Phase-field boundary conditions for the voxel finite cell method: surface-free stress analysis of CT-based bone structures

The voxel finite cell method employs unfitted finite element meshes and voxel quadrature rules to seamlessly transfer CT data into patient-specific bone discretizations. The method, however, still requires the explicit parametrization of boundary surfaces to impose traction and displacement boundary conditions, which constitutes a potential roadblock to automation. We explore a phase-field based formulation for imposing traction and displacement constraints in a diffuse sense. Its essential component is a diffuse geometry model generated from metastable phase-field solutions of the Allen-Cahn problem that assumes the imaging data as initial condition. Phase-field approximations of the boundary and its gradient are then employed to transfer all boundary terms in the variational formulation into volumetric terms. We show that in the context of the voxel finite cell method, diffuse boundary conditions achieve the same accuracy as boundary conditions defined over explicit sharp surfaces, if the inherent length scales, i.e., the interface width of the phase-field, the voxel spacing and the mesh size, are properly related. We demonstrate the flexibility of the new method by analyzing stresses in a human femur and a vertebral body

Computational performance of Free Mesh Method applied to continuum mechanics problems

The free mesh method (FMM) is a kind of the meshless methods intended for particle-like finite element analysis of problems that are difficult to handle using global mesh generation, or a node-based finite element method that employs a local mesh generation technique and a node-by-node algorithm. The aim of the present paper is to review some unique numerical solutions of fluid and solid mechanics by employing FMM as well as the Enriched Free Mesh Method (EFMM), which is a new version of FMM, including compressible flow and sounding mechanism in air-reed instruments as applications to fluid mechanics, and automatic remeshing for slow crack growth, dynamic behavior of solid as well as large-scale Eigen-frequency of engine block as applications to solid mechanics

Continuum elastic modeling of graphene resonators

Starting from an atomistic approach we have derived a hierarchy of successively more simplified continuum elasticity descriptions for modeling the mechanical properties of suspended graphene sheets. The descriptions are validated by applying them to square graphene-based resonators with clamped edges and studying numerically their mechanical responses. Both static and dynamic responses are treated. We find that already for deflections of the order of 0.5{\AA} a theory that correctly accounts for nonlinearities is necessary and that for many purposes a set of coupled Duffing-type equations may be used to accurately describe the dynamics of graphene membranes.Comment: 7 pages, 5 figure

Linear viscoelasticity - bone volume fraction relationships of bovine trabecular bone

Trabecular bone has been previously recognized as time-dependent (viscoelastic) material, but the relationships of its viscoelastic behaviour with bone volume fraction (BV/TV) have not been investigated so far. Therefore, the aim of the present study was to quantify the time-dependent viscoelastic behaviour of trabecular bone and relate it to BV/TV. Uniaxial compressive creep experiments were performed on cylindrical bovine trabecular bone samples ([Formula: see text] ) at loads corresponding to physiological strain level of 2000 [Formula: see text] . We assumed that the bone behaves in a linear viscoelastic manner at this low strain level and the corresponding linear viscoelastic parameters were estimated by fitting a generalized Kelvin–Voigt rheological model to the experimental creep strain response. Strong and significant power law relationships ([Formula: see text] ) were found between time-dependent creep compliance function and BV/TV of the bone. These BV/TV-based material properties can be used in finite element models involving trabecular bone to predict time-dependent response. For users’ convenience, the creep compliance functions were also converted to relaxation functions by using numerical interconversion methods and similar power law relationships were reported between time-dependent relaxation modulus function and BV/TV

Characterisation of time-dependent mechanical behaviour of trabecular bone and its constituents

Trabecular bone is a porous composite material which consists of a mineral phase (mainly hydroxyapatite), organic phase (mostly type I collagen) and water assembled into a complex, hierarchical structure. In biomechanical modelling, its mechanical response to loads is generally assumed to be instantaneous, i.e. it is treated as a time-independent material. It is, however, recognised that the response of trabecular bone to loads is time-dependent. Study of this time-dependent behaviour is important in several contexts such as: to understand energy dissipation ability of bone; to understand the age-related non-traumatic fractures; to predict implant loosening due to cyclic loading; to understand progressive vertebral deformity; and for pre-clinical evaluation of total joint replacement. To investigate time-dependent behaviour, bovine trabecular bone samples were subjected to compressive loading, creep, unloading and recovery at multiple load levels (corresponding to apparent strain of 2,000-25,000 με). The results show that: the time-dependent behaviour of trabecular bone comprises of both recoverable and irrecoverable strains; the strain response is nonlinearly related to applied load levels; and the response is associated with bone volume fraction. It was found that bone with low porosity demonstrates elastic stiffening followed by elastic softening, while elastic softening is demonstrated by porous bone at relatively low loads. Linear, nonlinear viscoelastic and nonlinear viscoelastic-viscoplastic constitutive models were developed to predict trabecular bone’s time-dependent behaviour. Nonlinear viscoelastic constitutive model was found to predict the recovery behaviour well, while nonlinear viscoelastic-viscoplastic model predicts the full creep-recovery behaviour reasonably well. Depending on the requirements all these models can be used to incorporate time-dependent behaviour in finite element models. To evaluate the contribution of the key constituents of trabecular bone and its microstructure, tests were conducted on demineralised and deproteinised samples. Reversed cyclic loading experiments (tension to compression) were conducted on demineralised trabecular bone samples. It was found that demineralised bone exhibits asymmetric mechanical response - elastic stiffening in tension and softening in compression. This tension to compression transition was found to be smooth. Tensile multiple-load-creep-unload-recovery experiments on demineralised trabecular samples show irrecoverable strain (or residual strain) even at the low stress levels. Demineralised trabecular bone samples demonstrate elastic stiffening with increasing load levels in tension, and their time-dependent behaviour is nonlinear with respect to applied loads . Nonlinear viscoelastic constitutive model was developed which can predict its recovery behaviour well. Experiments on deproteinised samples showed that their modulus and strength are reasonably well related to bone volume fraction. The study considers an application of time-dependent behaviour of trabecular bone. Time-dependent properties are assigned to trabecular bone in a bone-screw system, in which the screw is subjected to cyclic loading. It is found that separation between bone and the screw at the interface can increase with increasing number of cycles which can accentuate loosening. The relative larger deformation occurs when this system to be loaded at the higher loading frequency. The deformation at the bone-screw interface is related to trabecular bone’s bone volume fraction; screws in a more porous bone are at a higher risk of loosening

Failure initiation at V-notch tips in quasi-brittle materials

International audienceAt V-notched tips in specimens made of quasi-brittle materials a small damaged or plastic zone is evident that cannot not be neglected in terms of dissipated energy and stress state, although it is small. Herein, to predict the failure initiation at the notch tip, we extend the finite fracture mechanics (FFM) coupled criterion, which requires a simultaneously fulfillment of an energy and a stress criteria. In the small damaged zone, a damage model is introduced so to decrease the effective Young's modulus in a power law in terms of the distance to the notch tip in such a way that the stress field remains bounded. It seems particularly suited to quasi-brittle materials, since no diffuse damage can occur. This damage zone is coupled to the FFM criterion to provide the necessary condition for failure initiation. Under the assumption that the damaged zone and the virtual crack extension are small, matched asymptotic expansions are used. It is shown that the damaged zone grows first, proportionally to the square of the applied load and then, above a threshold, a virtual crack of a given length simultaneously satisfies the energy and stress criteria, and failure occurs. The approach allows taking into account varying tensile strength and material toughness in the damaged zone, as may reasonably be expected. Moreover, it is shown that the same coupled stress-energy criterion can directly be applied to quasi-brittle materials by appropriately using the actual material toughness as measured on a cracked specimen

A Parallel Spectral Element Method For Dynamic Three-Dimensional Nonlinear Elasticity Problems

We present a high-order method employing Jacobi polynomial-based shape functions, as an alternative to the typical Legendre polynomial-based shape functions in solid mechanics, for solving dynamic three-dimensional geometrically nonlinear elasticity problems. We demonstrate that the method has an exponential convergence rate spatially and a second-order accuracy temporally for the four classes of problems of linear/geometrically nonlinear elastostatics/elastodynamics. The method is parallelized through domain decomposition and message passing interface (MPI), and is scaled to over 2000 processors with high parallel performance