneity in NFM mechanics, which cannot be addressed by a parallel model. Distributions of maximum/minimum principal stresses in the NFM showed a transition from fiber-dominated to matrix-dominated behavior as the matrix shear modulus increased. The matrix-dominated behavior also included a shift in the fiber kinematics toward the affine limit. We conclude that if only gross averaged parameters are of interest, parallel-type models are suitable. If, however, one is concerned with phenomena, such as individual cell-fiber interactions or tissue failure that could be altered by local variations in the stress field, then the detailed model is necessary in spite of its higher computational cost.

Control of epoxy creep using graphene.

The creep behavior of epoxy-graphene platelet (GPL) nanocomposites with different weight fractions of filler is investigated by macroscopic testing and nanoindentation. No difference is observed at low stress and ambient temperature between neat epoxy and nanocomposites. At elevated stress and temperature the nanocomposite with the optimal weight fraction, 0.1 wt% GPLs, creeps significantly less than the unfilled polymer. This indicates that thermally activated processes controlling the creep rate are in part inhibited by the presence of GPLs. The phenomenon is qualitatively similar at the macroscale and in nanoindentation tests. The results are compared with the creep of epoxy-single-walled (SWNT) and multi-walled carbon nanotube (MWNT) composites and it is observed that creep in both these systems is similar to that in pure epoxy, that is, faster than creep in the epoxy-GPL system considered in this work.

Mechanics of three-dimensional, nonbonded random fiber networks.

The mechanical behavior of an ensemble of athermal fibers forming a nonbonded network subjected to triaxial compression is studied using a numerical model. The response exhibits a power law dependence of stress on the dilatation strain and hysteresis upon loading and unloading. A stable hysteresis loop results after the first loading and unloading cycle. In the early stages of compaction, strain energy is associated primarily with the bending of fibers, while at higher densities, it is stored primarily in the axial deformation mode. It is shown that the exponent of the power law, and the partition of energy in the axial and bending modes depends on the ratio of the bending to axial stiffness of the fibers. Accounting for interfiber friction does not change the functional form of the stress-strain relationship or the exponent. The central feature that distinguishes the mechanics of this system from that of bonded random networks is the relative sliding at contacts and the ensuing fiber rearrangements. We show that suppressing sliding leads to a much stiffer response. The results indicate that the value of the exponent of the stress-strain power law is determined not only by fiber bending and the formation of new contacts, but also by the relative sliding and axial deformation of fibers.