On the theoretical foundations of thin solid and liquid shells

Journal of Computational Physics

Duong

Mandadapu

et al. 2017

This paper presents a new finite element (FE) formulation for liquid shells that is based on an explicit, 3D surface discretization using C 1 -continuous finite elements constructed from NURBS interpolation. Both displacement-based and mixed displacement/pressure FE formulations are proposed. The latter is needed for area-incompressible material behavior, where penalty-type regularizations can lead to misleading results. In order to obtain quasi-static solutions for liquid shells devoid of shear stiffness, several numerical stabilization schemes are proposed based on adding stiffness, adding viscosity or using projection. Several numerical examples are considered in order to illustrate the accuracy and the capabilities of the proposed formulation, and to compare the different stabilization schemes. The presented formulation is capable of simulating non-trivial surface shapes associated with tube formation and protein-induced budding of lipid bilayers. In the latter case, the presented formulation yields non-axisymmetric solutions, which have not been observed in previous simulations. It is shown that those non-axisymmetric shapes are preferred over axisymmetric ones.

“…Likewise definitions follow for the reference surface. The variation and linearization of the above quantities can be found in Sauer and Duong [54].…”

Section: Thin Shell Kinematicsmentioning

confidence: 98%

Section: Discretized Weak Formmentioning

confidence: 99%

Section: Pure Bending and Stretching Of A Flat Stripmentioning

confidence: 99%

“…Following decomposition (43), f e intσ can be split into the in-plane and out-of-plane contributions [55] f e…”

Section: Discretized Weak Formmentioning

confidence: 99%

“…For thin shells the stored energy function (per reference surface area) takes the form W = W (a αβ , b αβ ), such that [61,54] …”

Section: Constitutionmentioning

confidence: 99%

See 3 more Smart Citations

A stabilized finite element formulation for liquid shells and its application to lipid bilayers

Journal of Computational Physics

Duong

Mandadapu

et al. 2017

A general isogeometric finite element formulation for rotation‐free shells with in‐plane bending of embedded fibers

Duong

Itskov

Numerical Meth Engineering

2022

This article presents a general, nonlinear isogeometric finite element formulation for rotation‐free shells with embedded fibers that captures anisotropy in stretching, shearing, twisting, and bending—both in‐plane and out‐of‐plane. These capabilities allow for the simulation of large sheets of heterogeneous and fibrous materials either with or without matrix, such as textiles, composites, and pantographic structures. The work is a computational extension of our earlier theoretical work that extends existing Kirchhoff‐Love shell theory to incorporate the in‐plane bending resistance of initially straight or curved fibers. The formulation requires only displacement degrees‐of‐freedom to capture all mentioned modes of deformation. To this end, isogeometric shape functions are used in order to satisfy the required C1$$ {C}^1 $$‐continuity for bending across element boundaries. The proposed formulation can admit a wide range of material models, such as surface hyperelasticity that does not require any explicit thickness integration. To deal with possible material instability due to fiber compression, a stabilization scheme is added. Several benchmark examples are used to demonstrate the robustness and accuracy of the proposed computational formulation.

An isogeometric finite element formulation for boundary and shell viscoelasticity based on a multiplicative surface deformation split

Paul

Numerical Meth Engineering

2022

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.