Bulk Elastic Energy Research Articles

The poker-chip experiments of synthetic elastomers explained

In a recent study, Kumar and Lopez-Pamies (J. Mech. Phys. Solids 150: 104359, 2021) have provided a complete quantitative explanation of the famed poker-chip experiments of Gent and Lindley (Proc. R. Soc. Lond. Ser. A 249: 195–205, 1959) on natural rubber. In a nutshell, making use of the fracture theory of Kumar, Francfort, and Lopez-Pamies (J. Mech. Phys. Solids 112: 523–551, 2018), they have shown that the nucleation of cracks in poker-chip experiments in natural rubber is governed by the strength — in particular, the hydrostatic strength — of the rubber, while the propagation of the nucleated cracks is governed by the Griffith competition between the bulk elastic energy of the rubber and its intrinsic fracture energy. The main objective of this paper is to extend the theoretical study of the poker-chip experiment by Kumar and Lopez-Pamies to synthetic elastomers that, as opposed to natural rubber: (i) may feature a hydrostatic strength that is larger than their uniaxial and biaxial tensile strengths and (ii) do not typically exhibit strain-induced crystallization. A parametric study, together with direct comparisons with recent poker-chip experiments on a silicone elastomer, show that these two different material characteristics have a profound impact on where and when cracks nucleate, as well as on where and when they propagate. In conjunction with the results put forth earlier for natural rubber, the results presented in this paper provide a complete description and explanation of the poker-chip experiments of elastomers at large. As a second objective, this paper also introduces a new fully explicit constitutive prescription for the driving force that describes the material strength in the fracture theory of Kumar, Francfort, and Lopez-Pamies.

A phase field fracture model for ultra-thin micro-/nano-films with surface effects

Surface effects usually remarkably affect the mechanical response of ultra-thin micro-/nano-structures. However, the mechanisms of surface effects on the fracture characteristics of ultra-thin films are still not fully understood. To this end, this paper develops a modeling framework to investigate the fracture of ultra-thin films at microscales or below. Such a framework couples the Gurtin–Murdoch theory with a phase-field fracture model, in which the former is adopted to introduce the surface effects, i.e., the surface residual stress and surface elasticity of a thin film, and the latter is able to model crack evolution without requiring predefined crack paths or any criteria. Furthermore, a novel crack driving force is introduced, which encompasses the tensile components of both bulk elastic energy and surface elastic energy. Several numerical examples including the biaxial tension test as well as the single-edge notched tension/shear test are performed. The simulation results indicate that the surface strain energy plays a major role in the total elastic strain energy of an ultra-thin film when its thickness is at a micro level, thus demonstrating the significance of surface effects. Moreover, the mode-I fracture test shows that the surface elasticity and surface residual stress have a remarkable influence on the displacement at failure, while for the mode-II fracture test, the surface residual stress significantly influences the fracture characteristics such as the crack path and failure displacement. The developed model paves the way for revealing the fracture mechanisms of ultra-thin micro-/nano-films and conducting their safety assessment.

A Numerical Study on the Performance of Liquid Crystal Biosensor Microdroplets

The numerical results from the modeling of liquid crystals dispersed in aqueous solutions in the form of axially symmetric droplets, with the aim of helping to facilitate the development of liquid crystal biosensors, were obtained. We developed a transient two-dimensional nonlinear model obtained via torque balance that incorporates Frank’s elastic free energy. In order to perform parametric studies, we defined the scaled parameters based on the surface viscosity and the homeotropic anchoring energy at the droplet interface. To evaluate the performance of the biosensor, the average angle and characteristic time were defined as performance criteria. Using these results, we studied the bulk reorientation of liquid crystal droplets in aqueous solutions caused by biomolecular interaction. Furthermore, we examined how surface viscosity affects the performance of a biosensor in the case of weak planar anchoring. The droplet interface ordering was modeled using the Euler–Lagrange equation. The droplets’ equilibrium was determined by minimizing their total distortion energy based on the interaction between their surface and bulk elastic energy. Two factors that contributed to the biosensor performance were homeotropic strength and surface viscosity. This highlights the importance of controlling the surface and physicochemical properties to achieve the desired liquid crystal orientation. In addition, our results provide insight into the role that surface viscosity plays in controlling radial configuration.

Weak-anchoring effects in a thin pinned ridge of nematic liquid crystal.

A theoretical investigation of weak-anchoring effects in a thin two-dimensional pinned static ridge of nematic liquid crystal resting on a flat solid substrate in an atmosphere of passive gas is performed. Specifically, we solve a reduced version of the general system of governing equationsrecently derived by Cousins etal. [Proc. R. Soc. A 478, 20210849 (2022)10.1098/rspa.2021.0849] valid for a symmetric thin ridge under the one-constant approximation of the Frank-Oseen bulk elastic energy with pinned contact lines to determine the shape of the ridge and the behavior of the director within it. Numerical investigations covering a wide range of parameter values indicate that the energetically preferred solutions can be classified in terms of the Jenkins-Barratt-Barbero-Barberi critical thickness into five qualitatively different types of solution. In particular, the theoretical results suggest that anchoring breaking occurs close to the contact lines. The theoretical predictions are supported by the results of physical experiments for a ridge of the nematic 4^{'}-pentyl-4-biphenylcarbonitrile (5CB). In particular, these experiments show that the homeotropic anchoring at the gas-nematic interface is broken close to the contact lines by the stronger rubbed planar anchoring at the nematic-substrate interface. A comparison between the experimental values of and the theoretical predictions for the effective refractive index of the ridge gives a first estimate of the anchoring strength of an interface between air and 5CB to be (9.80±1.12)×10^{-6}Nm^{-1} at a temperature of (22±1.5)^{∘}C.

Energies for Elastic Plates and Shells from Quadratic-Stretch Elasticity

We derive stretching and bending energies for isotropic elastic plates and shells. Through the dimensional reduction of a bulk elastic energy quadratic in Biot strains, we obtain two-dimensional bending energies quadratic in bending measures featuring a bilinear coupling of stretches and geometric curvatures. For plates, the bending measure is invariant under spatial dilations and naturally extends primitive bending strains for straight rods. For shells or naturally-curved rods, the measure is not dilation invariant, and contrasts with previous ad hoc postulated forms. The corresponding field equations and boundary conditions feature moments linear in the bending measures, and a decoupling of stretching and bending such that application of a pure moment results in isometric deformation of a unique neutral surface, primitive behaviors in agreement with classical linear response but not displayed by commonly used analytical models. We briefly comment on relations between our energies, those derived from a neo-Hookean bulk energy, and a commonly used discrete model for flat membranes. Although the derivation requires consideration of stretch and rotation fields, the resulting energy and field equations can be expressed entirely in terms of metric and curvature components of deformed and reference surfaces.

Bimodal polymorphic nanodomains in ferroelectric films for giant energy storage

Enhanced functional responses of ferroelectrics have often been achieved by complex compositional design of a morphotropic or polymorphic phase boundary. Here, we observe a pseudo-linear, ultra-slim polarization-electric field (P-E) hysteresis loop in simple composition, lead-free Ba(Zr,Ti)O3 of up to μm-thick. It features a low remnant polarization and a high maximum polarization after a delayed saturation process with a field-insensitive, thickness-scalable high energy efficiency of ∼90%. This giant energy storage performance is attributed to the self-assembled, bimodal polymorphic nanodomains consisting of two sets of coherent polymorphic nanodomains. The first set of domains has the best-matched, low index {110} interface. They manifest themselves as “entangled nanophases” and dominate in thinner or annealed films. The 2nd set of domains with a high-index {114} interface and an in-plane anisotropy leads to the largest reduction in bulk elastic energy. These domains show as “segregated nanophases” and prevail in as-grown thick films. The presence of multi-polar states in both sets of nanodomains helps reduce the remnant polarization and delay the polarization saturation. Furthermore, the segregated tetragonal phase in-plane nanodomains lead to a larger maximum polarization under a high poling field. Strain engineering of such nanodomain structures provides a promising alternative to chemical compositional design, for the optimization of dielectric thin films used in capacitive energy storage applications.

Curvature of the elastic deformations in a nematic sample.

In this work we study the geometry of the elastic deformations of the uniaxial nematic liquid crystals at the bulk. We will show that, at this region of the sample, the elastic terms of the free energy can be separated as the sum of two kinds of elastic deformations, the first is proportional to the Gaussian curvature obtained from the director field of a three-dimensional nematic sample and the second is composed by those terms that cannot be expressed as resulting from this curvature. To achieve these results we will construct the metric of an unixial nematic sample using the fact that the director gives the direction of the anisotropy of the system. With this approach we will give analytical and geometrical arguments to show that the elastic terms determined by [Formula: see text], [Formula: see text] and [Formula: see text] are contained in a curvature term, while the terms fixed by the splay elastic term, [Formula: see text], and the bend elastic term, [Formula: see text], are not. The novelty here is that while [Formula: see text] and [Formula: see text] do not contribute the bulk elastic energy of a nematic sample, they have an important contribution to the curvature of the system.

Computational modelling of tactoid dynamics in chromonic liquid crystals

ABSTRACTMotivated by recent experiments, the isotropic–nematic phase transition in chromonic liquid crystals is studied. As temperature decreases, nematic nuclei nucleate, grow and coalesce, giving rise to tactoid microstructures in an isotropic liquid. These tactoids produce topological defects at domain junctions (disclinations in the bulk or point defects on the surface). We simulate such tactoid equilibria and their coarsening dynamics with a model using degree of order, a variable length director and an interfacial normal as state descriptors. We adopt Ericksen’s work and introduce an augmented Oseen–Frank energy, with non-convexity in both interfacial energy and the dependence of the energy on the degree of order. A gradient flow dynamics of this energy does not succeed in reproducing some simple expected feature of tactoid dynamics. Therefore, a strategy is devised based on continuum kinematics and thermodynamics to represent such features. The model is used to predict tactoid nucleation, expansion and coalescence during the process of phase transition. We reproduce observed behaviours in experiments and perform an experimentally testable parametric study of the effect of bulk elastic and tactoid interfacial energy parameters on the interaction of interfacial and bulk fields in the tactoids.

A Variational Model for Dislocations at Semi-coherent Interfaces

We propose and analyze a simple variational model for dislocations at semi-coherent interfaces. The energy functional describes the competition between two terms: a surface energy induced by dislocations that compensate the lattice misfit at the interface, and a far field elastic energy, spent to decrease the amount of needed dislocations. We prove that the former scales like the surface area of the interface, the latter like its diameter. The proposed continuum model is deduced from some rigorous derivation from the semi-discrete theory of dislocations. Even if we deal with finite elasticity, linearized elasticity naturally emerges in our analysis since the far field strain vanishes as the interface size increases.

Anomalously Soft Non-Euclidean Springs

In this work we study the mechanical properties of a frustrated elastic ribbon spring-the non-Euclidean minimal spring. This spring belongs to the family of non-Euclidean plates: it has no spontaneous curvature, but its lateral intrinsic geometry is described by a non-Euclidean reference metric. The reference metric of the minimal spring is hyperbolic, and can be embedded as a minimal surface. We argue that the existence of a continuous set of such isometric minimal surfaces with different extensions leads to a complete degeneracy of the bulk elastic energy of the minimal spring under elongation. This degeneracy is removed only by boundary layer effects. As a result, the mechanical properties of the minimal spring are unusual: the spring is ultrasoft with a rigidity that depends on the thickness t as t^{7/2} and does not explicitly depend on the ribbon's width. Moreover, we show that as the ribbon is widened, the rigidity may even decrease. These predictions are confirmed by a numerical study of a constrained spring. This work is the first to address the unusual mechanical properties of constrained non-Euclidean elastic objects.

Beading instability in soft cylindrical gels with capillary energy: Weakly non-linear analysis and numerical simulations

Soft cylindrical gels can develop a long-wavelength peristaltic pattern driven by a competition between surface tension and bulk elastic energy. In contrast to the Rayleigh–Plateau instability for viscous fluids, the macroscopic shape in soft solids evolves toward a stable beading, which strongly differs from the buckling arising in compressed elastic cylinders.This work proposes a novel theoretical and numerical approach for studying the onset and the non-linear development of the elasto-capillary beading in soft cylinders, made of neo-Hookean hyperelastic material with capillary energy at the free surface, subjected to axial stretch. Both a theoretical study, deriving the linear and the weakly non-linear stability analyses for the problem, and numerical simulations, investigating the fully non-linear evolution of the beaded morphology, are performed. The theoretical results prove that an axial elongation can not only favour the onset of beading, but also determine the nature of the elastic bifurcation. The fully non-linear phase diagrams of the beading are also derived from finite element numerical simulations, showing two peculiar morphological transitions when varying either the axial stretch or the material properties of the gel. Since the bifurcation is found to be subcritical for very slender cylinders, an imperfection sensitivity analysis is finally performed. In this case, it is shown that a surface sinusoidal imperfection can resonate with the corresponding marginally stable solution, thus selecting the emerging beading wavelength.In conclusion, the results of this study provide novel guidelines for controlling the beaded morphology in different experimental conditions, with important applications in micro-fabrication techniques, such as electrospun fibres.

Relaxation of granular elasticity

In granular materials, particles constitute a complex force chains network through contact with each other, and elastic energies are stored due to deformation of particles. This elastic behavior is macroscopic manifestation of inter-particle contacts. Elastic constants or elastic moduli are of fundamental importance for granular material. Due to the hyper-static property of inter-particle forces, the bulk elastic energy stored in the contacts is metastable in the viewpoint of energy landscape, i.e. a high energy state may approaches a more stable state (i.e. relatively lower state) under the action of external perturbations or internal stress, resulting in the elastic modulus reduction. This process is the so-called elasticity relaxation. It may be more obvious in granular materials.The time-dependent behavior of granular materials, especially the creep, has been studied in experiments and numerical simulations, while the stress relaxation has few reported investigations. Stress relaxation is defined as the process in vohich the initial strain is maintained and the stress decays with the time. From energetic viewpoint, elastic energy is stored in the deformation of particles. The granular system is in a metastable state when confined in a state easy to break the balance. Generally speaking, the shape and grading of particles, volume fraction, surface friction properties, initial structure features, ageing time, loading strain rate will all play important roles in stress relaxation.In this work, it is believed that the elastic relaxation is the only mechanism to describe the stress relaxation, and the mechanism of it is analyzed from the viewpoint of the potential energy surface. Stress relaxation is calculated by means of the so-called two-granular temperature theory (TGT) we developed previously (Sun Q et al. 2015 Sci. Rep. 5 9652). The stress decays fast at the beginning, then decreases gradually slowly to a stable value. The logarithmic fit is first proposed to describe the stress decay in the compressed system. Calculated results of stress relaxation match well with the measured results in a recently published paper (Miksic A, Alava M J 2013 Phys. Rev. E 88 032207). Both elastic energy and granular temperature may be reduced with increasing time. It is found that the initial value of the granular temperature has a great influence on the stress relaxation, and at present its effect is input by trial and error. It would be a major problem how to determine the initial value of the granular temperature. Moreover, the relaxation coefficient of elastic stress is basically chosen as a function of granular temperature which is described by the Arrhenius equation that need be further investigated.

Wrinkle-to-fold transition in soft layers under equi-biaxial strain: A weakly nonlinear analysis

Soft materials can experience a mechanical instability when subjected to a finite compression, developing wrinkles which may eventually evolve into folds or creases. The possibility to control the wrinkling network morphology has recently found several applications in many developing fields, such as scaffolds for biomaterials, stretchable electronics and surface micro-fabrication. Albeit much is known of the pattern initiation at the linear stability order, the nonlinear effects driving the pattern selection in soft materials are still unknown. This work aims at investigating the nature of the elastic bifurcation undertaken by a growing soft layer subjected to a equi-biaxial strain. Considering a skin effect at the free surface, the instability thresholds are found to be controlled by a characteristic length, defined by the ratio between capillary energy and bulk elasticity. For the first time, a weakly nonlinear analysis of the wrinkling instability is performed here using the multiple-scale perturbation method applied to the incremental theory in finite elasticity. The Ginzburg–Landau equations are derived for different superposing linear modes. This study proves that a subcritical pitchfork bifurcation drives the observed wrinkle-to-fold transition in swelling gels experiments, favoring the emergence of hexagonal creased patterns, albeit quasi-hexagonal patterns might later emerge because of an expected symmetry break. Moreover, if the surface energy is somewhat comparable to the bulk elastic energy, it has the same stabilizing effect as for fluid instabilities, driving the formation of stable wrinkles, as observed in elastic bi-layered materials.

NONLINEAR THIN-WALLED BEAMS WITH A RECTANGULAR CROSS-SECTION — PART I

Our aim is to rigorously derive a hierarchy of one-dimensional models for thin-walled beams with rectangular cross-section, starting from three-dimensional nonlinear elasticity. The different limit models are distinguished by the different scaling of the elastic energy and of the ratio between the sides of the cross-section. In this paper we report the first part of our results. More precisely, denoting by h and δhthe length of the sides of the cross-section, with δh≪ h, and by [Formula: see text] the scaling factor of the bulk elastic energy, we analyze the cases in which δh/εh→ 0 (subcritical) and δh/εh→ 1 (critical).

Peristaltic patterns for swelling and shrinking of soft cylindrical gels

We propose a variational method for determining the surface patterns of cylindrical gels for both swelling and shrinking. Exact solutions are calculated for the initial stages of such peristaltic instabilities. The morphology and the formation mechanisms depend on a competition between bulk elastic energy and surface tension.

Macroscopic Torsional Strain and Induced Molecular Conformational Deracemization

A macroscopic helical twist is imposed on an achiral nematic liquid crystal by controlling the azimuthal alignment directions at the two substrates. On application of an electric field the director rotates in the substrate plane. This electroclinic effect, which requires the presence of chirality, is strongest at the two substrates and increases with increasing imposed twist distortion. We present a simple model involving a trade-off among bulk elastic energy, surface anchoring energy, and deracemization entropy that suggests the large equilibrium director rotation induces a deracemization of chiral conformations in the molecules-effectively "top-down" chiral induction-quantitatively consistent with experiment.

Experimental Study of Anchoring Characteristics and Splay-Bend Surface-Like Elasticity of Nematics with Using Non-Twisted Nematic Cells

Question of second-order terms in bulk elastic free energy functional for nematics is discussed and investigated experimentally. Under assumption that these surface-like terms contribute only to surface free energy functional, they were firstly substituted by boundary conditions. Several flat-parallel cells (of thickness from 1.9 µm to 31.2 µm) filled with 5CB nematics of planar initial orientation (induced by substrates covered with rubbed SE130 polyimide) were investigated in static electric fields by measurement of their optical transmittance influenced by planar deformation states. The splay and bend elastic constants and anchoring characteristics were determined from these experimental data. The polar anchoring for this nematics-substrate system can not be qualified uniquely as strong or weak. Simulated optical transmittance characteristics computed with using these parameters imitate results of experiment with good accuracy within measurement errors. It is demonstrated that precise determining of nematics-substrate coupling and nematics material parameters is sufficient for quantitative description of nematics deformations. Afterwards the values of components of the free energy of nematics layer corresponding to the bulk elastic deformation, bulk interaction with electric field, nematics interaction with boundary and splay-bend surface contribution are computed from experimental data in two cases which can be interpreted as weak and strong anchoring. The following comparison of them leads to conclusions on the negligible magnitude order of the splay-bend elastic constant.

Composite Liquid Crystalline Mixtures for Guided Wave Electro–Optic Applications

A low refractive index multifunctional composite liquid crystalline mixture is proposed to address the issues of in-guide tuning of optical properties. As the standard optical telecommunication line is made of fused silica, signal phase and amplitude modulation using liquid crystal requires that its refractive indices match the refractive index of silica, which is low compared to the majority of commercially available mixtures. In the present work, low molecular mass liquid crystals are mixed with fluorinated additives to adjust refractive index below a critical limit (n < 1.4584 at D-line) without impacting significantly on the nematic temperature range (TCN = − 24°C and TNI = 68°C). A method of reducing the surface anchoring energy of liquid crystals to glass surface is also proposed as a non contact alignment method around the reduced diameter optical fibers. The bulk elastic energy around the fiber acts as an alignment mechanism when LC-fused silica molecular interactions are minimized. The combination of these two properties makes possible the in-guide modulation of the light phase and amplitude with the liquid crystal used as an outer-cladding electro-controllable medium, significantly improving insertion losses.

Deformation of bilayer lipid membranes in bio‐inspired materials and systems

AbstractPlanar bilayer lipid membranes (BLMs) are currently employed to construct many bio‐inspired material systems and structures. In order to characterize the pressure effects on the equilibrium configurations of these biological membranes, a novel continuum model is proposed. The BLM is assumed to be a two‐layer Smectic A liquid crystal. The mean orientation of the amphiphilic molecules comprising the membrane is postulated to be perpendicular to the layers and each layer is idealized as a two dimensional liquid. Moreover, the BLM is modeled as a simply supported plate undergoing small deformations. It is subjected to a pressure load that acts perpendicularly to the layers. The equilibrium equations and boundary conditions are derived from the bulk elastic energy for Smectic A liquid crystals as described by de Gennes and Prost (1993) by using variational methods. The resulting fourth‐order linear partial differential equation is solved by employing cylindrical functions and the series solution is proved to be convergent. The solution is numerically computed for values of the model parameters that are reported in the literature. (© 2008 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

Pressure effects on the equilibrium configurations of bilayer lipid membranes**This paper is dedicated to the memory of our colleagues, Professors Kevin P Granata and Liviu Librescv, who lost their lives during the sensless tragedy on 16 April, 2007 at Virginia Tech.

Planar bilayer lipid membranes (BLMs) are currently employed to construct many bio-inspired material systems and structures. In order to characterize the pressure effects on the equilibrium configurations of these biological membranes, a novel continuum model is proposed. The BLM is assumed to be a two-layer smectic A liquid crystal. The mean orientation of the amphiphilic molecules comprising the membrane is postulated to be perpendicular to the layers and each layer is idealized as a two-dimensional liquid. Moreover, the BLM is modeled as a simply supported plate undergoing small deformations. It is subjected to a pressure load that acts perpendicularly to the layers. The equilibrium equations and boundary conditions are derived from the bulk elastic energy for smectic A liquid crystals as described by de Gennes and Prost (1993 The Physics of Liquid Crystals 2nd edn (Oxford Science Publications)) by using variational methods. The resulting fourth-order linear partial differential equation is solved by employing cylindrical functions and the series solution is proved to be convergent. The solution is numerically computed for values of the model parameters that are reported in the literature.