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We present a stable mixed isogeometric finite element formulation for geometrically and materially nonlinear beams in transient elastodynamics, where a Cosserat beam formulation with extensible directors is used. The extensible directors yields a linear configuration space incorporating constant in-plane cross-sectional strains. Higher-order (incompatible) strains are introduced to correct stiffness, whose additional degrees-of-freedom are eliminated by an element-wise condensation. Further, the present discretization of the initial director field leads to the objectivity of approximated strain measures, regardless of the degree of basis functions. For physical stress resultants and strains, we employ a global patch-wise approxiation using B-spline basis functions, whose higher-order continuity enables to use much less degrees-of-freedom, compared to element-wise approximation. For time-stepping, we employ an implicit energy-momentum consistent scheme, which exhibits superior numerical stability in comparison to standard trapezoidal and mid-point rules. Several numerical examples are presented to verify the present method., Comment: 64 pages, 30 figures

We respond to David Steigmann's discussion of our paper "A general theory for anisotropic Kirchhoff-Love shells with in-plane bending of embedded fibers, Math. Mech. Solids, 28(5):1274-1317" (arXiv:2101.03122). His discussion allows us to clarify two misleading statements in our original paper, and confirm that its formulation is fully consistent with the formulation of Steigmann. We also demonstrate that some of our original statements criticized by Steigmann are not wrong., Comment: 7 pages, 1 figure

This work presents a new hybrid discretization approach to alleviate membrane locking in isogeometric finite element formulations for Kirchhoff-Love shells. The approach is simple, and requires no additional dofs and no static condensation. It does not increase the bandwidth of the tangent matrix and is effective for both linear and nonlinear problems. It combines isogeometric surface discretizations with classical Lagrange-based surface discretizations, and can thus be run with existing isogeometric finite element codes. Also, the stresses can be recovered straightforwardly. The effectiveness of the proposed approach in alleviating, if not eliminating, membrane locking is demonstrated through the rigorous study of the convergence behavior of several classical benchmark problems. Accuracy gains are particularly large in the membrane stresses. The approach is formulated here for quadratic NURBS, but an extension to other discretization types can be anticipated. The same applies to other constraints and associated locking phenomena., Comment: 36 pages, 33 figures, 1 table

The effect of higher order continuity in the solution field by using NURBS basis function in isogeometric analysis (IGA) is investigated for an efficient mixed finite element formulation for elastostatic beams. It is based on the Hu-Washizu variational principle considering geometrical and material nonlinearities. Here we present a reduced degree of basis functions for the additional fields of the stress resultants and strains of the beam, which are allowed to be discontinuous across elements. This approach turns out to significantly improve the computational efficiency and the accuracy of the results. We consider a beam formulation with extensible directors, where cross-sectional strains are enriched to avoid Poisson locking by an enhanced assumed strain method. In numerical examples, we show the superior per degree-of-freedom accuracy of IGA over conventional finite element analysis, due to the higher order continuity in the displacement field. We further verify the efficient rotational coupling between beams, as well as the path-independence of the results., Comment: 58 pages, 23 figures

Understanding the interaction mechanics between graphene layers and co-axial carbon nanotubes (CNTs) is essential for modeling graphene and CNT-based nanoelectromechanical systems. This work proposes a new continuum contact model to study interlayer interactions between curved graphene sheets. The continuum model is calibrated and validated using molecular dynamics (MD) simulations. These are carried out employing the reactive empirical bond order (REBO)+Lennard-Jones (LJ) to model the interactions within a sheet, while the LJ, Kolmogorov-Crespi (KC), and Lebedeva potentials are used to model the interactions between sheets. The continuum contact model is formulated for separation distances greater than 0.29nm, when sheet deformations are sufficiently small and do not affect the sheet interactions substantially. This allows to treat the master contact surface as rigid, thus simplifying the contact formulation greatly. The model calibration is conducted systematically for a sequence of different stackings using existing and newly proposed ansatz functions. The calibrated continuum model is then implemented in a curvilinear finite element (FE) shell formulation to investigate the pull-out and twisting interactions between co-axial CNTs. The resisting pull-out forces and torques depend strongly on the chirality of the considered CNTs. The absolute differences between FE and MD results are very small, and can be attributed to model assumptions and loading conditions., Comment: 39 pages, 26 figures

This work proposes a new efficient approach for calculating the bending stiffness of two-dimensional materials using simple atomistic tests on small periodic unit cells. The tests are designed such that bending deformations are dominating and membrane deformations are minimized. Atomistic ab-initio simulations then allow for the efficient computation of bending energies. Density functional theory is used for this. Atomistic bending energies are then compared to classical models from structural mechanics. Two different models are considered for this -- one based on beam theory and one based on rigid linkage theory -- and their results are compared with each other. Four different materials with 2D hexagonal (honeycomb) structure are chosen as a case study: graphene, hexagonal boron nitride, silicene, and blue phosphorene. The calculated bending stiffnesses converge with increasing unit cell size, such that small unit cells already provide accurate results that are in good agreement with the literature. Using the same atomistic tests, it is shown that the bending stiffness of graphene can still be considered constant at moderately large deformations. Apart from being efficient and accurate, the proposed approach allows for various extensions.

This work proposes four novel hybrid quadrature schemes for the efficient and accurate evaluation of weakly singular boundary integrals (1/r kernel) on arbitrary smooth surfaces. Such integrals appear in boundary element analysis for several partial differential equations including the Stokes equation for viscous flow and the Helmholtz equation for acoustics. The proposed quadrature schemes apply a Duffy transform-based quadrature rule to surface elements containing the singularity and classical Gaussian quadrature to the remaining elements. Two of the four schemes additionally consider a special treatment for elements near to the singularity, where refined Gaussian quadrature and a new moment-fitting quadrature rule are used. The hybrid quadrature schemes are systematically studied on flat B-spline patches and on NURBS spheres considering two different sphere discretizations: An exact single-patch sphere with degenerate control points at the poles and an approximate discretization that consist of six patches with regular elements. The efficiency of the quadrature schemes is further demonstrated in boundary element analysis for Stokes flow, where steady problems with rotating and translating curved objects are investigated in convergence studies for both, mesh and quadrature refinement. Much higher convergence rates are observed for the proposed new schemes in comparison to classical schemes.

A new nonlinear hyperelastic bending model for shells formulated directly in surface form is presented, and compared to four prominently used bending models. Through an essential set of elementary nonlinear bending test cases, the stresses and moments of each model are examined analytically. Only the proposed bending model passes all the test cases while the other bending models either fail or only pass the test cases for small deformations. The proposed new bending model can handle large deformations and initially curved surfaces. It is based on the principal curvatures and their directions in the initial configuration, and it thus can have different bending moduli along those directions. These characteristics make it flexible in modeling a given material, while it does not suffer from the pathologies of existing bending models. Further, the bending models are compared computationally through four classical benchmark examples and one contact example. As the underlying shell theory is based on Kirchhoff-Love kinematics, isogeometric NURBS shape functions are used to discretize the shell surface. The linearization and efficient finite element implementation of the proposed new model are also provided.

This paper presents an isogeometric finite element formulation for nonlinear beams with impenetrability constraints, based on the kinematics of Cosserat rods with unconstrained directors. The beam cross-sectional deformation is represented by director vectors of an arbitrary order. For the frictionless lateral beam-to-beam contact, a surface-to-surface contact algorithm combined with an active set strategy and a penalty method is employed. The lateral boundary surface of the beam is parameterized by its axis and cross-sectional boundary curves with NURBS basis functions having at least $C^2$-continuity, which yields a continuous surface metric and curvature for the closest point projection. Three-dimensional constitutive laws of hyperelastic materials are considered. Several numerical examples verify the accuracy and efficiency of the proposed beam contact formulation in comparison to brick element solutions. The lateral contact pressure distribution of the beam formulation is in excellent agreement with the contact pressure of the brick element formulation while requiring much less degrees-of-freedom., Comment: 44 pages, 36 figures

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., Comment: In 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

Cementless implants have become widely used for total hip replacement surgery. The long-term stability of these implants is achieved by bone growing around and into the porous surface of the implant, a process called osseointegration. However, debonding of the bone-implant interface can still occur due to aseptic implant loosening and insufficient osseointegration, which may have dramatic consequences. The aim of this work is to describe a new 3D finite element frictional contact formulation for the debonding of partially osseointegrated implants. The contact model is based on a modified Coulomb's friction law (Immel et al. 2020, Biomech. Model. Mechanobiol.) that takes into account the tangential debonding of the bone-implant interface. This model is extended in the direction normal to the bone-implant interface by considering a cohesive zone model, to account for adhesion phenomena in the normal direction and for adhesive friction of partially bonded interfaces. The model is applied to simulate the debonding of an acetabular cup implant. The influence of partial osseointegration and adhesive effects on the long-term stability of the implant is assessed. The influence of different patient- and implant-specific parameters such as the friction coefficient, the trabecular Young's modulus and the interference fit is also analyzed, in order to determine the optimal stability for different configurations. Furthermore, this work provides guidelines for future experimental and computational studies, that are necessary for further parameter calibration., Comment: Biomech Model Mechanobiol (2022)

This paper 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 [1] 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 $C^1$-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., Comment: This version changes the title for a better clarity. It also updates the reference list and improves minor text editing. Results unchanged

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., Comment: 37 pages, 18 figures, 5 tables

This work presents a self-contained continuum formulation for coupled chemical, mechanical and thermal contact interactions. The formulation is very general and hence admits arbitrary geometry, deformation and material behavior. All model equations are derived rigorously from the balance laws of mass, momentum, energy and entropy in the framework of irreversible thermodynamics, thus exposing all the coupling present in the field equations and constitutive relations. In the process, the conjugated kinematic and kinetic variables for mechanical, thermal and chemical contact are identified, and the analogies between mechanical, thermal and chemical contact are highlighted. Particular focus is placed on the thermodynamics of chemical bonding distinguishing between exothermic and endothermic contact reactions. Distinction is also made between long-range, non-touching surface interactions and short-range, touching contact. For all constitutive relations examples are proposed and discussed comprehensively with particular focus on their coupling. Finally, three analytical test cases are presented that illustrate the thermo-chemo-mechanical contact coupling and are useful for verifying computational models. While the main novelty is the extension of existing contact formulations to chemical contact, the presented formulation also sheds new light on thermo-mechanical contact, since it is consistently derived from basic principles using only few assumptions., Comment: 40 pages, 11 figures, 3 tables

This work presents a generalized Kirchhoff-Love shell theory that can explicitly capture fiber-induced anisotropy not only in stretching and out-of-plane bending, but also in in-plane bending. This setup is particularly suitable for heterogeneous and fibrous materials such as textiles, biomaterials, composites and pantographic structures. The presented theory is a direct extension of classical Kirchhoff-Love shell theory to incorporate the in-plane bending resistance of fibers. It also extends existing second-gradient Kirchhoff-Love shell theory for initially straight fibers to initially curved fibers. To describe the additional kinematics of multiple fiber families, a so-called in-plane curvature tensor -- which is symmetric and of second order -- is proposed. The effective stress tensor and the in-plane and out-of-plane moment tensors are then identified from the mechanical power balance. These tensors are all second order and symmetric in general. Constitutive equations for hyperelastic materials are derived from different expressions of the mechanical power balance. The weak form is also presented as it is required for computational shell formulations based on rotation-free finite element discretizations., Comment: This version changes the title for a better clarity. It adds two appendixes for the proof in frame invariants and comparison with Cosserat theory. It improves further text editing. Results unchanged

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 ${\Bbb R}^3$, 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., Comment: 67 pages, 29 figures

Bayesian regularization-backpropagation neural network (BR-BPNN) model is employed to predict some aspects of the gecko spatula peeling viz. the variation of the maximum normal and tangential pull-off forces and the resultant force angle at detachment with the peeling angle. K-fold cross validation is used to improve the effectiveness of the model. The input data is taken from finite element (FE) peeling results. The neural network is trained with 75% of the FE dataset. The remaining 25% are utilized to predict the peeling behavior. The training performance is evaluated for every change in the number of hidden layer neurons to determine the optimal network structure. The relative error is calculated to draw a clear comparison between predicted and FE results. It is shown that the BR-BPNN model in conjunction with k-fold technique has significant potential to estimate the peeling behavior., Comment: 18 pages, 9 figures

A topology optimization approach for designing large deformation contact-aided shape morphing compliant mechanisms is presented. Such mechanisms can be used in varying operating conditions. Design domains are described by regular hexagonal elements. Negative circular masks are employed to perform dual task, i.e., to decide material states of each element and also, to generate rigid contact surfaces. Each mask is characterized by five design variables, which are mutated by a zero-order based hill-climbing optimizer. Geometric and material nonlinearities are considered. Continuity in normals to boundaries of the candidate designs is ensured using a boundary resolution and smoothing scheme. Nonlinear mechanical equilibrium equations are solved using the Newton-Raphson method. An updated Lagrange approach in association with segment-to-segment contact method is employed for the contact formulation. Both mutual and self contact modes are permitted. Efficacy of the approach is demonstrated by designing four contact-aided shape morphing compliant mechanisms for different desired curves. Performance of the deformed profiles is verified using a commercial software. The effect of frictional contact surface on the actual profile is also studied, Comment: 19 pages, 12 figures, 5 tables

A continuum-based computational contact model is employed to study coupled adhesion and friction in gecko spatulae. Nonlinear finite element analysis is carried out to simulate spatula peeling from a rigid substrate. It is shown that the "frictional adhesion" behavior, until now only observed from seta to toe levels, is also present at the spatula level. It is shown that for sufficiently small spatula pad thickness, the spatula detaches at a constant angle known as the critical detachment angle irrespective of the peeling and shaft angles. The spatula reaches the same energy states at the jump-off contact point, which directly relates to the invariance of the critical detachment angle. This study also reveals that there is an optimum pad thickness associated with the invariance of the critical detachment angle. It is further observed that the sliding of the spatula pad is essential for the invariance of the critical detachment angle., Comment: 18 pages, 13 figures

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.

This work presents numerical techniques to enforce continuity constraints on multi-patch surfaces for three distinct problem classes. The first involves structural analysis of thin shells that are described by general Kirchhoff-Love kinematics. Their governing equation is a vector-valued, fourth-order, nonlinear, partial differential equation (PDE) that requires at least $C^1$-continuity within a displacement-based finite element formulation. The second class are surface phase separations modeled by a phase field. Their governing equation is the Cahn-Hilliard equation - a scalar, fourth-order, nonlinear PDE - that can be coupled to the thin shell PDE. The third class are brittle fracture processes modeled by a phase field approach. In this work, these are described by a scalar, fourth-order, nonlinear PDE that is similar to the Cahn-Hilliard equation and is also coupled to the thin shell PDE. Using a direct finite element discretization, the two phase field equations also require at least a $C^1$-continuous formulation. Isogeometric surface discretizations - often composed of multiple patches - thus require constraints that enforce the $C^1$-continuity of displacement and phase field. For this, two numerical strategies are presented: For this, two numerical strategies are presented: A Lagrange multiplier formulation and a penalty method. The curvilinear shell model including the geometrical constraints is taken from Duong et al. (2017) and it is extended to model the coupled phase field problems on thin shells of Zimmermann et al. (2019) and Paul et al. (2020) on multi-patches. Their accuracy and convergence are illustrated by several numerical examples considering deforming shells, phase separations on evolving surfaces, and dynamic brittle fracture of thin shells., Comment: In this version, typos in the list of references were fixed, floating pictures in Chapter 6 were reduced, and the sign of the bending moments was changed (see text passages after Eqs. (63) and (71), footnotes 13-14 and 23, and Appendix A

In this paper, the validity and accuracy of three interatomic potentials and the continuum shell model of Ghaffari and Sauer [1] are investigated. The mechanical behavior of single-layered graphene sheets (SLGSs) under uniaxial stretching, biaxial stretching and pure bending is studied for this comparison. The validity of the molecular and continuum models is assessed by direct comparison with density functional theory (DFT) data available in the literature. The molecular simulations are carried out employing the MM3, Tersoff and REBO+LJ potentials. The continuum formulation uses an anisotropic hyperelastic material model in the framework of the geometrically exact Kirchhoff-Love shell theory and isogeometric finite elements. Results from the continuum model are in good agreement with those from DFT. The results from the MM3 potential agree well up to the point of material instability, whereas those from the REBO+LJ and Tersoff potentials agree only for small deformations. Only the Tersoff potential is found to yield auxetic response in SLGSs under uniaxial stretch. Additionally, the transverse vibration frequencies of a pre-stretched graphene sheet and a carbon nanocone are obtained using the continuum model and molecular simulations with the MM3 potential. The variations of the frequencies from these approaches agree within an error of 5%.

Cementless implants are widely used in orthopedic and oral surgery. However, debonding-related failure still occurs at the bone-implant interface. It remains difficult to predict such implant failure since the underlying osseointegration phenomena are still poorly understood. Especially in terms of friction and adhesion at the macro-scale, there is a lack of data and reliable models. The aim of this work is to present a new friction formulation that can model the tangential contact behavior between osseointegrated implants and bone tissue, with focus on debonding. The classical Coulomb's law is combined with a state variable friction law to model a displacement-dependent friction coefficient. A smooth state function, based on the sliding distance, is used to model implant debonding. The formulation is implemented in a 3D nonlinear finite element framework, and it is calibrated with experimental data and compared to an analytical model for mode III cleavage of a coin-shaped, titanium implant (Mathieu et al. 2012). Overall, the results show close agreement with the experimental data, especially the peak and the softening part of the torque curve with a relative error of less than 2.25 %. In addition, better estimates of the bone's shear modulus and the adhesion energy are obtained. The proposed model is particularly suitable to account for partial osseointegration, as is also shown.

We present an adaptive space-time phase field formulation for dynamic fracture of brittle shells. Their deformation is characterized by the Kirchhoff-Love thin shell theory using a curvilinear surface description. All kinematical objects are defined on the shell's mid-plane. The evolution equation for the phase field is determined by the minimization of an energy functional based on Griffith's theory of brittle fracture. Membrane and bending contributions to the fracture process are modeled separately and a thickness integration is established for the latter. The coupled system consists of two nonlinear fourth-order PDEs and all quantities are defined on an evolving two-dimensional manifold. Since the weak form requires $C^1$-continuity, isogeometric shape functions are used. The mesh is adaptively refined based on the phase field using Locally Refinable (LR) NURBS. Time is discretized based on a generalized-$\alpha$ method using adaptive time-stepping, and the discretized coupled system is solved with a monolithic Newton-Raphson scheme. The interaction between surface deformation and crack evolution is demonstrated by several numerical examples showing dynamic crack propagation and branching., Comment: In this version, typos were fixed, Fig. 16 is added, the literature review is extended and clarifying explanations and remarks are added at several places. Supplementary movies are available at https://av.tib.eu/series/641/supplemental+videos+of+the+paper+an+adaptive+space+time+phase+field+formulation+for+dynamic+fracture+of+brittle+shells+based+on+lr+nurbs

A new hyperelastic membrane material model is proposed for single layer blue phosphorus ($\beta\text{-P}$), also known as blue phosphorene. The model is fully nonlinear and captures the anisotropy of $\beta\text{-P}$ at large strains. The material model is calibrated from density functional theory (DFT) calculations considering a set of elementary deformation states. Those are pure dilatation and uniaxial stretching along the armchair and zigzag directions. The material model is compared and validated with additional DFT results and existing DFT results from the literature, and the comparison shows good agreement. The new material model can be directly used within computational shell formulations that are for example based on rotation-free isogeometric finite elements. This is demonstrated by simulations of the indentation and vibration of single layer blue phosphorus sheets at micrometer scales. The elasticity constants at small deformations are also reported.

The present work investigates the normal and tangential peeling behaviour of a gecko spatula using a coupled adhesion-friction model. The objective is to explain the strong attachment and easy detachment behaviour of the spatulae as well as to understand the principles behind their optimum design. Using nonlinear finite element computations, it is shown that during tangentially-constrained peeling the partial sliding of the spatula pad near the peeling front stretches the spatula, thus increasing the strain energy and leading to high pull-off forces. The model is used to investigate the influence of various parameters on the pull-off forces -- such as the peeling angle, spatula shaft angle, strip thickness, and material stiffness. The model shows that increasing the spatula pad thickness beyond a certain level does not lead to a significant increase in the attachment forces. Further, the easy detachment behaviour of geckos is studied under tangentially-free peeling conditions. It is found that the spatulae readily detach from the substrate by changing their shaft angle and eventually peel vertically like a tape. Since the present computational model is not limited by the geometrical, kinematical, and material restrictions of theoretical models, it can be employed to analyse similar biological adhesive systems., Comment: 30 pages, 20 figures

A thermomechanical, polar continuum formulation under finite strains is proposed for anisotropic materials using a multiplicative decomposition of the deformation gradient. First, the kinematics and conservation laws for three dimensional, polar and non-polar continua are obtained. Next, these kinematics are connected to their corresponding counterparts for surface continua based on Kirchhoff-Love kinematics. Likewise, the conservation laws for Kirchhoff-Love shells are derived from their three dimensional counterparts. From this, the weak forms are obtained for three dimensional non-polar continua and Kirchhoff-Love shells. These formulations are expressed in tensorial form so that they can be used in both curvilinear and Cartesian coordinates. They can be used to model anisotropic crystals and soft biological materials, and they can be extended to other field equations, like Maxwell's equations to model thermo-electro-magneto-mechanical materials.

An arbitrary Lagrangian--Eulerian (ALE) finite element method for arbitrarily curved and deforming two-dimensional materials and interfaces is presented here. An ALE theory is developed by endowing the surface with a mesh whose in-plane velocity need not depend on the in-plane material velocity, and can be specified arbitrarily. A finite element implementation of the theory is formulated and applied to curved and deforming surfaces with in-plane incompressible flows. Numerical inf--sup instabilities associated with in-plane incompressibility are removed by locally projecting the surface tension onto a discontinuous space of piecewise linear functions. The general isoparametric finite element method, based on an arbitrary surface parametrization with curvilinear coordinates, is tested and validated against several numerical benchmarks. A new physical insight is obtained by applying the ALE developments to cylindrical fluid films, which are computationally and analytically found to be stable to non-axisymmetric perturbations, and unstable with respect to long-wavelength axisymmetric perturbations when their length exceeds their circumference. A Lagrangian scheme is attained as a special case of the ALE formulation. Though unable to model fluid films with sustained shear flows, the Lagrangian scheme is validated by reproducing the cylindrical instability. However, relative to the ALE results, the Lagrangian simulations are found to have spatially unresolved regions with few nodes, and thus larger errors., Comment: 59 pages, 16 figures

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 concise theoretical and computational framework for the finite element formulation of frictional contact problems with arbitrarily large deformation and sliding. The aim of this work is to extend the contact theory based on surface potentials (Sauer and De Lorenzis, 2013) to account for friction. Coulomb friction under isothermal conditions is considered here. For a consistent friction formulation, we start with the first and second laws of thermodynamics and derive the governing equations at the contact interface. A so-called interacting gap can then be defined as a kinematic variable unifying both sliding/sticking and normal/tangential contact. A variational principle for the frictional system can then be formulated based on a purely kinematical constraint. The direct elimination approach applied to the tangential part of this constraint leads to the so-called moving friction cone approach of Wriggers and Haraldsson (2003). Compared with existing friction formulations, our approach reduces the theoretical and computational complexity. Several numerical examples are presented to demonstrate the accuracy and robustness of the proposed friction formulation., Comment: In this version, minor typos are removed. Eq. (25) in the previous version is removed since it is not needed in the final form of variation and linearization. Results are unchanged. Supplementary movies are available at https://av.tib.eu/series/529/

Surfactants play an important role in various physiological and biomechanical applications. An example is the respiratory system, where pulmonary surfactants facilitate the breathing and reduce the possibility of airway blocking by lowering the surface tension when the lung volume decreases during exhalation. This function is due to the dynamic surface tension of pulmonary surfactants, which depends on the concentration of surfactants spread on the liquid layer lining the interior surface of the airways and alveoli. Here, a finite membrane element formulation for liquids is introduced that allows for the dynamics of concentration-dependent surface tension, as is the particular case for pulmonary surfactants. A straightforward approach is suggested to model the contact line between liquid drops/menisci and planar solid substrates, which allows the presented framework to be easily used for drop shape analysis. It is further shown how line tension can be taken into account. Following an isogeometric approach, NURBS-based finite elements are used for the discretization of the membrane surface. The capabilities of the presented computational model is demonstrated by different numerical examples - such as the simulation of liquid films, constrained and unconstrained sessile drops, pendant drops and liquid bridges - and the results are compared with experimental data., Comment: Some typos are removed. Eqs. 13 and 105 are modified. Eqs. 64 and 73 are added; thus, the rest of equations is renumbered. All the numerical experiments are repeated. The example of Sec. 6.3 is slightly modified

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.

A new hyperelastic material model is proposed for graphene-based structures, such as graphene, carbon nanotubes (CNTs) and carbon nanocones (CNC). The proposed model is based on a set of invariants obtained from the right surface Cauchy-Green strain tensor and a structural tensor. The model is fully nonlinear and can simulate buckling and postbuckling behavior. It is calibrated from existing quantum data. It is implemented within a rotation-free isogeometric shell formulation. The speedup of the model is 1.5 relative to the finite element model of Ghaffari et al. [1], which is based on the logarithmic strain formulation of Kumar and Parks [2]. The material behavior is verified by testing uniaxial tension and pure shear. The performance of the material model is illustrated by several numerical examples. The examples include bending, twisting, and wall contact of CNTs and CNCs. The wall contact is modeled with a coarse grained contact model based on the Lennard-Jones potential. The buckling and post-buckling behavior is captured in the examples. The results are compared with reference results from the literature and there is good agreement.

Topologies of large deformation Contact-aided Compliant Mechanisms (CCMs), with self and mutual contact, exemplified via path generation applications, are designed using the continuum synthesis approach. Design domains are parameterized using honeycomb tessellation. Assignment of material to each cell, and generation of rigid contact surfaces, are accomplished via suitably sizing and positioning negative circular masks. To facilitate contact analysis, boundary smoothing is implemented. Mean value coordinates are employed to compute shape functions, as many regular hexagonal cells get degenerated into irregular, concave polygons as a consequence of boundary smoothing. Both, geometric and material nonlinearities are considered in the finite element analysis. The augmented Lagrange multiplier method in association with an active set strategy is employed to incorporate both self and mutual contact. CCMs are evolved using the stochastic hill climber search. Synthesized contact-aided compliant continua trace paths with single and importantly, multiple kinks and experience multiple contact interactions pertaining to both self and mutual contact modes., Comment: 26 pages, 30 figures

The nonlinear frequencies of pre-stressed graphene-based structures, such as flat graphene sheets and carbon nanotubes, are calculated. These structures are modeled with a nonlinear hyperelastic shell model. The model is calibrated with quantum mechanics data and is valid for high strains. Analytical solutions of the natural frequencies of various plates are obtained for the Canham bending model by assuming infinitesimal strains. These solutions are used for the verification of the numerical results. The performance of the model is illustrated by means of several examples. Modal analysis is performed for square plates under pure dilatation or uniaxial stretch, circular plates under pure dilatation or under the effects of an adhesive substrate, and carbon nanotubes under uniaxial compression or stretch. The adhesive substrate is modeled with van der Waals interaction (based on the Lennard-Jones potential) and a coarse grained contact model. It is shown that the analytical natural frequencies underestimate the real ones, and this should be considered in the design of devices based on graphene structures.

This paper presents a new isogeometric mortar contact formulation based on an extended finite element interpolation to capture physical pressure discontinuities at the contact boundary. The so called two-half-pass algorithm is employed, which leads to an unbiased formulation and, when applied to the mortar setting, has the additional advantage that the mortar coupling term is no longer present in the contact forces. As a result, the computationally expensive segmentation at overlapping master-slave element boundaries, usually required in mortar methods (although often simplified with loss of accuracy), is not needed from the outset. For the numerical integration of general contact problems, the so-called refined boundary quadrature is employed, which is based on adaptive partitioning of contact elements along the contact boundary. The contact patch test shows that the proposed formulation passes the test without using either segmentation or refined boundary quadrature. Several numerical examples are presented to demonstrate the robustness and accuracy of the proposed formulation., Comment: In this version, we have removed the patch test comparison with the classical mortar method and removed corresponding statements. They will be studied in further detail in future work, so that the focus is now entirely on the new IGA mortar formulation

Isogeometrically enriched finite elements offer efficient localized isogeometric analysis (IGA) enrichment for numerical simulations involving large computational domains. This is achieved by employing surface enriched elements to interface isogeometric elements with classical Langrangian finite elements. In this paper, we explore their applicability and merits for fluid-structure interaction (FSI) analysis. The implemented approach not only offers an enrichment of the finite element space, but also offers a framework for discretizing and analyzing fluid and structure with different finite element approaches, namely, classical Lagrange finite elements and IGA. In this context, a monolithic solution approach with an explicit grid update mechanism is implemented for FSI. The applicability and the impact of the isogeometric enrichment approach on the accuracy of the numerical solution is assessed by comparing the obtained results with existing reference solutions of FSI benchmark examples involving two- and three-dimensional incompressible fluid flow past hyper-elastic solids., Comment: FSI benchmarking, fluid-structure interaction, isogeometric analysis, isogeometric enrichment, monolithic solver, nonlinear finite elements

A unified fluid-structure interaction (FSI) formulation is presented for solid, liquid and mixed membranes. Nonlinear finite elements (FE) and the generalized-alpha 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., Comment: 38 pages, 17 figures

This paper presents a general theory and isogeometric finite element implementation for studying mass conserving phase transitions on deforming surfaces. The mathematical problem is governed by two coupled fourth-order nonlinear partial differential equations (PDEs) that live on an evolving two-dimensional manifold. For the phase transitions, the PDE is the Cahn-Hilliard equation for curved surfaces, which can be derived from surface mass balance in the framework of irreversible thermodynamics. For the surface deformation, the PDE is the (vector-valued) Kirchhoff-Love thin shell equation. Both PDEs can be efficiently discretized using $C^1$-continuous interpolations without derivative degrees-of-freedom (dofs). Structured NURBS and unstructured spline spaces with pointwise $C^1$-continuity are utilized for these interpolations. The resulting finite element formulation is discretized in time by the generalized-$\alpha$ scheme with adaptive time-stepping, and it is fully linearized within a monolithic Newton-Raphson approach. A curvilinear surface parameterization is used throughout the formulation to admit general surface shapes and deformations. The behavior of the coupled system is illustrated by several numerical examples exhibiting phase transitions on deforming spheres, tori and double-tori., Comment: fixed typos, extended literature review, added clarifying notes to the text, added supplementary movie files