Convergence of Hencky-type discrete beam model to euler inextensible elastica in large deformation: Rigorous proof

The present chapter concerns rigorous homogenization of a Hencky-type discrete beam model, which is useful for the numerical study of complex fibrous systems as pantographic sheets as well as woven fabrics. -convergence of the discrete model towards the inextensible Euler’s beam model is proven and the result is established for placements in Rd in large deformation regime

Electro-deformation of a moving boundary: a drop interface and a lipid bilayer membrane

This dissertation focuses on the deformation of a viscous drop and a vesicle immersed in a (leaky) dielectric fluid under an electric field. A number of mathematical tools, both analytical and numerical, are developed for these investigations. The dissertation is divided into three parts. First, a large-deformation model is developed to capture the equilibrium deformation of a viscous spheroidal drop covered with non-diffusing insoluble surfactant under a uniform direct current (DC) electric field. The large- deformation model predicts the dependence of equilibrium spheroidal drop shape on the permittivity ratio, conductivity ratio, surfactant coverage, and the elasticity number. Results from the model are carefully compared against the small-deformation (quasispherical) analysis, experimental data and numerical simulation results in the literature. Moreover, surfactant effects, such as tip stretching and surface dilution effects, are greatly amplified at large surfactant coverage and high electric capillary number. These effects are well captured by the spheroidal model, but cannot be described in the second-order small-deformation theory. The large-deformation spheroidal model is then extended to study the equilibrium deformation of a giant unilamellar vesicle (GUV) under an alternating current (AC) electric field. The vesicle membrane is modeled as a thin capacitive spheroidal shell and the equilibrium vesicle shape is computed from balancing the mechanical forces between the fluid, the membrane and the imposed electric field. Detailed comparison against both experiments and small-deformation theory shows that the spheroidal model gives better agreement with experiments in terms of the dependence on fluid conductivity ratio, electric field strength and frequency, and vesicle size. Asymptotic analysis is conducted to compute the crossover frequency where a prolate vesicle crosses over to an oblate shape, and comparisons show the spheroidal model gives better agreement with experimental observations. Finally, a numerical scheme based on immersed interface method for two-phase fluids is developed to simulate the time-dependent dynamics of an axisymmetric drop in an electric field. The second-order immersed interface method is applied to solving both the fluid velocity field and the electric field. To date this has not been done before in the literature. Detailed numerical studies on this new numerical scheme shows numerical convergence and good agreement with the large-deformation model. Dynamics of an axisymmetric viscous drop under an electric field is being simulated using this novel numerical code

An implicit non-ordinary state-based peridynamics for large deformation solid mechanics problems

The numerical simulation of the cracking process remains one of the most significant challenges in solid mechanics. Compared classical approaches, peridynamics(PD) has some attractive features because the basic equations remain applicable even when singularities appear in the deformation. Numerical time-integration plays a big role in any computational framework and unlike explicit time-integration, implicit time-integration methods can be much more efficient because of the ability to adopt fairly large time increments, making it a suitable option for PD analyses of large deformation problems. The objective of this thesis is to propose an implicit non-ordinary state-based peridynamics (NOSB PD) approach focusing on quasistatic analyses with large deformation mechanics. Firstly, the use of the adaptive dynamic relaxation (ADR) method as a solution strategy for quasi-static analyses with large deformation mechanics is discussed. Next, an analytical expression of the Jacobian matrix based on the equation of motion of NOSB PD is formulated to ensure optimum convergence of the global residual force. To address some instability issues in the existing “corresponding material” model, caused by zero-energy modes instability, recent approaches proposed by Silling (2017) are used to control the spurious deformation modes. An additional stabilisation term with respect to displacement is included in the derivatives for Jacobian formulation. This allows a more accurate NOSB PD approach to model material behaviour where correspondence materials have previously failed due to instability. Finally, to validate the proposed methodology, several numerical examples of 2D damage problems model using a stabilised correspondence model are verified, and suggestions are made for future implementation. The novelty of this thesis lies in providing theoretical development and numerical implementation of an implicit non-linear NOSB PD focusing on quasi-static analyses with large deformation mechanics. Findings from this thesis will interest researchers working in numerical methods, along with those solving discontinuous solid mechanics problems

Parallel three-dimensional simulations of quasi-static elastoplastic solids

Hypo-elastoplasticity is a flexible framework for modeling the mechanics of many hard materials under small elastic deformation and large plastic deformation. Under typical loading rates, most laboratory tests of these materials happen in the quasi-static limit, but there are few existing numerical methods tailor-made for this physical regime. In this work, we extend to three dimensions a recent projection method for simulating quasi-static hypo-elastoplastic materials. The method is based on a mathematical correspondence to the incompressible Navier-Stokes equations, where the projection method of Chorin (1968) is an established numerical technique. We develop and utilize a three-dimensional parallel geometric multigrid solver employed to solve a linear system for the quasi-static projection. Our method is tested through simulation of three-dimensional shear band nucleation and growth, a precursor to failure in many materials. As an example system, we employ a physical model of a bulk metallic glass based on the shear transformation zone theory, but the method can be applied to any elastoplasticity model. We consider several examples of three-dimensional shear banding, and examine shear band formation in physically realistic materials with heterogeneous initial conditions under both simple shear deformation and boundary conditions inspired by friction welding.Comment: Final version. Accepted for publication in Computer Physics Communication