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

TL;DR

This paper introduces a novel isogeometric finite element formulation for modeling the viscoelastic behavior of membranes and thin shells, incorporating a multiplicative surface deformation split within a curvilinear coordinate framework.

Contribution

It presents a new numerical approach combining isogeometric analysis with a multiplicative surface deformation split for viscoelastic shells and membranes.

Findings

Accurate modeling of large deformations and viscoelastic effects like creep and relaxation.

Verification of membrane and bending viscosity with analytical solutions.

Effective numerical implementation demonstrating convergence and applicability.

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.