Non-affine Deformation Research Articles

Nanoparticles reorganizations in polymer nanocomposites under large deformation

The structural evolution of nanoparticles (NPs) dispersion under uniaxial stretching and extension/retraction deformation cycles above Tg was investigated in model silica/polymethylmethacrylate (PMMA) nanocomposites (PNCs) by a combination of Small Angle X-ray Scattering (SAXS) and Transmission Electronic Microscopy (TEM). The different structure displacements and reorganizations can be quantitatively characterized as a function of elongation ratio, silica volume fraction and NP size. At low NP volume fraction, a rotation/orientation of non-connected aggregates is observed along the stretching direction, while the reinforcement is low and might be limited by the large-scale aggregates. At high volume fraction, the stress–strain curves exhibit three regimes. (i) At low stretching ratio, in the linear deformation regime, reinforcement is driven by the primary network filler structure. (ii) Above a few percent of deformation, a yield is observed and can be associated to the network breakdown as revealed by cyclic extension/retraction experiments. (iii) As a result of this yield, at larger deformation, the stress curve appears as shifted upward with respect to the one of pure polymer. A persistence of this vertical shift (constant value up to large deformation) might be related with SAXS measurements to a non-affine deformation of the NPs network due to new structural arrangements, while in a second case, the decrease of stress to the pure polymer value with increasing deformation is related with observation of affine deformation after the yield. Finally, affinity and non-affinity after yielding are discussed for all the systems according to the strength of the NP–NP interaction.

Simulation of planar soft tissues using a structural constitutive model: Finite element implementation and validation

Computational implementation of physical and physiologically realistic constitutive models is critical for numerical simulation of soft biological tissues in a variety of biomedical applications. It is well established that the highly nonlinear and anisotropic mechanical behaviors of soft tissues are an emergent behavior of the underlying tissue microstructure. In the present study, we have implemented a structural constitutive model into a finite element framework specialized for membrane tissues. We noted that starting with a single element subjected to uniaxial tension, the non-fibrous tissue matrix must be present to prevent unrealistic tissue deformations. Flexural simulations were used to set the non-fibrous matrix modulus because fibers have little effects on tissue deformation under three-point bending. Multiple deformation modes were simulated, including strip biaxial, planar biaxial with two attachment methods, and membrane inflation. Detailed comparisons with experimental data were undertaken to insure faithful simulations of both the macro-level stress–strain insights into adaptations of the fiber architecture under stress, such as fiber reorientation and fiber recruitment. Results indicated a high degree of fidelity and demonstrated interesting microstructural adaptions to stress and the important role of the underlying tissue matrix. Moreover, we apparently resolve a discrepancy in our 1997 study (Billiar and Sacks, 1997. J. Biomech. 30 (7), 753–756) where we observed that under strip biaxial stretch the simulated fiber splay responses were not in good agreement with the experimental results, suggesting non-affine deformations may have occurred. However, by correctly accounting for the isotropic phase of the measured fiber splay, good agreement was obtained. While not the final word, these simulations suggest that affine fiber kinematics for planar collagenous tissues is a reasonable assumption at the macro level. Simulation tools such as these are imperative in the design and simulation of native and engineered tissues.

Synergistic Stiffening in Double-Fiber Networks

Many biological materials are composite structures, interpenetrating networks of different types of fibers. The composite nature of such networks leads to superior mechanical properties, but the origin of this mechanical synergism is still poorly understood. Here we study soft composite networks, made by mixing two self-assembling fiber-forming components. We find that the elastic moduli of the composite networks significantly exceed the sum of the moduli of the two individual networks. This mechanical enhancement is in agreement with recent simulations, where it was attributed to a suppression of non-affine deformation modes in the most rigid fiber network due to the reaction forces in the softer network. The increase in affinity also causes a loss of strain hardening and an increase in the critical stress and stain at which the network fails.

Soft Modes and Nonaffine Rearrangements in the Inherent Structures of Supercooled Liquids

We find that the hierarchical organization of the potential energy landscape in a model supercooled liquid can be related to a change in the spatial distribution of soft normal modes. For groups of nearby minima, between which fast relaxation processes typically occur, the localization of the soft modes is very similar. The spatial distribution of soft regions changes, instead, for minima between which transitions relevant to structural relaxation occur. This may be the reason why the soft modes are able to predict spatial heterogeneities in the dynamics. Nevertheless, the very softest modes are only weakly correlated with dynamical heterogeneities and instead show higher statistical overlap with regions in the local minima that would undergo nonaffine rearrangements if subjected to a shear deformation. This feature of the supercooled liquid is reminiscent of the behavior of nonaffine deformations in amorphous solids, where the very softest modes identify the loci of plastic instabilities.

Scale-Dependent Nonaffine Elasticity of Semiflexible Polymer Networks

The cytoskeleton of eukaryotic cells provides mechanical support and governs intracellular transport. These functions rely on the complex mechanical properties of networks of semiflexible protein filaments. We study the impact of local network deformations on the scale-dependent mobility of probe particles in entangled networks of actin filaments using high-bandwidth microrheology. We find that micron-sized particles in these networks experience two opposing noncontinuum elastic effects: entropic depletion reduces the effective network rigidity, while local nonaffine deformations of the network substantially enhance the rigidity at low frequencies, eventually leading to a size-independent response and strong violation of the generalized Stokes formula. We show that a simple model of lateral bending of filaments embedded in a viscoelastic background leads to an intermediate scaling regime for the apparent elastic modulus ${G}^{\ensuremath{'}}(\ensuremath{\omega})\ensuremath{\sim}{\ensuremath{\omega}}^{9/16}$, closely matching the experiments. These results demonstrate that nonaffine bending deformations can be dominant for the mobility of objects of the size of vesicles and organelles in the cell.

Analytical Model of Fractal Aggregate Stability and Restructuring in Shear Flows

We combine the analytical theory of nonaffine deformations of noncrystalline solids with numerical Stokesian dynamical simulations to obtain analytical closed-form expressions for the shear modulus of fractal aggregates in shear flows. The proposed framework also provides analytical predictions for the evolution of the fractal dimension df of the aggregate during the aggregation process. This leads to a lower bound on df below which aggregates are mechanically unstable (they possess floppy modes) and cannot survive without restructuring into more compact, higher-df configurations. In the limit of large aggregates, the predicted lower bound is df = 2.407. This result provides the long-sought explanation as to why all experimental and simulation studies in the past consistently reported df ≳ 2.4 for shear-induced colloidal aggregation. The analytical expressions derived here can be used within population balance calculations of colloidal aggregation in shear flows whereby until now the fractal dimension evolution was treated as a free parameter. These results may open up the possibility of developing new microscopic mechanical manipulation techniques to control nanoparticle and colloidal aggregates at the nanoscale.

Short-range correlations control the G/K and Poisson ratios of amorphous solids and metallic glasses

The bulk modulus of many amorphous materials, such as metallic glasses, behaves nearly in agreement with the assumption of affine deformation, namely that the atoms are displaced just by the amount prescribed by the applied strain. In contrast, the shear modulus behaves as for nonaffine deformations, with additional displacements due to the structural disorder which induce a marked material softening to shear. The consequence is an anomalously large ratio of the bulk modulus to the shear modulus for disordered materials characterized by dense atomic packing, but not for random networks with point atoms. We explain this phenomenon with a microscopic derivation of the elastic moduli of amorphous solids accounting for the interplay of nonaffinity and short-range particle correlations due to excluded volume. Short-range order is responsible for a reduction of the nonaffinity which is much stronger under compression, where the geometric coupling between nonaffinity and the deformation field is strong, whilst under shear this coupling is weak. Predictions of the Poisson ratio based on this model allow us to rationalize the trends as a function of coordination and atomic packing observed with many amorphous materials.

Migration, Force Generation and Mechanosensing of Cells in Collagen Gels

Collagen gels are frequently used to study cell migration in a three-dimensional environment. Mechanical properties of collagen gels are governed by non-affine deformation of the collagen fibrils, such as buckling and tautening, resulting at the macroscopic scale in strain stiffening under shear and a strong lateral contraction under stretch. It is currently unknown how these macroscopic properties play out at the scale of a migrating cell, and how this depends on the cell geometry. To explore this question, we develop a non-linear elastic material model for collagen gels based on observations from confocal microscopy that fibrils can evade mechanical stress using their internal degrees of freedom. This non-affine behavior results in a non-linear force length relationship of fibril segments and leads to a macroscopic strain stiffening and lateral contraction. In particular, we show that tautening of fibrils results in a strong material stiffening against expanding forces, e.g. from a migrating cell with a diameter larger than the network pore diameter. By this mechanism, even a soft collagen gel can sterically constrain a migrating cell. Using our material model, we compute cell traction forces, induced stresses as well as material stiffening, from collagen fiber displacements during the migration of MDA-MB 231 breast carcinoma cells through dilute (0.6 mg/ml, Young's modulus 85 Pa) and dense (2.4 mg/ml, Young's modulus 1100 Pa) collagen gels. We find that cells exert highly localized forces onto the matrix, leading to a localized ∼2-fold material stiffening. However, the average traction force magnitude increases with collagen concentration by ∼3-fold, from 40nN in dilute gels to 120nN in dense gels. This observation may help explain why cells can migrate more efficiently in stiffer gels, despite their narrower pore diameter.

Non-affine fields in solid–solid transformations: the structure and stability of a product droplet

We describe the microstructure, morphology, and dynamics of growth of a droplet of martensite nucleating in a parent austenite during a solid–solid transformation, using a Landau theory written in terms of both conventional affine elastic deformations and non-affine deformations. Non-affineness, ϕ, serves as a source of strain incompatibility and screens long-ranged elastic interactions. It is produced wherever the local stress exceeds a threshold and anneals diffusively thereafter. Using a variational calculation, we find three types of stable solution (labeled I, II, and III) for the structure of the product droplet, depending on the stress threshold and the scaled mobilities of ϕ parallel and perpendicular to the parent–product interface. The profile of the non-affine field ϕ is different in these three solutions: I is characterized by a vanishingly small ϕ, II admits large values of ϕ localized in regions of high stress within the parent–product interface, and III is a structure in which ϕ completely wets the parent–product interface. The width l and size W of the twins follow the relation in solution I; this relation does not hold for II or III. We obtain a dynamical phase diagram featuring these solutions, and argue that they represent specific solid-state microstructures.

A physically based structural model for a textile prosthetic mesh

This study is aimed at a physically based model formulation for a textile mesh implant to describe its mechanical behavior at a mesoscopic length scale. A structural model of a representative unit cell of the knitted prosthetic mesh is proposed based on the theory of multibody systems. The model geometry and the constitutive laws of the force elements are defined based on physical considerations. The parameters determining the force laws, are adjusted to fit the experimentally observed force response and the macro- and mesoscale kinematics. A comparison between the experimental data and the model response show its excellent descriptive capabilities. The level of non-affine deformations within a unit cell is proposed as a mesoscale criterion to quantify the mechanical biocompatibility of textile mesh implants. This view might help to understand clinical observations and complications associated with a local mismatch of deformation between the implant and the host tissue.

Normal stresses in elastic networks

When loaded in simple shear deformation, polymeric materials may develop so-called normal stresses: stresses perpendicular to the direction of the applied shear. These normal stresses are intrinsically nonlinear: basic symmetry considerations dictate they may only enter at O(γ(2)), with γ the dimensionless shear strain. There is no fundamental restriction on their sign, and normal stresses may be positive (pushing outward) or negative (pulling inward). Most materials tend to dilate in the normal direction, but a wide variety of biopolymer networks including fibrin and actin gels have been reported to present anomalously large, negative normal stresses-a feature which has been ascribed to the intrinsic elastic nonlinearity of semiflexible fibers. In this work, we present analytical results on a model nonlinear network, which we expand to the required nonlinear order to show that due to geometric, rather than elastic, nonlinearities (negative) normal stresses generically arise in filamentous networks-even in networks composed of linear, Hookean springs. We investigate analytically and numerically how the subsequent addition of elastic nonlinearities, nonaffine deformations, and filament persistence through cross-linkers augment this basic behavior.

Measuring spatial distribution of the local elastic modulus in glasses

Glasses exhibit spatially inhomogeneous elastic properties, which can be investigated by measuring their elastic moduli at a local scale. Various methods to evaluate the local elastic modulus have been proposed in the literature. A first possibility is to measure the local stress-local strain curve and to obtain the local elastic modulus from the slope of the curve or, equivalently, to use a local fluctuation formula. Another possible route is to assume an affine strain and to use the applied global strain instead of the local strain for the calculation of the local modulus. Most recently, a third technique has been introduced, which is easy to be implemented and has the advantage of low computational cost. In this contribution, we compare these three approaches by using the same model glass and reveal the differences among them caused by the nonaffine deformations.

Field-stiffening effect of magneto-rheological elastomers

Magneto-rheological elastomers (MREs) are a class of soft active materials known for their tunable stiffness. Dispersed with magnetic particles, these polymer-based composites tend to be stiffer under a magnetic field. Such a stiffening effect is often attributed to the magnetic interaction among filler particles, but the well-acknowledged dipole-interaction model fails to explain the stiffening effect in tension/compression, which was observed in experiments. Other mechanisms, such as the effect of non-affine deformation, have been proposed, but there is no conclusive evidence on the dominating mechanism for the field-stiffening effect. This paper investigates various filler-chain structures, and seeks to identify the ultimate origin of the field-stiffening effect in MREs. Two different methods are used for cross verification: a dipole-interaction model and a finite-element simulation based on continuum field theories. This paper studies both the shear and axial deformation of the material, with a magnetic field applied in the particle-chain direction. It is found that while the magnetic interaction between particles is indeed the major cause of the stiffening effect, the wavy chain structure is the key to the modulus increase. Besides, chain–chain interaction and non-affine deformation are shown to be insignificant. In addition, the dependence of the stiffening effect on filler concentration is calculated, and the results qualitatively agree with experimental observations. The models also predict some interesting results that could be easily verified by future experiments.

Nonaffine displacements in crystalline solids in the harmonic limit

A systematic coarse graining of microscopic atomic displacements generates a local elastic deformation tensor D as well as a positive definite scalar χ measuring nonaffinity, i.e., the extent to which the displacements are not representable as affine deformations of a reference crystal. We perform an exact calculation of the statistics of χ and D and their spatial correlations for solids at low temperatures, within a harmonic approximation and in one and two dimensions. We obtain the joint distribution P(χ,D) and the two-point spatial correlation functions for χ and D. We show that nonaffine and affine deformations are coupled even in a harmonic solid, with a strength that depends on the size of the coarse-graining volume Ω and dimensionality. As a corollary to our work, we identify the field h(χ) conjugate to χ and show that this field may be tuned to produce a transition to a state where the ensemble average <χ> and the correlation length of χ diverge. Our work should be useful as a template for understanding nonaffine displacements in realistic systems with or without disorder and as a means for developing computational tools for studying the effects of nonaffine displacements in melting, plastic flow, and the glass transition.

Flow in Foams and Flowing Foams

Aqueous foams are complex fluids composed of gas bubbles tightly packed in a surfactant solution. Even though they generally consist only of Newtonian fluids, foam flow obeys nonlinear laws. This can result from nonaffine deformations of the disordered bubble packing as well as from a coupling between the surface flow in the surfactant monolayers and the bulk liquid flow in the films, channels, and nodes. A similar coupling governs the permeation of liquid through the bubble packing that is observed when foams drain due to gravity. We review the experimental state of the art as well as recent models that describe the interplay of the processes at multiple length scales involved in foam drainage and rheology.

Granular materials: Bridging damaged solids and turbulent fluids

Granular materials exhibit abundant dissipations due to fluctuations in both granular motions and configurations (i.e., granular skeleton) evolutions. Twin granular temperatures Tk and Te are introduced accounting for two types of fluctuations, and the so-called twin granular temperatures theory is established as an extension of granular solid hydrodynamics. By using simulations, the nonaffine deformations in a 2D assembly are simulated by using discrete element methods. By analogizing with microdamages in deformed solids, double scalar damage variables, DP and Dq, are proposed to describe the deformed granular solid under triaxial compressions. Granular flows are found intrinsically turbulent due to the presence of Tk and the Naiver–Stokes equation is obtained for granular flows.

An Improved Non-Affine Arruda-Boyce Type Constitutive Model for Collagen Networks

This work investigates, by means of computational modeling, the mechanical properties of soft collagen tissues on the basis of elasticity theory. Bio-polymer networks are structurally disordered and thus compelled to deform non-affine. To capture that in our computational modeling, we supplement the well-known affine Arruda-Boyce model with positional disorder and compute the resultant changes in mechanical response. We characterize this mechanical behavior as a response to various homogeneous deformations in 3D networks, assuming different constitutive behavior for the individual fibers (in the small deformations linear regime, hookean springs under the entropic elasticity assumption, and in the nonlinear regime freely-jointed and worm-like chains), as well as different coordination numbers (4, 6 or 8 chains connecting at each cross linking point) of the resulting fiber networks. Furthermore we compare the moduli of the simulated networks with their affine deformed counterparts. Previous work has clearly demonstrated that non-affine deformation modes in elastic (bio)polymer networks greatly affect their mechanics. As the original Arruda-Boyce model can be represented with a particular form of strain-energy function that is micro-mechanically motivated, incorporation of the non-affinity yields amended predictions of the macroscopic mechanical behavior of soft fibrous networks, based on an improved representation of microscopic network structure and deformations. We show that shear and bulk moduli in the Arruda-Boyce model can be as off as 30% when compared with the shear and bulk moduli in the non-affine model. This entire evaluation of the ways non-affinity enhances the well known Arruda-Boyce model sets the groundwork for developing accurate constitutive relations for fibrous biological materials, for use in finite element analysis of soft tissues.

Stress relaxation through crosslink unbinding in cytoskeletal networks

The mechanical properties of cells are dominated by the cytoskeleton, an interconnected network of long elastic filaments. The connections between the filaments are provided by crosslinking proteins, which constitute, next to the filaments, the second important mechanical element of the network. An important aspect of cytoskeletal assemblies is their dynamic nature, which allows remodeling in response to external cues. The reversible nature of crosslink binding is an important mechanism that underlies these dynamical processes. Here, we develop a theoretical model that provides us insight into how the mechanical properties of cytoskeletal networks may depend on their underlying constituting elements. We incorporate three important ingredients: non-affine filament deformations in response to network strain; the interplay between filament and crosslink mechanical properties; and reversible crosslink (un)binding in response to the imposed stress. With this we are able to self-consistently calculate the nonlinear modulus of the network as a function of deformation amplitude and crosslink as well as filament stiffnesses. During loading, crosslink unbinding processes lead to a relaxation of stress and therefore to a reduction of the network modulus and eventually to network failure, when all crosslinks are unbound. This softening due to crosslink unbinding generically competes with an inherent stiffening response, which may be due to either filament or crosslink nonlinear elasticity.

Meso-mechanical analysis of deformation characteristics for dynamically triggered slip in a granular medium

The deformation characteristics of a sheared granular layer during stick–slip are studied from a meso-mechanical viewpoint, both in the absence and in the presence of externally applied vibration. The ultimate goal is to characterize the physics of dynamic earthquake triggering, where one earthquake, i.e., slip on one fault, is triggered via the seismic waves radiated by another spatially and temporally distant seismic event. Toward this goal, we performed Discrete Element Method simulations of a two-dimensional packing of disks, mimicking a mature geologic fault. These simulations were used to investigate the affine and non-affine deformations inside the granular layer and their spatial–temporal evolution across the stick–slip cycle. The simulation results show that slip in general is accompanied by the appearance of localized regions with high values of both affine and non-affine deformations. These regions are temporally correlated and are mainly concentrated in a shear zone at the interface between the granular layer and the driving block. Dynamic triggering is found to initiate slip when vibration is applied late in the stick–slip cycle, when the system is close to a critical state. It is also found that vibration itself introduces a large amount of affine and non-affine strains, which leads to the initiation of slip at lower shear stress than an equivalent slip event without vibration.

Evaluating Mohr–Coulomb yield criterion for plastic flow in model metallic glasses

The yielding of metallic glasses (MGs) is generally considered to be dependent on pressure or normal stress, which is usually described by the Mohr–Coulomb yield criterion. Experimentally, the orientation of the shear band angle or fracture angles in MG is roughly 40° to 42° in uniaxial compression and 50° to 59° in uniaxial tension, both with respect to the loading direction. However, the amounts of deviation from the maximum shear stress direction (45°) under uniaxial tension and compression are quite different, which seems to be inconsistent with the Mohr–Coulomb yield criterion. Here we carried out molecular dynamics simulations to study the incipient plasticity of model Dzugutov glasses subjected to various uniaxial and biaxial stress states. The yield points are defined as the sharp rise of the population of atoms participating in non-affine deformations, in addition to the traditional offset method. The yield surfaces thus created agree well with the Mohr–Coulomb yield criterion. Furthermore, the orientations of the embryonic shear bands (instead of the fully-grown shear bands, or fracture planes) are measured, which also follow predictions based on the Mohr–Coulomb criterion. Our simulation results call for orientational measurement of incipient shear bands in experiments to critically examine the Mohr–Coulomb yield criterion in MG systems.