Non-affine Deformation Research Articles

The Theories of Rubber Elasticity and the Goodness of Their Constitutive Stress–Strain Equations

One of the most important challenges in polymer science is a rigorous understanding of the molecular mechanisms of rubber elasticity by relating macroscopic deformation to molecular changes and deriving the constitutive stress–strain equation for the elastomeric network. The models developed from the last century to today describe many aspects of the physics of rubber elasticity; although these theories are successful, they are not complete. In this review we analyze the main theoretical and phenomenological models of rubber elasticity, including their assumptions, main characteristics, and stress–strain equations. Then, we compare the predictions of the theories to our experimental data of polydimethylsiloxane (PDMS) rubber, in order to highlight the goodness of the reviewed models. The nonaffine and phenomenological deformation models verify the experimental curves in tension and compression in the whole investigated deformation range λ≤2. On the contrary, the affine deformation hypothesis is rigorously verified only in the deformation range λ≤1.

Lightweight silicon and glass composites with submicron viscoelastic interlayers and unconventional combinations of stiffness and damping

The necessity of stiff structural materials with advanced damping characteristics has arisen naturally along with the technological evolution. However, despite ever-growing demand, the achievement of stiff materials with high damping factor remains challenging because of the antagonistic nature of the two properties. Here, this challenge is accomplished by exploiting the non-affine deformation of laminated composites paired with the dissipative nature of the viscoelastic phase. Guided by a finite element design analysis, composites with flat submillimeter stiff layers and submicron viscoelastic interlayers are fabricated. The viscoelastic component consists of a prudently chemically architectured comb-like polydimethylsiloxane (PDMS) elastomer that properly adheres to the stiff silicon or glass layers, strong enough to withstand repeated dynamic cycles. The composites are fabricated using an unconventional but simple stacking route based on the diffusion of a platinum catalyst precursor into a reactive solvent-free PDMS melt. The fabricated composites, Si/PDMS and glass/PDMS, exhibit an elastic modulus higher than common monolithic glass, they are as light as glass but have about four orders of magnitude higher loss factor. The composites markedly outperform numerous customary materials, they escape the Ashby limit for mechanical damping – stiffness trade-off, and their exceptional combinations of properties are maintained over a broad range of temperatures and frequencies.

Perspective on the description of viscoelastic flows via continuum elastic dumbbell models

Non-Newtonian fluid mechanics and computational rheology widely exploit elastic dumbbell models such as Oldroyd-B and FENE-P for a continuum description of viscoelastic fluid flows. However, these constitutive equations fail to accurately capture some characteristics of realistic polymers, such as the steady extension in simple shear and extensional flows, thus questioning the ability of continuum-level modeling to predict the hydrodynamic behavior of viscoelastic fluids in more complex flows. Here, we present seven elastic dumbbell models, which include different microstructurally inspired terms, i.e., (i) the finite polymer extensibility, (ii) the conformation-dependent friction coefficient, and (iii) the conformation-dependent non-affine deformation. We provide the expressions for the steady dumbbell extension in shear and extensional flows and the corresponding viscosities for various elastic dumbbell models incorporating different microscopic features. We show the necessity of including these microscopic features in a constitutive equation to reproduce the experimentally observed polymer extension in shear and extensional flows, highlighting their potential significance in accurately modeling viscoelastic channel flow with mixed kinematics.

Investigating Temperature-Dependent Microscopic Deformation in Tough and Self-Healing Hydrogel Using Time-Resolved USAXS.

Tough and self-healing hydrogels are typically sensitive to loading rates or temperatures due to the dynamic nature of noncovalent bonds. Understanding the structure evolution under varying loading conditions can provide valuable insights for developing new tough soft materials. In this study, polyampholyte (PA) hydrogel with a hierarchical structure is used as a model system. The evolution of the microscopic structure during loading is investigated under varied loading temperatures. By combining ultra-small angle X-ray scattering (USAXS) and Mooney-Rivlin analysis, it is elucidated that the deformation of bicontinuous hard/soft phase networks is closely correlated with the relaxation dynamics or strength of noncovalent bonds. At high loading temperatures, the gel is soft and ductile, and large affine deformation of the phase-separated networks is observed, correlated with the fast relaxation dynamics of noncovalent bonds. At low loading temperatures, the gel is stiff, and nonaffine deformation occurs from the onset of loading due to the substantial breaking of noncovalent bonds and limited chain mobility as well as weak adaptation of phase deformation to external stretch. This work provides an in-depth understanding of the relationship between structure and performance of tough and self-healing hydrogels.

Applicability of Kroon network theory for biaxial mechanics in rubber through multiscale approach

In this paper, we address the problem of the multiscale prediction of the multiaxial hyperelastic behavior of rubber. Our goal is to test the predictive capabilities of a hyperelastic model grounded in the principles of continuum mechanics, bridging the gap between continuum-level mechanics and molecular-level descriptions. This constitutive model posits that macroscopic behavior results from the non-affine deformation of the molecular network, influenced by topological constraints. We employ molecular dynamics methods to measure the mechanical behavior of a rubber network structure under various deformation modes. The study includes an analysis of the molecular network deformation during the straining process, shedding light on the deformation mechanisms at the molecular level. By examining how the total network stress in our specifically designed molecular structure responds to biaxial strain ratios, we are able to assess the effectiveness of the hyperelastic constitutive model.

Methodological Aspects and Mesh Convergence in Numerical Analysis of Athermal Fiber Network Material Deformation

A balance between model complexity, accuracy, and computational cost is a central concern in numerical simulations. In particular, for stochastic fiber networks, the non-affine deformation of fibers, related non-linear geometric features due to large global deformation, and size effects can significantly affect the accuracy of the computer experiment outputs and increase the computational cost. In this work, we systematically investigate methodological aspects of fiber network simulations with a focus on the output accuracy and computational cost in models with cellular (Voronoi) and fibrous (Mikado) network architecture. We study both p and h-refinement of the discretizations in finite element solution procedure, with uniform and length-based adaptive h-refinement strategies. The analysis is conducted for linear elastic and viscoelastic constitutive behavior of the fibers, as well as for networks with initially straight and crimped fibers. With relative error as the determining criterion, we provide recommendations for mesh refinement, comment on the necessity of multiple realizations, and give an overview of associated computational cost that will serve as guidance toward minimizing the computational cost while maintaining a desired level of solution accuracy.

Non-affine deformation analysis and 3D packing defects: A new way to probe membrane heterogeneity in molecular simulations.

Here, we discuss a new framework developed over the last 5 years in our group to probe nanoscale membrane heterogeneity. The framework is based on the idea of characterizing lateral heterogeneity through non-affine deformation (NAD) measurements, transverse heterogeneity through three dimensional (3D) lipid packing defects, and using these approaches to formalize the seemingly trivial correlation between lateral organization and lipid packing in biological membranes. We find that measurements from NAD analysis, a prescription which is borrowed from Physics of glasses and granular material, can faithfully distinguish between liquid-ordered and disordered phases in membranes at molecular length scalesand, can also be used to identify phase boundaries with high precision. Concomitantly, 3D-packing defects can not only distinguish between the two co-existing fluid phases based on their molecular scale packing (or membrane free volume), but also provide a route to connect the membrane domains to their functionality, such as exploring the molecular origins of inter-leaflet domain registration and peptide partitioning. The correlation between lateral membrane order and transverse packing presents novel molecular design-level features that can explain functions such as protein/peptide partitioning and small-molecule permeation dynamics in complex and heterogeneous membranes with high-fidelity. The framework allows us to explore the nature of lateral organization and molecular packing as a manifestation of intricate molecular interactions among a chemically rich variety of lipids and other molecules in a membrane with complex membrane composition and asymmetry across leaflets.

Elastic–plastic intermittent re-arrangements of frictionless, soft granular matter under very slow isotropic deformations

How do soft granular materials (or dense amorphous systems) respond to externally applied deformations at different rates and for different system sizes? This long-standing question was intensively studied for shear deformations but only more recently for isotropic deformations, like compression–decompression cycles. For moderate strain rates, in the solid-like state, above jamming, the system appears to evolve more or less smoothly in time/strain, whereas for slow enough deformations, the material flips intermittently between the elastic, reversible base state and plastic, dynamic “events.” Only during the latter events, the microstructure re-arranges irreversibly. The reversible base state involves both affine and non-affine deformations, while the events are purely non-affine. The system size and rate dependence of the events are studied, providing reference data for comparison in future studies evaluating materials like hydrogel particles.

Coarse-grained molecular dynamics simulation on strain-hardening and fracture behaviors of polycarbonate: Effect of polydispersity and spatial distribution

Coarse-grained molecular dynamics simulation of polycarbonate is utilized to investigate the relationship between molecular structure (i.e., polydispersity and molecular spatial distribution) and strain-hardening and fracture behavior of polycarbonate. We find that strain-hardening modulus and chain extensibility, which are the constitutive parameters of the Eindhoven Glassy Polymer model are highly affected by spatial distribution but are insensitive to polydispersity. This is attributed to the higher rate of nonaffine deformation in the structure with a high radius of gyration. On the other hand, maximum stress at fracture is highly influenced by both spatial distribution and polydispersity due to the ability to sustain entanglements at larger strain. We suggest the phenomenological expression of maximum stress as a function of the radius of gyration, the number of entanglements, and polydispersity.

Elasticity of Highly Entangled Polymer Networks and Gels: Review of Models and Theory of Nonaffine Deformations

The main models of phantom and topologically entangled polymer networks are surveyed. A theory of anisotropic and nonaffine deformation of both swollen and deswollen (with partial solvent removal) strongly entangled polymer networks in athermal and θ-solvents has been developed. It is shown that under weak anisotropic deformations of the deswollen network, the entanglement tube consists of fractal loopy globules. In a θ-solvent, slight deformations of the network lead to a decrease in the overlap of loopy globules without changing their sizes. Deformations of swollen networks, as well as strong deformations of deswollen networks, are described in terms of the slip-tube model. An effective Hamiltonian has been derived that determines the entropy of fractal loopy globules. Based on the Hamiltonian, it is shown that topological constraints can be described using the polymer–quantum diffusion analogy. The connection between topological and quantum entanglements is demonstrated.

Brain tissue mechanics is governed by microscale relations of the tissue constituents

Local mechanical tissue properties are a critical regulator of cell function in the central nervous system (CNS) during development and disorder. However, we still don't fully understand how the mechanical properties of individual tissue constituents, such as cell nuclei or myelin, determine tissue mechanics. Here we developed a model predicting local tissue mechanics, which induces non-affine deformations of the tissue components. Using the mouse hippocampus and cerebellum as model systems, we show that considering individual tissue components alone, as identified by immunohistochemistry, is not sufficient to reproduce the local mechanical properties of CNS tissue. Our results suggest that brain tissue shows a universal response to applied forces that depends not only on the amount and stiffness of the individual tissue constituents but also on the way how they assemble. Our model may unify current incongruences between the mechanics of soft biological tissues and the underlying constituents and facilitate the design of better biomedical materials and engineered tissues. To this end, we provide a freely-available platform to predict local tissue elasticity upon providing immunohistochemistry images and stiffness values for the constituents of the tissue.

Elasticity of Highly Entangled Polymer Networks and Gels: Review of Models and Theory of Nonaffine Deformations

Elasticity of Highly Entangled Polymer Networks and Gels: Review of Models and Theory of Nonaffine Deformations

Univariate Gauss quadrature for structural modelling of tissues and materials with distributed fibres

This work presents a method for computing the averaged free energy and constitutive relations in hyperelastic material models with distributed fibres, as they apply to soft fibre-reinforced materials and biological tissues. While these models are currently implemented through either spherical cubature of the fibre free energy or its Taylor series, we here propose a new method based on a univariate Gauss quadrature rule with integration points and weights informed by the statistical moments of the distribution of fibre stretch. As an intrinsic property, the new approach separates the integration of the fibre constitutive law from the integration of the orientation distribution, the latter leading to structural tensors of even order. Provided the latter 2n−1 tensors are computed accurately up to tensorial order 2(2n−1), the method integrates exactly any polynomial of order 2n−1 that agrees with the fibre law at the n integration points. After formally introducing the quadrature method for generally non-affine fibre deformations and arbitrary order, we focus on the important special case of affine fibre kinematics and discuss the rules with n≤3 integration points, for which the corresponding positions and weights are determined analytically. At a computational cost comparable to the existing approaches, the new method does not require the fibre law to be analytic and can thus robustly deal with piece-wise definitions of the fibre energy, in contrast to Taylor-series approaches, and it does not induce additional anisotropy as it can occur with spherical cubature rules. The 3-point rule is further investigated and illustrated in numerical examples relating to soft collagenous tissues based on a Fortran implementation of the method suitable for use in finite element analyses.

Lipid packing in biological membranes governs protein localization and membrane permeability

Plasma membrane (PM) heterogeneity has long been implicated in various cellular functions. However, mechanistic principles governing functional regulations of lipid environment are not well understood due to the inherent complexities associated with the relevant length and timescales that limit both direct experimental measurements and their interpretation. In this context, computer simulations hold immense potential to investigate molecular-level interactions and mechanisms that lead to PM heterogeneity and its functions. Herein, we investigate spatial and dynamic heterogeneity in model membranes with coexisting liquid ordered and liquid disordered phases and characterize the membrane order in terms of the local topological changes in lipid environment using the nonaffine deformation framework. Furthermore, we probe the packing defects in these membranes, which can be considered as the conjugate of membrane order assessed in terms of the nonaffine parameter. In doing so, we formalize the connection between membrane packing and local membrane order and use that to explore the mechanistic principles behind their functions. Our observations suggest that heterogeneity in mixed phase membranes is a consequence of local lipid topology and its temporal evolution, which give rise to disparate lipid packing in ordered and disordered domains. This in turn governs the distinct nature of packing defects in these domains that can play a crucial role in preferential localization of proteins in mixed phase membranes. Furthermore, we observe that lipid packing also leads to contrasting distribution of free volume in the membrane core region in ordered and disordered membranes, which can lead to distinctive membrane permeability of small molecules. Our results, thus, indicate that heterogeneity in mixed phase membranes closely governs the membrane functions that may emerge from packing-related basic design principles.

Addressing the Folding of Intermolecular Springs in Particle Simulations: Fixed Image Convention

Mesoscopic simulations of long polymer chains and soft matter systems are conducted routinely in the literature in order to assess the long-lived relaxation processes manifested in these systems. Coarse-grained chains are, however, prone to unphysical intercrossing due to their inherent softness. This issue can be resolved by introducing long intermolecular bonds (the so-called slip-springs) which restore these topological constraints. The separation vector of intermolecular bonds can be determined by enforcing the commonly adopted minimum image convention (MIC). Because these bonds are soft and long (ca 3–20 nm), subjecting the samples to extreme deformations can lead to topology violations when enforcing the MIC. We propose the fixed image convention (FIC) for determining the separation vectors of overextended bonds, which is more stable than the MIC and applicable to extreme deformations. The FIC is simple to implement and, in general, more efficient than the MIC. Side-by-side comparisons between the MIC and FIC demonstrate that, when using the FIC, the topology remains intact even in situations with extreme particle displacement and nonaffine deformation. The accuracy of these conventions is the same when applying affine deformation. The article is accompanied by the corresponding code for implementing the FIC.

On the tensile response of formed fiber networks with low areal density

Fiber networks are ubiquitous. The networks of interest to this study are distinguished from the widely studied planar random fiber networks (RFN) by their low areal density (grammage), preferential orientation, and fiber curl. Oriented networks are created through a high speed manufacturing process called forming in paper making. A dilute suspension of elastic fibers in water impinges on a moving wire to create a formed network. A discrete element method (DEM) based forming model is developed that incorporates the qualitative effects of fluid shear (preferred fiber orientation) and viscous drag during drainage (fiber conformation to the wire topography). Virtual tensile tests performed on thus formed networks reveal that the tensile response is initially softer due to non-affine deformation, and anisotropic due to preferential orientation. Consequently, an orientation dependent power law with a non-unique exponent emerges for the scaling of modulus and strength with the network density. Tensile test simulations reveal the re-alignment of fibers along the loading direction, and strong strain localization resulting in a few fibers (∼1%) carrying the imposed strain energy. Fiber curl and orientation are observed to govern the degree of non-affine response, stiffness, and strain localization, suggesting that controlling the degree of anisotropy and fiber curl offers the possibility to design networks for specific stiffness and specific strength.

An elastic–viscoplastic model with non-affine deformation and rotation of a distribution of embedded fibres

A large deformation continuum model for an elastically isotropic visco-plastic matrix reinforced by a distribution of fibres has been formulated. The fibre distribution is a symmetric positive-definite tensor with unit trace that is determined by an evolution equation. This evolution equation includes a competition between tendencies for affine evolution of the distribution and non-affine evolution causing the distribution to align with the elastic distortional deformation tensor or to become isotropic. Two elastic deformation measures are introduced associated with the fibre distribution. Specifically, non-affine measures of elastic distribution stretch and non-coaxiality of the distribution cause anisotropic stress response. The resulting model describes a number of effects relevant to growth and remodelling of biological tissues as well as to additive manufacturing of artificial materials. For example, the equations model the smooth transition from isotropic to anisotropic response seen in many biological tissues.

Nonaffine Deformation of Semiflexible Polymer and Fiber Networks

Networks of semiflexible or stiff polymers such as most biopolymers are known to deform inhomogeneously when sheared. The effects of such nonaffine deformation have been shown to be much stronger than for flexible polymers. To date, our understanding of nonaffinity in such systems is limited to simulations or specific 2D models of athermal fibers. Here, we present an effective medium theory for nonaffine deformation of semiflexible polymer and fiber networks, which is general to both 2D and 3D and in both thermal and athermal limits. The predictions of this model are in good agreement with both prior computational and experimental results for linear elasticity. Moreover, the framework we introduce can be extended to address nonlinear elasticity and network dynamics.

Non-affine dissipation in polymer fracture

Viscoelasticity is a universal property in most soft solids. Specifically, for the fracture of polymeric materials, viscoelasticity has often been regarded as a major mechanism of toughening, and the excess fracture energy beyond the intrinsic value correlated to the stress–strain hysteresis, which is commonly used to characterize viscoelasticity in the framework of continuum mechanics. Through systematically planned experiments, the current study shows that a noticeable package of energy dissipation exists besides that associated with the hysteretic deformation. This package of energy is attributed to the highly non-affine deformation near the crack tip — the rearrangement and mutual sliding of the neighboring chains caused by the rupture of a polymer chain, which are insignificant in samples under homogeneous deformation. Various means of inter-chain-friction reduction, e.g., by applying lateral stretches, diluting the chains with solvent, or subject the material to dynamic or static fatigue, can mitigate the effect of non-affine dissipation, and bring the fracture energy down to the intrinsic threshold. In contrast, by making an elastomer more elastic, e.g., by training through cyclic loading, the non-affine dissipation remains effective, and the limiting fracture energy is significantly higher than the threshold even when the hysteresis is negligible. The identification of the non-affine-dissipation mechanism paves the way to the development of new soft materials with both excellent elasticity and significant fracture toughness.

The role of non-affine deformations in the elastic behavior of the cellular vertex model.

The vertex model of epithelia describes the apical surface of a tissue as a tiling of polygonal cells, with a mechanical energy governed by deviations in cell shape from preferred, or target, area, A0, and perimeter, P0. The model exhibits a rigidity transition driven by geometric incompatibility as tuned by the target shape index, . For with p*(6) the perimeter of a regular hexagon of unit area, a cell can simultaneously attain both the preferred area and preferred perimeter. As a result, the tissue is in a mechanically soft compatible state, with zero shear and Young's moduli. For p0 < p*(6), it is geometrically impossible for any cell to realize the preferred area and perimeter simultaneously, and the tissue is in an incompatible rigid solid state. Using a mean-field approach, we present a complete analytical calculation of the linear elastic moduli of an ordered vertex model. We analyze a relaxation step that includes non-affine deformations, leading to a softer response than previously reported. The origin of the vanishing shear and Young's moduli in the compatible state is the presence of zero-energy deformations of cell shape. The bulk modulus exhibits a jump discontinuity at the transition and can be lower in the rigid state than in the fluid-like state. The Poisson's ratio can become negative which lowers the bulk and Young's moduli. Our work provides a unified treatment of linear elasticity for the vertex model and demonstrates that this linear response is protocol-dependent.