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In various biological processes such as endocytosis and caveolae formation, the cell membrane is locally deformed into curved morphologies. Previous models to study membrane morphologies resulting from locally induced curvature often only consider the possibility of axisymmetric shapes-an indeed unphysical constraint. Past studies predict that the cell membrane buds at low resting tensions and stalls at a flat pit at high resting tensions. In this work, we lift the restriction to axisymmetry to study all possible membrane morphologies. Only if the resting tension of the membrane is low, we reproduce axisymmetric membrane morphologies. When the resting tension is moderate to high, we show that 1) axisymmetric membrane pits are unstable and 2) nonaxisymmetric ridge-shaped structures are energetically favorable. Furthermore, we find the interplay between intramembrane viscous flow and the rate of induced curvature affects the membrane's ability to transition into nonaxisymmetric ridges and axisymmetric buds. In particular, we show that axisymmetric buds are favored when the induced curvature is rapidly increased, whereas nonaxisymmetric ridges are favored when the curvature is slowly increased. Our results hold relevant implications for biological processes such as endocytosis and physical phenomena like phase separation in lipid bilayers.

We present an adaptive space-time phase field formulation for dynamic fracture of brittle shells. Their deformation is characterized by the Kirchhoff–Love thin shell theory using a curvilinear surface description. All kinematical objects are defined on the shell’s mid-plane. The evolution equation for the phase field is determined by the minimization of an energy functional based on Griffith’s theory of brittle fracture. Membrane and bending contributions to the fracture process are modeled separately and a thickness integration is established for the latter. The coupled system consists of two nonlinear fourth-order PDEs and all quantities are defined on an evolving two-dimensional manifold. Since the weak form requires C1-continuity, isogeometric shape functions are used. The mesh is adaptively refined based on the phase field using Locally Refinable (LR) NURBS. Time is discretized based on a generalized-α method using adaptive time-stepping, and the discretized coupled system is solved with a monolithic Newton–Raphson scheme. The interaction between surface deformation and crack evolution is demonstrated by several numerical examples showing dynamic crack propagation and branching.

An arbitrary Lagrangian–Eulerian (ALE) finite element method for arbitrarily curved and deforming two-dimensional materials and interfaces is presented here. An ALE theory is developed by endowing the surface with a mesh whose in-plane velocity need not depend on the in-plane material velocity, and can be specified arbitrarily. A finite element implementation of the theory is formulated and applied to curved and deforming surfaces with in-plane incompressible flows. Numerical inf–sup instabilities associated with in-plane incompressibility are removed by locally projecting the surface tension onto a discontinuous space of piecewise linear functions. The general isoparametric finite element method, based on an arbitrary surface parametrization with curvilinear coordinates, is tested and validated against several numerical benchmarks. A new physical insight is obtained by applying the ALE developments to cylindrical fluid films, which are computationally and analytically found to be stable to non-axisymmetric perturbations, and unstable with respect to long-wavelength axisymmetric perturbations when their length exceeds their circumference. A Lagrangian scheme is attained as a special case of the ALE formulation. Though unable to model fluid films with sustained shear flows, the Lagrangian scheme is validated by reproducing the cylindrical instability. However, relative to the ALE results, the Lagrangian simulations are found to have spatially unresolved regions with few nodes, and thus larger errors.

This paper presents a general theory and isogeometric finite element implementation for studying mass conserving phase transitions on deforming surfaces. The mathematical problem is governed by two coupled fourth-order nonlinear partial differential equations (PDEs) that live on an evolving two-dimensional manifold. For the phase transitions, the PDE is the Cahn–Hilliard equation for curved surfaces, which can be derived from surface mass balance in the framework of irreversible thermodynamics. For the surface deformation, the PDE is the (vector-valued) Kirchhoff–Love thin shell equation. Both PDEs can be efficiently discretized using C 1 -continuous interpolations without derivative degrees-of-freedom (dofs). Structured NURBS and unstructured spline spaces with pointwise C 1 -continuity are utilized for these interpolations. The resulting finite element formulation is discretized in time by the generalized-α scheme with adaptive time-stepping, and it is fully linearized within a monolithic Newton–Raphson approach. A curvilinear surface parameterization is used throughout the formulation to admit general surface shapes and deformations. The behavior of the coupled system is illustrated by several numerical examples exhibiting phase transitions on deforming spheres, tori and double-tori.

The given text discusses various mathematical equations and concepts related to the bending behavior of fibers. It addresses issues with certain expressions and proposes alternative models that take into account the change in curvature of the fibers. The text also responds to criticisms raised by another researcher and provides clarification on certain equations and variables. Overall, the text provides a technical and detailed analysis of the topic of fiber bending. [Extracted from the article]

The genome inside the eukaryotic cells is guarded by a unique shell structure, called the nuclear envelope (NE), made of lipid membranes. This structure has an ultra torus topology with thousands of torus-shaped holes that imparts the structure a high flexural stiffness. Inspired from this biological design, here we present a novel "torene" architecture to design lightweight shell structures with ultra-stiffness for engineering applications. We perform finite element analyses on classic benchmark problems to investigate the mechanics of torene shells. This study reveals that the torene shells can achieve one order of magnitude or higher flexural stiffness than traditional shells with the same amount of material. This novel geometric strategy opens new avenues to exploit additive manufacturing to design lightweight shell structures for extreme mechanical environments. [ABSTRACT FROM AUTHOR]

We present an isogeometric finite element formulation of frictionless beam‐to‐beam contact, using a Cosserat beam formulation with unconstrained directors. The beam's cross‐sectional deformation can be efficiently represented by director vectors. For the first order beam kinematics, Poisson locking due to the inability to represent linear in‐plane strain field in the cross‐section is alleviated by an enhanced assumed strain (EAS) method [1]. We employ a Gauss point‐to‐surface contact formulation combined with an active set iteration and a penalty method. The beam's cross‐sectional boundary as well as the axis is parameterized by a NURBS (non‐uniform rational B‐spline) curve, so that the lateral boundary surface has at least C2‐continuity, which yields a continuous surface metric and curvature in the closest point projection [2]. Since we do not employ zero stress conditions, a three‐dimensional nonlinear constitutive law can be straightforwardly utilized. We present several numerical examples, where a compressible Neo‐Hookean material is particularly considered. [ABSTRACT FROM AUTHOR]

This work presents a generalized Kirchhoff–Love shell theory that can explicitly capture fiber-induced anisotropy not only in stretching and out-of-plane bending, but also in in-plane bending. This setup is particularly suitable for heterogeneous and fibrous materials such as textiles, biomaterials, composites and pantographic structures. The presented theory is a direct extension of classical Kirchhoff–Love shell theory to incorporate the in-plane bending resistance of fibers. It also extends existing second-gradient Kirchhoff–Love shell theory for initially straight fibers to initially curved fibers. To describe the additional kinematics of multiple fiber families, a so-called in-plane curvature tensor—which is symmetric and of second order—is proposed. The effective stress tensor and the in-plane and out-of-plane moment tensors are then identified from the mechanical power balance. These tensors are all second order and symmetric in general. Constitutive equations for hyperelastic materials are derived from different expressions of the mechanical power balance. The weak form is also presented as it is required for computational shell formulations based on rotation-free finite element discretizations. The proposed theory is illustrated by several analytical examples. [ABSTRACT FROM AUTHOR]

A Bayesian regularization-backpropagation neural network (BR-BPNN) model is employed to predict some aspects of the gecko spatula peeling, viz. the variation of the maximum normal and tangential pull-off forces and the resultant force angle at detachment with the peeling angle. K -fold cross validation is used to improve the effectiveness of the model. The input data is taken from finite element (FE) peeling results. The neural network is trained with 75 % of the FE dataset. The remaining 25 % are utilized to predict the peeling behavior. The training performance is evaluated for every change in the number of hidden layer neurons to determine the optimal network structure. The relative error is calculated to draw a clear comparison between predicted and FE results. It is shown that the BR-BPNN model in conjunction with the k -fold technique has significant potential to estimate the peeling behavior. [ABSTRACT FROM AUTHOR]

This work presents a numerical formulation to model isotropic viscoelastic material behavior for membranes and thin shells. The surface and the shell theory are formulated within a curvilinear coordinate system, which allows the representation of general surfaces and deformations. The kinematics follow from Kirchhoff–Love theory and the discretization makes use of isogeometric shape functions. A multiplicative split of the surface deformation gradient is employed, such that an intermediate surface configuration is introduced. The surface metric and curvature of this intermediate configuration follow from the solution of nonlinear evolution laws—ordinary differential equations—that stem from a generalized viscoelastic solid model. The evolution laws are integrated numerically with the implicit Euler scheme and linearized within the Newton–Raphson scheme of the nonlinear finite element framework. The implementation of membrane and bending viscosity is verified with the help of analytical solutions and shows ideal convergence behavior. The chosen numerical examples capture large deformations and typical viscoelasticity behavior, such as creep, relaxation, and strain rate dependence. It is also shown that the proposed formulation can be straightforwardly applied to model boundary viscoelasticity of 3D bodies. [ABSTRACT FROM AUTHOR]