This paper presents a general non-linear computational formulation for rotation-free thin shells based on isogeometric finite elements. It is a displacement-based formulation that admits general ...material models. The formulation allows for a wide range of constitutive laws, including both shell models that are extracted from existing 3D continua using numerical integration and those that are directly formulated in 2D manifold form, like the Koiter, Canham and Helfrich models. Further, a unified approach to enforce the G1-continuity between patches, fix the angle between surface folds, enforce symmetry conditions and prescribe rotational Dirichlet boundary conditions, is presented using penalty and Lagrange multiplier methods. The formulation is fully described in the natural curvilinear coordinate system of the finite element description, which facilitates an efficient computational implementation. It contains existing isogeometric thin shell formulations as special cases. Several classical numerical benchmark examples are considered to demonstrate the robustness and accuracy of the proposed formulation. The presented constitutive models, in particular the simple mixed Koiter model that does not require any thickness integration, show excellent performance, even for large deformations.
This work presents a general unified theory for coupled nonlinear elastic and inelastic deformations of curved thin shells. The coupling is based on a multiplicative decomposition of the surface ...deformation gradient. The kinematics of this decomposition is examined in detail. In particular, the dependency of various kinematical quantities, such as area change and curvature, on the elastic and inelastic strains is discussed. This is essential for the development of general constitutive models. In order to fully explore the coupling between elastic and different inelastic deformations, the surface balance laws for mass, momentum, energy and entropy are examined in the context of the multiplicative decomposition. Based on the second law of thermodynamics, the general constitutive relations are then derived. Two cases are considered: Independent inelastic strains, and inelastic strains that are functions of temperature and concentration. The constitutive relations are illustrated by several nonlinear examples on growth, chemical swelling, thermoelasticity, viscoelasticity and elastoplasticity of shells. The formulation is fully expressed in curvilinear coordinates leading to compact and elegant expressions for the kinematics, balance laws and constitutive relations.
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.
Contact involving soft materials often combines dry adhesion, sliding friction, and large deformations. At the local level, these three aspects are rarely captured simultaneously, but included in the ...theoretical models by Mergel et al., (2019). We here develop a corresponding finite element framework that captures 3D finite-strain contact of two deformable bodies. This framework is suitable to investigate sliding friction even under tensile normal loads. First, we demonstrate the capabilities of our finite element model using both 2D and 3D test cases, which range from compliant tapes to structures with high stiffness, and include deformable–rigid and deformable–deformable contact. We then provide new results on the onset of sliding of smooth elastomer–glass interfaces, a setup that couples nonlinear material behavior, adhesion, and large frictional stresses. Our simulations not only agree well with both experimental and theoretical findings, they also provide new insights into the current debate on the shear-induced reduction of the contact area in elastomeric contact.
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•Our computational model incorporates adhesion, friction, and large deformations.•It is suited for contact of soft materials like rubber, adhesives, or biomaterials.•It also captures shear-induced contact area reduction in Hertzian rubber contact.•This reduction originates from large deformations, and is enhanced by adhesion.•Further applications include 2D and 3D tape peeling and asperity–asperity contact.
This work presents a Finite Element Model Updating inverse methodology for reconstructing heterogeneous material distributions based on an efficient isogeometric shell formulation. It uses nonlinear ...hyperelastic material models suitable for describing incompressible material behavior as well as initially curved shells. The material distribution is discretized by bilinear elements such that the nodal values are the design variables to be identified. Independent FE analysis and material discretization, as well as flexible incorporation of experimental data, offer high robustness and control. Three elementary test cases and one application example, which exhibit large deformations and different challenges, are considered: uniaxial tension, pure bending, sheet inflation, and abdominal wall pressurization. Experiment-like results are generated from high-resolution simulations with the subsequent addition of up to 4% noise. Local optimization based on the trust-region approach is used. The results show that with a sufficient number of experimental measurements, design variables and analysis elements, the algorithm is capable to reconstruct material distributions with high precision even in the presence of large noise. The proposed formulation is very general, facilitating its extension to other material models, optimization algorithms and meshing approaches. Adapted material discretizations allow for an efficient and accurate reconstruction of material discontinuities by avoiding overfitting due to superfluous design variables. For increased computational efficiency, the analytical sensitivities and Jacobians are provided.
We develop two new continuum contact models for coupled adhesion and friction, and discuss them in the context of existing models proposed in the literature. Our new models are able to describe ...sliding friction even under tensile normal forces, which seems reasonable for certain adhesion mechanisms. In contrast, existing continuum models for combined adhesion and friction typically include sliding friction only if local contact stresses are compressive. Although such models work well for structures with sufficiently strong local compression, they fail to capture sliding friction for soft and compliant systems (like adhesive pads), for which the resistance to bending is low. This can be overcome with our new models. For further motivation, we additionally present experimental results for the onset of sliding of a smooth glass plate on a smooth elastomer cap under low normal loads. As shown, the findings from these experiments agree well with the results from our models. In this paper we focus on the motivation and derivation of our continuum contact models, and provide a corresponding literature survey. Their implementation in a nonlinear finite element framework as well as the algorithmic treatment of adhesion and friction will be discussed in future work.
An isogeometric finite element formulation for geometrically and materially nonlinear Timoshenko beams is presented, which incorporates in-plane deformation of the cross-section described by two ...extensible director vectors. Since those directors belong to the space R3, a configuration can be additively updated. The developed formulation allows direct application of nonlinear three-dimensional constitutive equations without zero stress conditions. Especially, the significance of considering correct surface loads rather than applying an equivalent load directly on the central axis is investigated. Incompatible linear in-plane strain components for the cross-section have been added to alleviate Poisson locking by using an enhanced assumed strain (EAS) method. In various numerical examples exhibiting large deformations, the accuracy and efficiency of the presented beam formulation is assessed in comparison to brick elements. We particularly use hyperelastic materials of the St.Venant-Kirchhoff and compressible Neo-Hookean types.
•The extensible directors belong to the space R3, and describe in-plane cross-sectional deformations.•Enhanced assumed strain method is employed to alleviate Poisson locking.•A straightforward interface to general three-dimensional constitutive law is obtained.•The significance of considering correct surface loads rather than equivalent central axis loads is shown.•The developed beam formulation is verified by comparison with the results from brick elements.
A unified fluid–structure interaction (FSI) formulation is presented for solid, liquid and mixed membranes. Nonlinear finite elements (FE) and the generalized-α scheme are used for the spatial and ...temporal discretization. The membrane discretization is based on curvilinear surface elements that can describe large deformations and rotations, and also provide a straightforward description for contact. The fluid is described by the incompressible Navier–Stokes equations, and its discretization is based on stabilized Petrov–Galerkin FE. The coupling between fluid and structure uses a conforming sharp interface discretization, and the resulting non-linear FE equations are solved monolithically within the Newton–Raphson scheme. An arbitrary Lagrangian–Eulerian formulation is used for the fluid in order to account for the mesh motion around the structure. The formulation is very general and admits diverse applications that include contact at free surfaces. This is demonstrated by two analytical and three numerical examples exhibiting strong coupling between fluid and structure. The examples include balloon inflation, droplet rolling and flapping flags. They span a Reynolds-number range from 0.001 to 2000. One of the examples considers the extension to rotation-free shells using isogeometric FE.
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