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

Abstract

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 (ODEs) - 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.

arXiv commentIn this version, parts of the introduction and conclusion are rewritten, remarks 1.1, 1.2 and 3.1 are added, the title is slightly modified, the list of highlights is updated, and minor typos are fixed

References in corpus (14)

• Isogeometric analysis: an overview and computer implementation aspects
• A NURBS-based Inverse Analysis for Reconstruction of Nonlinear Deformations of Thin Shell Structures
• A new rotation-free isogeometric thin shell formulation and a corresponding continuity constraint for patch boundaries
• A stabilized finite element formulation for liquid shells and its application to lipid bilayers
• An explicit solution for implicit time stepping in finite strain viscoelasticity
• An adaptive space-time phase field formulation for dynamic fracture of brittle shells based on LR NURBS
• An isogeometric finite element formulation for phase transitions on deforming surfaces
• The multiplicative deformation split for shells with application to growth, chemical swelling, thermoelasticity, viscoelasticity and elastoplasticity
• Efficient isogeometric thin shell formulations for soft biological materials
• A new shell formulation for graphene structures based on existing ab-initio data
• Isogeometric continuity constraints for multi-patch shells governed by fourth-order deformation and phase field models
• Nonlinear material identification of heterogeneous isogeometric Kirchhoff-Love shells
• A general isogeometric finite element formulation for rotation-free shells with in-plane bending of embedded fibers
• A finite membrane element formulation for surfactants

Cited by in corpus (2)

• A simple and efficient hybrid discretization approach to alleviate membrane locking in isogeometric thin shells
• A new anisotropic bending model for nonlinear shells: Comparison with existing models and isogeometric finite element implementation