Fusing the Seth-Hill strain tensors to fit compressible elastic material responses in the nonlinear regime

Strain energy densities based on the Seth-Hill strain tensors are often used to describe the hyperelastic mechanical behaviours of isotropic, transversely isotropic and orthotropic materials for relatively large deformations. Since one parameter distinguishes which strain tensor of the Seth-Hill family is used, one has in theory the possibility to t the material response in the nonlinear regime. Most often for compressible deformations however, this parameter is selected such that the Hencky strain tensor is recovered, because it yields rather physical stress-strain responses. Hence, the response in the nonlinear regime is in practise not often tailored to match experimental data. To ensure that elastic responses in the nonlinear regime can more accurately be controlled, this contribution proposes three generalisations that combine several Seth-Hill strain tensors. The generalisations are formulated such that the stress-strain responses for in finitesimal deformations remain unchanged. Consequently, the identifi cation of the Young's moduli, Poisson's ratios and shear moduli is not a ffected. 3D fi nite element simulations are performed for isotropy and orthotropy, with an emphasis on the identifi cation of the new material parameters

A Variational Formulation of Dissipative Quasicontinuum Methods

Lattice systems and discrete networks with dissipative interactions are successfully employed as meso-scale models of heterogeneous solids. As the application scale generally is much larger than that of the discrete links, physically relevant simulations are computationally expensive. The QuasiContinuum (QC) method is a multiscale approach that reduces the computational cost of direct numerical simulations by fully resolving complex phenomena only in regions of interest while coarsening elsewhere. In previous work (Beex et al., J. Mech. Phys. Solids 64, 154-169, 2014), the originally conservative QC methodology was generalized to a virtual-power-based QC approach that includes local dissipative mechanisms. In this contribution, the virtual-power-based QC method is reformulated from a variational point of view, by employing the energy-based variational framework for rate-independent processes (Mielke and Roub\'i\v{c}ek, Rate-Independent Systems: Theory and Application, Springer-Verlag, 2015). By construction it is shown that the QC method with dissipative interactions can be expressed as a minimization problem of a properly built energy potential, providing solutions equivalent to those of the virtual-power-based QC formulation. The theoretical considerations are demonstrated on three simple examples. For them we verify energy consistency, quantify relative errors in energies, and discuss errors in internal variables obtained for different meshes and two summation rules.Comment: 38 pages, 21 figures, 4 tables; moderate revision after review, one example in Section 5.3 adde

Fatigue phase-field damage modeling of rubber using viscous dissipation: Crack nucleation and propagation

By regularizing sharp cracks within a pure continuum setting, phase-damage models offer the ability to capture crack nucleation as well as crack propagation. Crack branching and coalescence can furthermore be described without any additional efforts, as geometrical descriptions of the cracks are not required. In this contribution, we extend our previous phase-field model for rate-dependent fracture of rubbers in a finite strain setting (Loew et al., 2019) to describe damage under cyclic loading. The model is derived from the balance of mechanical energy and introduces a fatigue damage source as a function of the accumulated viscous dissipation under cyclic loading. We use uniaxial cyclic tension to present the influence of the fatigue material parameters and to confirm the model’s energy balance. The parameters are subsequently identified using monotonic and cyclic experiments of a plane stress nature. Finally, the model is validated by separate experiments, which demonstrate that the model accurately predicts (fatigue) crack nucleation as well as propagation

Non-localised contact between beams with circular and elliptical cross-sections

The key novelty of this contribution is a dedicated technique to e fficiently determine the distance (gap) function between parallel or almost parallel beams with circular and elliptical cross-sections. The technique consists of parametrizing the surfaces of the two beams in contact, fixing a point on the centroid line of one of the beams and searching for a constrained minimum distance between the surfaces (two variants are investigated). The resulting unilateral (frictionless) contact condition is then enforced with the Penalty method, which introduces compliance to the, otherwise rigid, beams' cross-sections. Two contact integration schemes are considered: the conventional slave-master approach (which is biased as the contact virtual work is only integrated over the slave surface) and the so-called two-half-pass approach (which is unbiased as the contact virtual work is integrated over the two contacting surfaces). Details of the finite element formulation which is suitably implemented using Automatic Di fferentiation techniques are presented. A set of numerical experiments shows the overall performance of the framework and allows a quantitative comparison of the investigated variants

Estimating fibres' material parameter distributions from limited data with the help of Bayesian inference

Numerous materials are essentially structures of discrete fibres, yarns or struts. Considering these materials at their discrete scale, one may distinguish two types of intrinsic randomness that affect the structural behaviours of these discrete structures: geometrical randomness and material randomness. Identifying the material randomness is an experimentally demanding task, because many small fibres, yarns or struts need to be tested, which are not easy to handle. To avoid the testing of hundreds of constituents, this contribution proposes an identification approach that only requires a few dozen of constituents to be tested (we use twenty to be exact). The identification approach is applied to articially generated measurements, so that the identified values can be compared to the true values. Another question this contribution aims to answer is how precise the material randomness needs to be identified, if the geometrical randomness will also influence the macroscale behaviour of these discrete networks. We therefore also study the effect of the identified material randomness to that of the actual material randomness for three types of structures; each with an increasing level of geometrical randomness

Discrete mechanical models and upscaling techniques for discrete materials

Numerous natural and man-made materials are essentially discrete structures at the mesoscale or microscale (see Fig. 1). Discrete mechanical models can be formulated to capture typical mechanical phenomena arising from this discreteness. Failure in these materials, which often starts with the fracture of an individual bond, can be predicted based on the small-scale mechanics with these models. For failure, but also for non-local mechanics, no phenomenological descriptions are required in these models. This makes them more predictive than constitutive material models for this type of materials

Beam-inside-beam contact: Mechanical simulations of slender medical instruments inside the human body

Background and Objective This contribution presents a rapid computational framework to mechanically simulate the insertion of a slender medical instrument in a tubular structure such as an artery, the cochlea or another slender instrument. Methods Beams are employed to rapidly simulate the mechanical behaviour of the medical instrument and the tubular structure. However, the framework’s novelty is its capability to handle the mechanical contact between an inner beam (representing the medical instrument) embedded in a hollow outer beam (representing the tubular structure). This “beam-inside-beam” contact framework, which forces two beams to remain embedded, is the first of its kind since existing contact frameworks for beams are “beam-to-beam” approaches, i.e. they repel beams from each other. Furthermore, we propose contact kinematics such that not only instruments and tubes with circular cross-sections can be considered, but also those with elliptical cross-sections. This provides flexibility for the optimization of patient-specific instruments. Results The results demonstrate that the framework’s robustness is substantial, because only a few increments per simulation and a few iterations per increment are required, even though large deformations, large rotations and large curvature changes of both the instrument and tubular structure occur. The stability of the framework remains high even if the modulus of the inner tube is thousand times larger than that of the outer tube. A mesh convergence study furthermore exposes that a relatively small number of elements is required to accurately approach the reference solution. Conclusions The framework’s high simulation speed originates from the exploitation of the rigidity of the beams’ cross-sections to quantify the exclusion between the inner and the hollow outer beam. This rigidity limits the accuracy of the framework at the same time, but this is unavoidable since simulation accuracy and simulation speed are two competing interests. Hence, the framework is particularly attractive if simulation speed is preferred over accuracy