This paper presents three different constitutive approaches to model thin rotation-free shells based on the Kirchhoff–Love hypothesis. One approach is based on numerical integration through the shell ...thickness while the other two approaches do not need any numerical integration and so they are computationally more efficient. The formulation is designed for large deformations and allows for geometrical and material nonlinearities, which makes it very suitable for the modeling of soft tissues. Furthermore, six different isotropic and anisotropic material models, which are commonly used to model soft biological materials, are examined for the three proposed constitutive approaches. Following an isogeometric approach, NURBS-based finite elements are used for the discretization of the shell surface. Several numerical examples are investigated to demonstrate the capabilities of the formulation. Those include the contact simulation during balloon angioplasty.
This paper presents a new finite element (FE) formulation for liquid shells that is based on an explicit, 3D surface discretization using C1-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.
An existing hyperelastic membrane model for graphene calibrated from ab-initio data (Kumar and Parks, 2014) is adapted to curvilinear coordinates and extended to a rotation-free shell formulation ...based on isogeometric finite elements. Therefore, the membrane model is extended by a hyperelastic bending model that reflects the ab-inito data of Kudin et al. (2001). The proposed formulation can be implemented straight-forwardly into an existing finite element package, since it does not require the description of molecular interactions. It thus circumvents the use of interatomic potentials that tend to be less accurate than ab-initio data. The proposed shell formulation is verified and analyzed by a set of simple test cases. The results are in agreement to analytical solutions and satisfy the FE patch test. The performance of the shell formulation for graphene structures is illustrated by several numerical examples. The considered examples are indentation and peeling of graphene and torsion, bending and axial stretch of carbon nanotubes. Adhesive substrates are modeled by the Lennard–Jones potential and a coarse grained contact model. In principle, the proposed formulation can be extended to other 2D materials.
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.
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.
•It is simpler to implement and thus 1.5 faster than the model of Ghaffari et al. 1.•It is fully nonlinear and can capture buckling and post-buckling behavior.•It is suitable to simulate and study carbon nanocones under large deformations.•It is applied to simulate contact of CNTs and CNCs with a Lennard-Jones wall.•The latter example demonstrates that CNCs are ideal candidates for AFM tips.
This work presents a new frictional sliding algorithm for liquid menisci in contact with solid substrates. In contrast to solid–solid contact, the liquid–solid contact behavior is governed by the ...contact line, where a contact angle forms and undergoes hysteresis. The new algorithm admits arbitrary meniscus shapes and arbitrary substrate roughness, heterogeneity and compliance. It is discussed and analyzed in the context of droplet contact, but it also applies to liquid films and solids with surface tension. The droplet is modeled as a stabilized membrane enclosing an incompressible medium. The contact formulation is considered rate-independent such that hydrostatic conditions apply. Three distinct contact algorithms are needed to describe the cases of frictionless surface contact, frictionless line contact and frictional line contact. For the latter, a predictor–corrector algorithm is proposed in order to enforce the contact conditions at the contact line and thus distinguish between the cases of advancing, pinning and receding. The algorithms are discretized within a monolithic finite element formulation. Several numerical examples are presented to illustrate the numerical and physical behavior of sliding droplets.
NURBS-enriched contact finite elements Corbett, Callum J.; Sauer, Roger A.
Computer methods in applied mechanics and engineering,
06/2014, Letnik:
275
Journal Article
Recenzirano
A novel enrichment of finite elements for contact computations based on isogeometric analysis is presented. Each body is divided into two parts, an enriched contact surface and the bulk domain ...together with surfaces that are not in contact. The latter part comprises the large majority of the domain and is treated in the usual manner with standard linear basis function, preserving the efficiency of classical finite element techniques. The enriched contact surface is discretized using NURBS basis functions of at least second order, allowing for a locally differentiable surface representation. This avoids the problem of suddenly changing normal vectors between element boundaries on the contact surface. Following the concept of isogeometric analysis, the smooth basis functions are not only used to describe the surface geometry, but also to approximate the solution on the surface. This leads to higher accuracy in the contact integral evaluation.
Numerical results are presented for 2D and 3D contact computations including frictionless sliding, adhesive peeling, and cohesive debonding. The presented contact element enrichment exhibits a major gain in numerical accuracy and stability without loss of efficiency compared to standard linear finite elements. The enrichment technique offers some advantages over Hermite and higher-order Lagrangian contact element enrichment techniques, such as locally differentiable surface representations in 3D, while featuring competitive accuracy and performance.
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