Shape Equation Research Articles

A damage localization technique using wave front shapes in composite laminates without knowing the velocity profile

Composite laminates are widely used in various fields, but their structures are prone to cracks and damage. Due to the difference in angles of the instantaneous direction of the wave front propagation and the direction of the energy flow in an anisotropic material, the use of Lamb waves for damage localization in composite laminates is a challenging task. Establishing the wave front shape equation can overcome the difficulty of damage localization caused by anisotropy, but this usually requires a priori knowledge of the acoustic velocity distribution of the laminates, which is not convenient for efficient damage localization. In this paper, a damage localization method based on wave front shapes for composite laminates without any knowledge of the velocity profile is presented. Numerical simulation and experimental results show that the proposed method works. This method shows good damage localization accuracy and has broad application prospects in non-destructive testing for plate structures with strong anisotropy.

Shape of the membrane neck around a hole during plasma membrane repair

Plasma membrane damage and rupture occurs frequently in cells, and holes must be sealed rapidly to ensure homeostasis and cell survival. The membrane repair machinery is known to involve recruitment of curvature-inducing annexin proteins, but the connection between membrane remodeling and hole closure is poorly described. The induction of curvature by repair proteins leads to the possible formation of a membrane neck around the hole as a key intermediate structure before sealing. We formulate a theoretical model of equilibrium neck shapes to examine the potential connection to a repair mechanism. Using variational calculus, the shape equations for the membrane near a hole are formulated and solved numerically. The system is described under a condition of fixed area, and a shooting approach is applied to fulfill the boundary conditions at the free membrane edge. A state diagram of neck shapes is produced describing the variation in neck morphology with respect to the membrane area. Two distinct types of necks are predicted, one with conformations curved beyond π existing at positive excess area, whereas flat neck conformations (curved below π) have negative excess area. The results indicate that in cells, the supply of additional membrane area and a change in edge tension is linked to the formation of narrow and curved necks. Such necks may be susceptible to passive or actively induced membrane fission as a possible mechanism for hole sealing during membrane repair in cells.

Evaluation of beach response due to construction of submerged detached breakwater

Submerged detached breakwaters (SDBWs) have increasingly been used in recent times as an alternative against their emergent counterpart (EDBWs) to mitigate erosion because the former do not spoil the seascape. Both of these structures are (usually) constructed using precast concrete blocks or natural granite rocks, hence becoming permeable structures. For an EDBW, a parabolic bay shape equation can be readily used to estimate the planar shape of the shoreline behind the structure, but there is still no approach to estimate how the shoreline behind the SDBW is formed. In this study, we estimated how the shoreline is balanced by examining how the dominant wave direction changes due to the diffraction of the transmitted wave generated after the installation of the SDBW from the long-term wave directional spectrum. The change in dominant wave direction was determined under the shoreline gradient condition where littoral drift does not occur, considering the diffraction phenomenon due to the difference in transmitted waves. This means that the shape of the equilibrium shoreline changes to face perpendicular to the dominant wave direction. As a meaningful result, when the transmittance is 0, it converges to the well-known empirical equation of EDBW. The present methodology is validated by comparing the observed data (wave and shoreline change) from two beaches (Anmok and Bongpo-Cheonjin Beaches) on the eastern coast of Korea. This rational approach to shoreline changes behind permeable SDBWs will help in proactive review work for coastal management as well as beach erosion mitigation.

Curvature controls beading in soft coated elastic cylinders: Finite wavemode instability and localized modulations

Axisymmetric beading instabilities in soft, elongated cylinders have been observed in a plethora of scenarios, ranging from cellular nanotunnels and nerves in biology to swollen cylinders and electrospun fibers in polymer physics. One of the common geometrical features that can be seen in these systems is the finite wavelength of the emerging pattern. However, modelling studies often predict that the instability has an infinite wavelength, which can be associated with localized necking or bulging. In this paper, we consider a soft elastic cylinder with a thin coating that resists bending, as described by the Helfrich free energy functional. The bending stiffness and natural mean curvature of the coating are two novel features whose competition against bulk elasticity and capillarity is investigated. For intermediate values of the bending stiffness, a linear stability analysis reveals that the mismatch between the current and natural mean curvature of the coating can lead to patterns emerging with a finite wavelength. This analysis creates a continuous bridge between the classical solutions of the shape equation obtained from the Helfrich functional and a curvature-controlled zero-wavemode instability, similar to the one induced by the competition between bulk elasticity and capillarity. A weakly non-linear analysis predicts that the criticality of the bifurcation depends on the controlling parameter, with both supercritical and subcritical bifurcations possible. When capillarity is introduced, the criticality of the bifurcation changes in a non-trivial way.

Non-linear electro-rheological model of a membrane immersed in Tanner-Power law fluids applied to outer hair cells: Shear-thinning mechanisms

Flexoelectric actuation employs an applied electric field to induce membrane curvature, which is the mechanism utilized by the outer hair cells (OHC) present in the inner ear. The model developed for this study, representing the OHC, integrates two key components: (i) an approximation of the flexoelectric membrane shape equation for circular membranes attached to the inner surface of a circular capillary, and (ii) the coupled capillary flow of contacting liquid viscoelastic phases characterized by the Tanner-Power law rheological equation of state. A second-order non-linear differential equation for average curvature has been derived, and a robust numerical method has been programmed. This model simplifies to a linear model used previously. The main challenge involves identifying and describing the enhancement in curvature change rate. It was observed that low symmetry, low viscosity, and soft membrane and shear-thickening behavior of the phases enhance the curvature change rate. Additionally, there exists a critical electric field frequency value that maximizes the curvature change rate (resonance effect). The current theory, model, and computational simulations add to the ongoing development comprehension of how biological membrane shape actuation through electromechanical couplings.

The mechanics of thin elastic sheet reinforced with fiber mesh subjected to combined flexure and extension

We present a three-dimensional continuum model aimed at investigating the mechanical performance of elastomeric materials reinforced with bidirectional fibers undergoing the combined out-of-plane bending moment and bilateral extension. To achieve this, we consider the Neo-Hookean strain energy model for the matrix material of the fiber composite and incorporate the strain energy of bidirectional fibers into the strain energy potential of the fiber composite. The bidirectional fibers’ strain energy is rigorously modeled by configuring the fiber kinematics on the fiber composite surface using differential geometry and accounting for stretching, bending, and twisting responses of the fibers via the computation of the first-order and second-order gradient of deformation. To establish equilibrium equations, we apply the variational principle to derive the constitutive relations, resulting in the normal and tangential shape equations of the fiber composite, and boundary conditions of bending and stretching. The model implementation emphasizes analyzing the combined out-of-plane bending and bilateral stretching effects on the fiber composite material, and the numerical results include the out-of-plane and in-plane deformation, strain-loading responses, bending, twisting, and stretching of fibers within the matrix material, as well as the deformation of the fiber network. Specifically, it is found that a larger bending moment results in intensified out-of-plane deformation while larger bilateral stretch decreases the out-of-plane deformation of fiber composite. In particular, bilateral stretches reduce the Lagrange strain (ɛ1) near the bilateral boundaries, unlike the intensity distribution of Lagrange strain (ɛ2) maintains almost unchanged due to no external tension applied on the upper and lower boundaries of the fiber while bending moments persist. These findings highlight the role of bilateral stretching acting as the prestress in reducing deflection, enhancing tensile strength and efficient use, and controlling crack generation of fiber-reinforced polymer (FRP) composites. More importantly, despite the scarcity of experimental data on fiber composite deformation at the microscopic, the theoretical demonstration of the extension and flexure of fiber units provide reasonable explanations for the resulting deformation of the overall fiber meshwork, validating the mechanism that the fiber microstructure deformation determines the overall deformation of fiber composites when subjected to combined bending moment and tension. Special attention should be given to the potential dislocation between the embedded meshwork and matrix material located in the center of the domain. Furthermore, the proposed model provides qualitatively reasonable explanations on the shaping of Bamboo Polylactic acid (PLA) composites, out-of-plane deformation of woven fabric, Sikken-type stiffener deformation, and fiber-reinforced thermoplastics via a series of numerical results.

Periodic cylindrical bilayers self-assembled from biblock polymers.

Amphiphilic polymers in aqueous solutions can self-assemble to form bilayer membranes, and their elastic properties can be captured using the well-known Helfrich model involving several elastic constants. In this paper, we employ the self-consistent field model to simulate sinusoidal bilayers self-assembled from diblock copolymers where an appropriate constraint term is introduced to stabilize periodic bilayers with prescribed amplitudes. Then, we devise several methods to extract the shape of these bilayers and examine the accuracy of the free energy predicted by the Helfrich model. Numerical results show that when the bilayer curvature is small, the Helfrich model predicts the excess free energy more accurately. However, when the curvature is large, the accuracy heavily depends on the method used to determine the shape of the bilayer. In addition, the dependence of free energy on interaction strength, constraint amplitude, and constraint period are systematically studied. Moreover, we have devised a method for attaining equilibrium states through the adjustment of constraints. Within the self-consistent field model, these equilibrium states manifest as distinct periodic cylindrical bilayers, which are consonant with the theoretical predictions formulated using the shape equations.

Analytical description of elastocapillary membranes held by needles.

Fluid objects bounded by elastocapillary membranes display intriguing physical properties due to the interplay of capillary and elastic stresses arising upon deformation. Increasingly exploited in foam or emulsion science, the mechanical properties of elastocapillary membranes are commonly characterised by the shape analysis of inflating/deflating bubbles or drops held by circular needles. These impose complex constraints on the membrane deformation, requiring the shape analysis to be done using elaborate numerical fitting procedures of the shape equations. While this approach has proven quite reliable, it obscures insight into the underlying physics of the problem. We therefore propose here the first fully theoretical approach to this problem using the elastic theory for a membrane with additive contributions of capillary and Hookean-type elastic stresses. We exploit this theory to discuss some of the key features of the predicted pressure-deformation relations. Interestingly, we highlight a breakdown of the quadratic approximation at a well-defined value of the elastocapillary parameter depending on the shape of the reference state, which is regularized by the non-quadratic terms. Additionally, we provide an analytical relationship which allows experimentalists to obtain the elastocapillary properties of a membrane by simple measurement of the height and the width of a deformed bubble (or a drop).

Inferring biophysical properties of membranes during endocytosis using machine learning.

Endocytosis is a fundamental cellular process in eukaryotic cells that facilitates the transport of molecules into the cell. With the help of fluorescence microscopy and electron tomography, researchers have accumulated extensive geometric data of membrane shapes during endocytosis. These data contain rich information about the mechanical properties of membranes, which are hard to access via experiments due to the small dimensions of the endocytic patch. In this study, we propose an approach that combines machine learning with the Helfrich theory of membranes to infer the mechanical properties of membranes during endocytosis from a dataset of membrane shapes extracted from electron tomography. Our results demonstrate that machine learning can output solutions that both match the experimental profile and satisfy the membrane shape equations derived from Helfrich theory. The learning results show that during the early stage of endocytosis, the inferred membrane tension is negative, indicating the presence of strong compressive forces at the boundary of the endocytic invagination. Our method presents a generic framework for extracting membrane information from super-resolution imaging.

Inside vesicle adhesion between two vesicles in 2-D case

A theoretic model is proposed to study the adhesion behavior of a vesicle adhering inside another vesicle in 2-D case. The equilibrium shape equations and boundary conditions are investigated. With zero pressure, the integral case of the shape equations with dimensionless parameters is derived. When a soft vesicle adheres inside a rigid ring, the bifurcation between two phases is revealed, the critical adhesion condition and phase diagram are obtained. The finite element method (FEM) is used to study the deformation processes induced by the change of work of adhesion and bending rigidity. The relationships between the energy, adhesion length, cell volume, bending rigidity and work of adhesion are obtained.

NEW PROCESS BASED EQUATION FOR A STATIC EQUILIBRIUM BEACH PLANFORM

Equilibrium planform formulations of beaches are engineering tools for providing solutions for beach erosion problems. They aim at defining the final orientation of the beach on a scale of years and thus they are used for evaluating the shoreline response due to human interference (ports, breakwaters, protection works…etc). These formulations create engineering solution to numerous coastal stability problems. The Parabolic Bay Shape Equation (PBSE) proposed by Hsu and Evans (1989) is the most used equation and allow to solve problems. However, these shape equations are valid under certain simplifications. Therefore, they are unable to determine the beach planform when the bathymetry is complex, with rocky areas or/and islands. The aim of this study is to present a new equilibrium beach planform equation (Gainza et al. 2018) that overcomes the problem previously mentioned. What is more, an extension of the model is presented in order to modeled the shape of a shoreline on a given day.

Spring-back behaviors of Ti-6Al-4V sheet under effect of strain rate

By using Impact Hydro-bending (IHB), the room-temperature spring-back of Ti-6Al-4V sheet can be reduced significantly. However, the existing analytical models for spring-back prediction do not take into account complex space-time deformation history, and thus cannot predict the spring-back behavior under IHB accurately. Although numerical simulation can achieve high accuracy of spring-back prediction, it cannot obtain the values and distributions of tangential strain/stress efficiently, and it is also unable to quantitatively distinguish the influence of space-time deformation history on the level of spring-back. In this paper, a theoretical model considering both the effects of the space-time deformation history and the strain rate is creatively established to predict the spring-back behaviors. In the proposed model, a novel shape equation is established to describe the space-time deformation history of sheet. When using the shape equation, only two parameters (kv and pv) need to be adjusted to accurately obtain the desired shape history of sheet. Specifically, both the effects of strain rate and cyclic loading can be considered in the proposed constitutive equation used for the analytical analysis. By utilizing the analytical model, the effect of space-time deformation history and strain rate on the spring-back reduction can be quantitatively characterized, and the result shows that the space-time deformation history is the primary factor affecting the spring-back reduction under high-strain-rate bending. The analytical model is further utilized to reveal the mechanism of spring-back reduction. According to the analytical results, cyclic loading occurred at the sidewall of sheet under IHB. Additionally, the plastic zone widens and the plastic level improves significantly under IHB compared to conventional punch bending (CPB). By observing the distributions of tangential stress at the end of deformation, it is found that in certain regions under IHB, the compressive stress in the outer layer is greater than that in the inner layer, which makes the sheet to bend towards its inner side after unloading, and thus promotes the reduction of spring-back.

Seiberg-Witten Geometry of Four-Dimensional $\mathcal N=2$ Quiver Gauge Theories

Seiberg-Witten geometry of mass deformed $\mathcal N=2$ superconformal ADE quiver gauge theories in four dimensions is determined. We solve the limit shape equations derived from the gauge theory and identify the space $\mathfrak M$ of vacua of the theory with the moduli space of the genus zero holomorphic (quasi)maps to the moduli space ${\rm Bun}_{\mathbf G} (\mathcal E)$ of holomorphic $G^{\mathbb C}$-bundles on a (possibly degenerate) elliptic curve $\mathcal E$ defined in terms of the microscopic gauge couplings, for the corresponding simple ADE Lie group $G$. The integrable systems $\mathfrak P$ underlying the special geometry of $\mathfrak M$ are identified. The moduli spaces of framed $G$-instantons on ${\mathbb R}^{2} \times {\mathbb T}^{2}$, of $G$-monopoles with singularities on ${\mathbb R}^{2} \times {\mathbb S}^{1}$, the Hitchin systems on curves with punctures, as well as various spin chains play an important role in our story. We also comment on the higher-dimensional theories.

On equilibrium states of fluid membranes

The paper studies the equilibrium configurations of inextensible elastic membranes exhibiting lateral fluidity. Using a continuum description of the membrane's motions based on the surface Navier–Stokes equations with bending forces, the paper derives differential equations governing the mechanical equilibrium. The equilibrium conditions are found to be independent of lateral viscosity and relate tension, pressure, and tangential velocity of the fluid. These conditions suggest that either the lateral fluid motion ceases or non-decaying stationary flow of mass can only be supported by surfaces with Killing vector fields, such as axisymmetric shapes. A shape equation is derived that extends the classical Helfrich model with an area constraint to membranes of non-negligible mass. Furthermore, the paper suggests a simple numerical method to compute solutions of the shape equation. Numerical experiments conducted reveal a diverse family of equilibrium configurations. The stability of equilibrium states involving lateral flow of mass remains an unresolved question.

Pattern formation, structure and functionalities of wrinkled liquid crystal surfaces: A soft matter biomimicry platform

This review presents an integrated theoretical and computational characterization and analysis of surface pattern formation in chiral and achiral liquid crystal self-assembly and the mechanical/optical/tribological/tissue engineering surface functionalities that emerge from various wrinkling processes. Strategies to target surface patterns include linear, non-linear, multidirectional and multiscale wrinkling phenomena. The focus of the review is to show the unique surface structure-functionalities that emerge from anisotropic liquid crystal soft matter, eliminating or reducing the need of aggressive solvents, extreme pressure/temperature conditions, erosion and other surface morphing approaches. The surface pattern formation theoretical-modelling- computational results are then connected and validated with actual biological surfaces that are considered solid liquid crystal analogues, such as exocuticles of insects, fish scales, and flowers. A unique feature of thein silicosurface pattern formation platform used throughout this review is the generalized liquid crystal shape equation that includes surface anchoring elasticity, membrane elasticity, and stress loads from liquid crystals orientation gradients. Clear characterization of surface shapes, curvatures, roughness, that are behind surface functionalities are introduced and applied to strengthen validation of predictions with actual nature’s surfaces. Wrinkling scaling laws, and the dependence of material properties on morphing mechanisms are elucidated. The predictions capture very well the two-scale wrinkling patterns in tulips, wrinkling gradients that display water sensor capabilities, egg carton shapes in rose petals and their potential for cell alignment, and the ability to create surface roughness with targeted kurtosis and skewness to control and optimize friction and tribological functionalities. The results are summarized in terms of surface geometry (open or closed) mechanisms and phenomena (anchoring, membrane elasticity), material properties (anchoring coefficients, membrane bending modulus, Frank elasticity), wrinkling scales and scaling laws (amplitude, wave-lengths, skewness, kurtosis) and functionalities (optical iridescence, friction, wettability, structural color, curvature-driven cell alignment and differentiation). Taken together, the range of surface geometries and surface functionalities captured by the liquid crystal biomimeticin silicoplatform provides a foundation for future experimental green manufacturing pathways based on anisotropic soft matter.

Tipologi Arsitektur Vernakular Melayu Uma Kabuong Limo di Kampung Pulau Belimbing, Kabupaten Kampar

Title: Typology of Malay Vernacular Architecture Uma Kabuong Limo in Pulau Belimbing Village, Kampar District One form of residential culture can be seen in the architecture as a residence for the people of Belimbing Island Village, Kampar regency. The house as a residential function always develops according to the needs and image of the owner, so changes in the Uma Kabuong Limo building are found. Uma Kabuong Limo can be categorized as vernacular architecture because Uma Kabuong Limo is a product and name that has existed for several generations with a development process that adapts the socio-cultural and cultural of the Malay community. The nature of this simplicity is not only a residential building but also becomes the identity of the community. Through the study of the architectural typology of Uma Kabuong Limo was studied through the approach of the computational model of heritage building information modeling (HBIM). This modeling is expected to get an in-depth and precise analysis to describe the typology process more accurately. This assessment process begins with field data observation, modeling, and discussion of building typologies. The Uma Kabuong Limo buildings observed are six buildings representing the distribution of facilities in The Belimbing Island Village. The findings were the existence of a slightly different basic shape equation in the kitchen. One of the buildings changed the shape of the roof, called the paranginan roof, a multi-storey roof. The more engravings and tidiness of home making depict, the higher the level of the social status of the building owner.

Geometry-structure models for liquid crystal interfaces, drops and membranes: wrinkling, shape selection and dissipative shape evolution.

We review our recent contributions to anisotropic soft matter models for liquid crystal interfaces, drops and membranes, emphasizing validations with experimental and biological data, and with related theory and simulation literature. The presentation aims to illustrate and characterize the rich output and future opportunities of using a methodology based on the liquid crystal-membrane shape equation applied to static and dynamic pattern formation phenomena. The geometry of static and kinetic shapes is usually described with dimensional curvatures that co-mingle shape and curvedness. In this review, we systematically show how the application of a novel decoupled shape-curvedness framework to practical and ubiquitous soft matter phenomena, such as the shape of drops and tactoids and bending of evolving membranes, leads to deeper quantitative insights than when using traditional dimensional mean and Gaussian curvatures. The review focuses only on (1) statics of wrinkling and shape selection in liquid crystal interfaces and membranes; (2) kinetics and dissipative dynamics of shape evolution in membranes; and (3) computational methods for shape selection and shape evolution; due to various limitations other important topics are excluded. Finally, the outlook follows a similar structure. The main results include: (1) single and multiple wavelength corrugations in liquid crystal interfaces appear naturally in the presence of surface splay and bend orientation distortions with scaling laws governed by ratios of anchoring-to-isotropic tension energy; adding membrane elasticity to liquid crystal anchoring generates multiple scales wrinkling as in tulips; drops of liquid crystals encapsulates in membranes can adopt, according to the ratios of anchoring/tension/bending, families of shapes as multilobal, tactoidal, and serrated as observed in biological cells. (2) Mapping the liquid crystal director to a membrane unit normal. The dissipative shape evolution model with irreversible thermodynamics for flows dominated by bending rates, yields new insights. The model explains the kinetic stability of cylinders, while spheres and saddles are attractors. The model also adds to the evolving understanding of outer hair cells in the inner ear. (3) Computational soft matter geometry includes solving shape equations, trajectories on energy and orientation landscapes, and shape-curvedness evolutions on entropy production landscape with efficient numerical methods and adaptive approaches.

Geometric design and optimization of scissor-type deployable structures

Geometric design is the essential first step to ensure transformation of a structure from an undeployed to a deployed state and vice-versa. Typically, this is achieved using motion criteria to prevent interference of parts. This paper describes the geometric design procedure of axisymmetric deployable structures made of planar angulated scissor units. The geometric design parameters are optimized not just using motion criteria but also structural performance measures. The first part of this paper discusses the refinement of geometric parameters, namely, shape equation, number of vertical layers, number of sectors in plan, and rise-to-span ratio, to ensure minimum change in span during deployment. Analysis results showed that more number of vertical layers, more sectors, and a lower rise-to-span ratio led to minimal changes in base radius during deployment. A finite element analysis of a simplified slice model was used to validate the results by comparing with the full three-dimensional model results. Genetic Algorithm was used to find the optimum number of vertical layers and sectors in plan based on structural performance. This paper presents results of concurrent optimization of layout, elevation, and member sizes of deployable domes to meet stiffness criteria and to reduce self-weight.

Exact general solutions for the mode shapes of longitudinally vibrating non-uniform rods via Lie symmetries

A Lie symmetry method-based approach is proposed for systematically computing general solutions in closed-form for the mode shape equation of non-uniform and unconventional vibrating rods. The mode shape equation is modeled by the elementary rod theory, addressing polynomial, exponential, trigonometric, and hyperbolic cross-section variations. The method provides algorithmic order-reduction steps for solving the investigated mode shape equation, producing a first-order Riccati equation whose integration reveals the aimed solutions for the problem. Illustrative examples are presented, including original solutions in closed-form as well as solutions previously obtained in the literature by other approaches. Mode shapes from general solutions with appropriate rod boundary conditions are also considered for different examples. • Original general solutions establish new exact mode shapes of non-uniform rods. • The Lie symmetry method allows addressing new cross-section variations systematically. • Mode shapes derive from the general solutions with rod boundary conditions. • Lie symmetry applications have been highlighted for further problems of interest.

Prediction of droplets characteristic diameters and polydispersity index induced by a bifluid spraying nozzle by the means of dimensional analysis

This work focuses on the study of sprays generated through a bifluid nozzle and the modelling of characteristic spray properties (two characteristic diameters and a polydispersity index) using dimensional analysis. Two types of dimensionless models were identified for each spray target property from the 75 experimental points considered. The first type used a conventional monomial-exponential shape equation, and the second applied shape identification through machine-learning. Although conventional models of the first type were mostly satisfactory when considering the characteristic diameters, they nevertheless showed clear limitations addressed by the machine-learning identified models. The conventional approach also failed to identify a satisfactory equation for the polydispersity index. The machine-learning approach provided an equation identifying this index to the main dimensionless parameters governing atomization. This identification provides a foundation for proposing a two-parameters dimensionless model that predicts spray particle size distribution. The combination of dimensional analysis with machine-learning equation identification thus paves the way to physically rigorous and easy-to-use models capable of predicting characteristic properties and full distributions.