Isotropic Elastic Shell Research Articles

Numerical simulation of nanoneedle-cell membrane collision: minimum magnetic force and initial kinetic energy for penetration

Aims and objectives: This research aims to develop a kinetic model that accurately captures the dynamics of nanoparticle impact and penetration into cell membranes, specifically in magnetically-driven drug delivery. The primary objective is to determine the minimum initial kinetic energy and constant external magnetic force necessary for successful penetration of the cell membrane. Model Development: Built upon our previous research on quasi-static nanoneedle penetration, the current model development is based on continuum mechanics. The modeling approach incorporates a finite element method and explicit dynamic solver to accurately represent the rapid dynamics involved in the phenomenon. Within the model, the cell is modeled as an isotropic elastic shell with a hemiellipsoidal geometry and a thickness of 200 nm, reflecting the properties of the lipid membrane and actin cortex. The surrounding cytoplasm is treated as a fluid-like Eulerian body. Scenarios and Results: This study explores three distinct scenarios to investigate the penetration of nanoneedles into cell membranes. Firstly, we examine two scenarios in which the particles are solely subjected to either a constant external force or an initial velocity. Secondly, we explore a scenario that considers the combined effects of both parameters simultaneously. In each scenario, we analyze the critical values required to induce membrane puncture and present comprehensive diagrams illustrating the results. Findings and significance: The findings of this research provide valuable insights into the mechanics of nanoneedle penetration into cell membranes and offer guidelines for optimizing magnetically-driven drug delivery systems, supporting the design of efficient and targeted drug delivery strategies.

Electrostatic interactions and structural transformations in viral shells.

Structural transformations occurring in proteinaceous viral shells (capsids) can be induced by changing the pH of bathing solution, thus modifying the dissociation equilibrium of ionizable amino acids in proteins. To analyze the effects of electrostatic interactions on viral capsids, we construct a model of 2D isotropic elastic shells with embedded point charges located in the centers of mass of individual proteins. We find that modification of the electrostatic interactions between proteins affects not only the size and shape of capsids, but in addition induces substantial deformations of hexamers in capsid structures. Using bacteriophage P22 and Nudarelia capensis omega virus (NωV) as examples, we analyze the capsid faceting and propose an explanation as to why the hexamers in spherical procapsid are skewed, while they acquire a regular shape in the faceted state. Also, we examine the electrostatic and elastic effects that can explain different shapes of coronavirus shells decorated with spikes, which are often localized in compact areas over the shell surface. The proposed mechanism of local curvature generation is supported by the remarkable correspondence between the shell shape and the distribution of spikes in model and observed shells.

Comparative analysis of the stability and natural vibrations of shallow panels under the action of thermomechanical loads

The work is a continuation of research devoted to substantiating the reliability of solutions obtained by the finite element method for the analysis of nonlinear deformation, buckling and vibrations of thin elastic shells under the action of thermomechanical loads. The method is based on geometrically nonlinear relations of the three-dimensional theory of thermoelasticity and the principles of the moment finite element scheme. A thin elastic shell of an inhomogeneous structure is modeled by a universal spatial isoparametric finite element. The modal analysis of the shell is implemented at each step of the static thermomechanical load. The subspace iteration method is used to determine the spectrum of the lowest frequencies of natural vibrations of shells. A shallow spherical panel with a square plan is considered. The effect of preheating on the loss of stability and vibrations of an elastic isotropic shell under uniform pressure loading is investigated. The behavior of the shell weakened by two pairs of cross-channels is analyzed. The weakening of the panel by narrow and wide channels, which can be eccentrically located relative to the middle surface of the shell, is considered. The effectiveness and adequacy of the method is confirmed by a comparative analysis of solutions with results obtained using modern multifunctional software systems LIRA-SAPR and SCAD. The features of using the systems for solving the problems under consideration are given. Analysis of the results made it possible to evaluate the possibilities of using these software systems to substantiate the reliability of solutions to certain classes of problems of geometrically nonlinear deformation, buckling and vibrations of elastic shells.

ON THE EQUILIBRIUM EQUATIONS OF LINEAR 6-PARAMETER ELASTIC SHELLS

We consider the linearized theory of 6-parameter elastic shells with general anisotropy. We derive the equilibrium equations from the virtual power statement and formulate the corresponding variational problem in the suitable functional framework. Then, using a Korn-type inequality for the linearized strain measures we prove the existence and uniqueness of weak solutions. Finally, we show that our general theo­rem can be applied to obtain existence results in the case of isotropic elastic shells. We illustrate this procedure by investigating three differ­ent linear shell models established previously in the literature, namely the simplified isotropic 6-parameter shell, the Cosserat isotropic model, and the higher-order 6-parameter Cosserat model.

Analytical solution for torsion of cylindrical orthotropic annular wedge-shaped bars reinforced by thin shells

This paper deals with the Saint-Venant torsion of elastic, cylindrically orthotropic bar whose cross section is a sector of a circular ring shaped bar. The cylindrically orthotropic homogeneous elastic wedge-shaped bar strengthened by on its curved boundary surfaces by thin isotropic elastic shells. An analytical method is presented to obtain the Prandtl’s stress function, torsion function, torsional rigidity and shearing stresses. A numerical example illustrates the application of the developed analytical method.

Derivation of a refined six-parameter shell model: descent from the three-dimensional Cosserat elasticity using a method of classical shell theory

Starting from the three-dimensional Cosserat elasticity, we derive a two-dimensional model for isotropic elastic shells. For the dimensional reduction, we employ a derivation method similar to that used in classical shell theory, as presented systematically by Steigmann (Koiter’s shell theory from the perspective of three-dimensional nonlinear elasticity. J Elast 2013; 111: 91–107). As a result, we obtain a geometrically nonlinear Cosserat shell model with a specific form of the strain energy density, which has a simple expression, with coefficients depending on the initial curvature tensor and on three-dimensional material constants. The explicit forms of the stress–strain relations and the local equilibrium equations are also recorded. Finally, we compare our results with other six-parameter shell models and discuss the relation to the classical Koiter shell model.

Numerical simulation of pressure therapy glove by using Finite Element Method

Pressure therapy garments apply pressure to suppress the growth and flatten hypertrophic scars caused by serious burns. The amount of pressure given by the pressure garments is critical to the treatment adherence and outcomes. In the present study, a biomechanical model for simulating the pressure magnitudes and distribution over hand dorsum given by a pressure glove was developed by using finite element method. In this model, the shape geometry of the hand, the mechanical properties of the glove and human body tissues were incorporated in the numerical stress analyses. The geometry of the hand was obtained by a 3D laser scanner. The material properties of two warp knitted fabrics were considered in the glove fabric model that developed from the glove production pattern with 10% size reduction in circumferential dimensions. The glove was regarded an isotropic elastic shell and the hand was assumed to be a homogeneous, isotropic and linearly elastic body. A glove wearing process was carried in the finite element analysis and the surface-to-surface contact pressure between hand and glove fabric was hence obtained. Through validation, the simulated contact pressure showed a good agreement with the experimental interface pressure measurement. The simulation model can be used to predict and visualise the pressure distribution exerted by a pressure therapy glove onto hand dorsum. It can provide information for optimising the material mechanical properties in pressure garment design and development, give a clue to understand the mechanisms of pressure action on hypertrophic scars and ultimately improve the medical functions of pressure garment.

The Effect of Internal Flowing Fluid on the Non-Linear Behavior of Orthotropic Circular Cylindrical Shells

The great use of circular cylindrical shells for conveying fluid in modern industrial applications has made of them an important research area in applied mechanics. Many researchers have studied this problem, however just a reduced number of these works have as object the analysis of orthotropic shells. Although most investigations deal with the analysis of elastic isotropic shells in contact with internal and external quiescent or flowing fluid, several modern and natural materials display orthotropic properties and also stiffened cylindrical shells can be treated as equivalent uniform orthotropic shells. In this work, the influence of internal flowing fluid on the dynamic instability and non-linear vibrations of a simply supported orthotropic circular cylindrical shell subjected to axial and lateral time-dependent loads is studied. To model the shell, the Donnell’s non-linear shallow shell theory without considering the effect of shear deformations is used. A model with eight degrees of freedom is used to describe the lateral displacements of the shell. The fluid is assumed to be incompressible and non-viscous and the flow to be isentropic and irrotational. The Galerkin method is applied to derive the set of coupled non-linear ordinary differential equations of motion which are, in turn, solved by the Runge-Kutta method. The obtained results show that the presence of the internal fluid and material properties have a great influence on the vibration characteristics of the shell.

On the existence of solutions to geometrically nonlinear problems for shallow Timoshenko-type shells with free edges

In the paper we investigate the solvability of the boundary-value problems for shallow isotropic elastic shells within the framework of Timoshenko’s shearmodel. The considered problems are nonlinear geometrically and linear physically. The method of studying consists in reducing the initial system of equilibrium equations to one nonlinear differential equation with respect to deflections. In doing so integral representations for the tangential displacements and angles of rotation play a significant role. The representations are deduced by making use of general solutions to the inhomogeneous Cauchy-Riemann equation. The solvability is established by the principle of contracting mappings.

Drilling couples and refined constitutive equations in the resultant geometrically non-linear theory of elastic shells

It is well known that distribution of displacements through the shell thickness is non-linear, in general. We introduce a modified polar decomposition of shell deformation gradient and a vector of deviation from the linear displacement distribution. When strains are assumed to be small, this allows one to propose an explicit definition of the drilling couples which is proportional to tangential components of the deviation vector. The consistent second approximation to the complementary energy density of the geometrically non-linear theory of isotropic elastic shells is constructed. From differentiation of the density we obtain the consistently refined constitutive equations for 2D surface stretch and bending measures. These equations are then inverted for 2D stress resultants and stress couples. The second-order terms in these constitutive equations take consistent account of influence of undeformed midsurface curvatures. The drilling couples are explicitly expressed by the stress couples, undeformed midsurface curvatures, and amplitudes of quadratic part of displacement distribution through the thickness. The drilling couples are shown to be much smaller than the stress couples, and their influence on the stress and strain state of the shell is negligible. However, such very small drilling couples have to be admitted in non-linear analyses of irregular multi-shell structures, e.g. shells with branches, intersections, or technological junctions. In such shell problems six 2D couple resultants are required to preserve the structure of the resultant shell theory at the junctions during entire deformation process.

Analysis of the Blank Holder Force Effect on the Preforming Process Using a Simple Discrete Approach

Simulation of the dry reinforcement preforming, first step of the Resin Transfer Moulding process, become necessary to determine the feasibility of the forming process, compute the fiber directions in the final composite component, and optimize process parameters during this step. Contrary to geometrical approaches, based on fishnet algorithms, finite element methods can take into account the actual physical parameters, the real boundary conditions and the mechanical behavior of the textile reinforcement. The fabric can be modeled either as continuum media with specific material behavior [5, 6], or using discrete structural elements to describe the textile structure at the mesoscopic scale. A semi-discrete approach, which is a compromise between the above continuous and discrete approaches, is also used for simulation. A discrete approach for the simulation of the preforming of dry woven reinforcement has been proposed and presented in a previous paper. This modelling is based on a “unit cell” formulated with elastic isotropic shells coupled to axial connectors. The connectors, which replace bars or beams largely studied in other discrete approaches, reinforce the structure in the yarn directions and naturally capture the specific anisotropic behavior of fabric. Shell elements are used to take into account the in-plane shear stiffness and to manage contact phenomena with the punch and die. The linear characteristic of the connectors, has been extended to a non linear behaviour in the present paper to better account for fabric undulation. Using this numerical model, we propose, in this work to study the effect of process parameters on the woven fabric deformation during the performing step. The emphasis will be placed on the analysis of the influence of the blank holder pressure on the shear angle distribution.

EQUILIBRIUM SHAPES FOR ISOTROPIC ELASTIC TUBES IN THE PLANAR CASE

When making an isotropic elastic shell into a curving tube, the crimp energy and bending energy determine the equilibrium shapes of the tube. In this study, we established a model to explore the elastic behavior of a tube made of an elastic shell. Two typical shapes: torus shape and periodic shape are discussed by studying the equilibrium shape equations in the planar case. Our study reveals that the crimp energy for an isotropic elastic tube is innegligible and will induce abundant shapes. It also reveals that varicose vein is more likely to occur when the blood vessels become thicker, which is in accordance with the clinic experiments.

Mathematical model of spiral waves propagating in the aorta

The spiral waves in the viscous incompressible fluid flow within an arterial vessel modeled by a thin elastic isotropic shell are studied. Asymptotic expansions are constructed for two types of spiral waves. The first type is spiral long wall waves generated (owing to the viscous fluid no-slip at the inner shell wall) by the longitudinal and twist harmonic waves that propagate along the wall. For these waves the amplitude distribution over the vessel cross-section has the form of a boundary layer localized near the inner shell surface. The second is short small-amplitude waves that practically fill the entire vessel cross-section. It is shown that for the short waves the transfer mechanismis the steady-state flow, the role of the longitudinal wall waves and the elastic characteristics of the shell being in this case insignificant.

A glance on the effects of temperature on axisymmetric dynamic behavior of multiwall carbon nanotubes

In this paper the effects of temperature on the radial breathing modes (RBMs) and radial wave propagation in multiwall carbon nanotubes (MWCNTs) are investigated using a continuum model of multiple elastic isotropic shells. The van der Waals forces between tubes are simulated as a nonlinear function of interlayer spacing of MWCNTs. The governing equations are solved using a finite element method. A wide range of innermost radius-to-thickness ratio of MWCNTs is considered to enhance the investigation. The presented solution is verified by comparing the results with those reported in the literature. The effects of temperature on the van der Waals interaction coefficient between layers of MWCNTs are examined. It is found that the variation of the van der Waals interaction coefficient at high temperature is sensible. Subsequently, variations of RBM frequencies and radial wave propagation in MWCNTs with temperatures up to 1 600 K are illustrated. It is shown that the thick MWCNTs are more sensible to temperature than the thin ones.

Achieving a broadband plasmonic-type acoustic cloak using multilayered, spherically isotropic elastic shells

Previous work [Guild, Alù and Haberman, Wave Motion 48, 468-482 (2011)] has shown promising results for the cloaking of a sphere using non-resonant scattering cancellation. Originally developed for electromagnetic waves using plasmonic materials, this plasmonic cloaking technique differs from transformation-based cloaking, canceling the scattered field within the surrounding medium while allowing the incident wave to interact with the object. In contrast to a transformation-based cloak, construction of a plasmonic-type acoustic cloak can be achieved using ordinary fluid and isotropic elastic shells. Although such designs have been shown to be extremely effective, the bandwidth is fundamentally limited in this configuration due to the flexural shell resonances occurring within the isotropic elastic layers. To mitigate these effects, spherically isotropic elastic layers can be utilized, which allow for independent control of the transverse (tangential) layer properties. In this paper, full-wave analytic expressions are developed for a multilayered plasmonic-type acoustic cloak, consisting of an arbitrary number of spherically isotropic elastic layers. Using these expressions, examples for materials of practical interest in acoustic applications will be presented and discussed.

A Simple Discrete Method for the Simulation of the Preforming of Woven Fabric Reinforcement

In this paper, a discrete approach for the simulation of the preforming of dry woven reinforcement is proposed. A “unit cell” is built using elastic isotropic shells and axial connectors instead of bars and beams used in previous studies. Shell elements are used to take into account the in-plane shear stiffness and to manage contact phenomenon with the punch and die. Connectors reinforce the structure in the yarn directions and naturally capture the specific behavior of the fabric. To identify the material parameters, uniaxial tensile tests and bias tests have been employed. A numerical algorithm, coupling Matlab and Abaqus/Explicit, is used to determine the shear parameters by an inverse method. The model has been implemented in Abaqus to simulate hemispherical stamping. Experimental results are compared to numerical simulations, good agreement between both results is shown.

Development of Intrinsic Formulation of W.-Z. Chien of the Geometrically Nonlinear Theory of Thin Elastic Shells

Summary Wei-zhang Chien (1944) derived the equilibrium equations and the compatibility conditions of the nonlinear theory of thin, isotropic elastic shells entirely in terms of the surface stress and strain measures associated with the shell base surface. This allowed Him to divide the complex boundary value problem (BVP) of nonlinear shell analysis into two disjoint and supposedly simpler steps: I) finding the surface stress and strain measures, and II) establishing displacements from already known surface strain measures. In the present paper some achievements of this formulation obtained during the last 66 years are reviewed, with special account of the results obtained by the author. In the first part, using the error of the constitutive equations, we remind some consistent intrinsic BVPs proposed in the literature. These are, in particular: 1) the intrinsic BVP in terms of the surface strain measures, 2) the refined intrinsic BVP in terms of the surface stress resultants and the surface bendings, 3) the almost inextensional bending BVP, 4) the almost membrane BVP, and 5) the intrinsic bending BVP reduced to two PDE for the stress and deformation functions. The alternative set of the refined intrinsic shell equations formulated in the rotated surface base is also presented. All discussed BVPs contain the corresponding intrinsic natural and kinematic boundary conditions. In the second part, recent achievements on determination of position of the deformed shell base surface from the surface strains and bendings are reviewed. Three methods of finding such a position are presented: a) by direct solving some vector ODE, b) through establishing the surface deformation gradient, and c) applying the right polar decomposition to the deformation gradient and then solving some ODE for the rotation tensor field. Finally, we briefly discuss some problems related to the intrinsic formulation of thin shell theory. These are: i) determination of position of a surface from components of its two fundamental forms, ii) bifurcation buckling of the axially compressed circular cylinder, iii) determination of position of the deformed base surface from its surface strains and the height function, and iv) basic assumptions of the special class of flexible shells.

Study on Bi-Stable Behaviors of Isotropic Shell Structures and Numerical Simulation

Bi-stable structure can be stable in both its extended and coiled forms. For the elastic isotropic shell, a theoretical model in terms of only two parameters is proposed. Based on the principle of minimum potential energy, the conditions for bi-stability of isotropic shell are derived. The results indicate that, a pre-stressed shell will have its two stable states if the acting pre-stress satisfies some confined conditions. Further, the bi-stable model is improved taking into account boundary effect of the shell. The results show the rolled-up radius for the second stable state becomes larger. Meanwhile, using shell element in ABAQUS code, numerical simulation on bi-stable process is carried out. The numerical results of rolled-up radius characterizing the bi-stability agree well with the predicted results.

Effects of dimensional parameters and various boundary conditions on axisymmetric vibrations of multi-walled carbon nanotubes using a continuum model

Axisymmetric vibrations of multi-walled carbon nanotubes (MWCNTs) with finite length are investigated in this paper. A multi-walled carbon nanotube is modeled with multiple elastic isotropic shells. Based on a continuum model and considering van der Waals forces between tubes, a two-dimensional finite element model is developed to obtain axisymmetric natural frequencies and mode shapes of MWCNTs. First, the axisymmetric vibrational characteristics of single-walled carbon nanotubes are investigated, and then, they are compared with those of MWCNTs. The effects of van der Waals forces on radial vibration of MWCNTs are also explained. Moreover, influences of end conditions on radial and axial natural frequencies of carbon nanotubes (CNTs) with wide range of length-to-thickness ratio are studied by considering the free-free (F-F) and simply supported (S-S) end conditions. Besides, it is focused on dependence of axisymmetric mode frequencies on dimensional parameters such as length, diameter, as well as number of layers of MWCNTs. Through this, explicit expressions are found for calculating the radial breathing mode and longitudinal mode frequencies of MWCNTs.

Displacement time history analysis and radial wave propagation velocity in pressurized multiwall carbon nanotubes

In this paper, radial wave propagation velocity in multiwall carbon nanotubes (MWCNTs) is investigated using continuum model as the multiple elastic isotropic shells. The effect of van der Waals forces between tubes is taken into account as a nonlinear function of inter-layer spacing of MWCNTs. To solve the governing equations, the finite element method is used. By using time history of displacement, the mean velocity of radial wave propagation from outermost tube to innermost tube is calculated. Also, the effects of geometric parameters such as inner radius and thickness of nanotubes on dynamic behaviors are investigated. Furthermore, dynamic load factors for each tube of MWCNTs are calculated using presented solution. The effects of hydrostatic pressures applied on bounding surfaces of MWCNTs on radial wave propagation are discussed. The presented solution and results are verified by comparing against those reported in published literatures.