In this work we present a novel computational method for embedding arbitrary curved one-dimensional (1D) fibers into three-dimensional (3D) solid volumes, as e.g. in fiber-reinforced materials. The fibers are explicitly modeled with highly efficient 1D geometrically exact beam finite elements, based on various types of geometrically nonlinear beam theories. The surrounding solid volume is modeled with 3D continuum (solid) elements. An embedded mortar-type approach is employed to enforce the kinematic coupling constraints between the beam elements and solid elements on non-matching meshes. This allows for very flexible mesh generation and simple material modeling procedures in the solid, since it can be discretized without having to account for the reinforcements, while still being able to capture complex nonlinear effects due to the embedded fibers. Several numerical examples demonstrate the consistency, robustness and accuracy of the proposed method, as well as its applicability to rather complex fiber-reinforced structures of practical relevance.

Summary This contribution proposes the first three‐dimensional (3D) beam‐beam interaction model for molecular interactions between curved slender fibers undergoing large deformations. While the general model is not restricted to a specific beam formulation, in the present work, it is combined with the geometrically exact beam theory and discretized via the finite element method. A direct evaluation of the total interaction potential for general 3D bodies requires the integration of contributions from molecule or charge distributions over the volumes of the interaction partners, leading to a six‐dimensional integral (two nested 3D integrals) that has to be solved numerically. Here, we propose a novel strategy to formulate reduced section‐section interaction laws for the resultant interaction potential between a pair of cross‐sections of two slender fibers such that only two one‐dimensional integrals along the fibers' length directions have to be solved numerically. This section‐section interaction potential (SSIP) approach yields a significant gain in efficiency, which is essential to enable the simulation of relevant time and length scales for many practical applications. In a first step, the generic structure of SSIP laws, which is suitable for the most general interaction scenario (eg, fibers with arbitrary cross‐section shape and inhomogeneous atomic/charge density within the cross‐section) is presented. Assuming circular, homogeneous cross‐sections, in a next step, specific analytical expressions for SSIP laws describing short‐range volume interactions (eg, van der Waals (vdW) or steric interactions) and long‐range surface interactions (eg, Coulomb interactions) are proposed. Besides ready‐to‐use expressions for the total interaction potential, also the resulting virtual work contributions, its finite element discretizations, as well as a suitable numerical regularization for the limit of zero separation are derived. The validity of the SSIP laws, as well as the accuracy and robustness of the general SSIP approach to beam‐beam interactions, is thoroughly verified by means of a set of numerical examples considering steric repulsion, electrostatic, or vdW adhesion.

This work focuses on finite element formulations for the accurate modeling and efficient simulation of the implicit dynamics of slender fiber-or rod-like components and their contact interaction when being embedded in complex systems of fiber-based materials and structures. Recently, the authors have proposed a novel all-angle beam contact (ABC) formulation that combines the advantages of existing point and line contact models in a variationally consistent manner. However, the ABC formulation has so far only been applied in combination with a special torsion-free beam model, which yields a very simple and efficient finite element formulation, but which is restricted to initially straight beams with isotropic cross-sections. In order to abstain from these restrictions, the current work combines the ABC formulation with a geometrically exact Kirchhoff-Love beam element formulation that is capable of treating even the most general cases of slender beam problems in terms of initial geometry and external loads. While the neglect of shear deformation that is inherent to this formulation has been shown to provide considerable numerical advantages in the range of high beam slenderness ratios, alternative shear-deformable beam models are required for examples with thick beams. For that reason, the current contribution additionally proposes a novel geometrically exact beam element based on the Simo-Reissner theory. Similar to the torsion-free and the Kirchhoff-Love beam elements, also this Simo-Reissner element is based on a C 1 -continuous Hermite interpolation of the beam centerline, which will allow for smooth contact kinematics. For this Hermitian Simo-Reissner element, a consistent spatial convergence behavior as well as the successful avoidance of membrane and shear locking will be demonstrated numerically. All in all, the combination of the ABC formulation with these different beam element variants (i.e. the torsion-free element, the Kirchhoff-Love element and the Simo-Reissner element) results in a very flexible and modular simulation framework that allows to choose the optimal element formulation for any given application in terms of accuracy, efficiency and robustness. Based on several practically relevant examples, the different variants are compared numerically, and, eventually, a general recommendation concerning the optimal choice of beam elements is made.

Living soft tissues appear to promote the development and maintenance of a preferred mechanical state within a defined tolerance around a so-called set point. This phenomenon is often referred to as mechanical homeostasis. In contradiction to the prominent role of mechanical homeostasis in various (patho)physiological processes, its underlying micromechanical mechanisms acting on the level of individual cells and fibers remain poorly understood, especially how these mechanisms on the microscale lead to what we macroscopically call mechanical homeostasis. Here, we present a novel computational framework based on the finite element method that is constructed bottom up, that is, it models key mechanobiological mechanisms such as actin cytoskeleton contraction and molecular clutch behavior of individual cells interacting with a reconstructed three-dimensional extracellular fiber matrix. The framework reproduces many experimental observations regarding mechanical homeostasis on short time scales (hours), in which the deposition and degradation of extracellular matrix can largely be neglected. This model can serve as a systematic tool for future in silico studies of the origin of the numerous still unexplained experimental observations about mechanical homeostasis.

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