Christian Hellmich

Ductile sliding between mineral crystals followed by rupture of collagen crosslinks: experimentally supported micromechanical explanation of bone strength.

There is an ongoing discussion on how bone strength could be explained from its internal structure and composition. Reviewing recent experimental and molecular dynamics studies, we here propose a new vision on bone material failure: mutual ductile sliding of hydroxyapatite mineral crystals along layered water films is followed by rupture of collagen crosslinks. In order to cast this vision into a mathematical form, a multiscale continuum micromechanics theory for upscaling of elastoplastic properties is developed, based on the concept of concentration and influence tensors for eigenstressed microheterogeneous materials. The model reflects bones hierarchical organization, in terms of representative volume elements for cortical bone, for extravascular and extracellular bone material, for mineralized fibrils and the extrafibrillar space, and for wet collagen. In order to get access to the stress states at the interfaces between crystals, the extrafibrillar mineral is resolved into an infinite amount of cylindrical material phases oriented in all directions in space. The multiscale micromechanics model is shown to be able to satisfactorily predict the strength characteristics of different bones from different species, on the basis of their mineral/collagen content, their intercrystalline, intermolecular, lacunar, and vascular porosities, and the elastic and strength properties of hydroxyapatite and (molecular) collagen.

Micromechanics-Based Conversion of CT Data into Anisotropic Elasticity Tensors, Applied to FE Simulations of a Mandible

Computer Tomographic (CT) image data have become a standard basis for structural analyses of bony organs. In this context, regression functions between stiffness components and Hounsfields units (HU) from CT, related to X-ray attenuation coefficients, are widely used for the definition of the (actually inhomogeneous and anisotropic) material behavior inside the organ. Herein, we suggest to derive the functional dependence of the fully orthotropic stiffness tensors on the Hounsfield units from the physical information contained in the X-ray attenuation coefficients: (i) Based on voxel average rules for the X-ray attenuation coefficients, we assign to each voxel the volume fraction occupied by water (marrow) and that occupied by solid bone matrix. (ii) By means of a continuum micromechanics representation for bone, which is based on voxel-invariant (species and whole bone-specific) stiffness properties of solid bone matrix and of water, we convert the aforementioned volume fractions into voxel-specific orthotropic stiffness tensor components. The micromechanics model, in combination with the average rule for X-ray attenuation coefficients, predicts a quasi-linear relationship between axial Young’s modulus and HU, and highly nonlinear relationships for both circumferential and radial Young’s moduli as well as for the shear moduli in all principal material directions. Corresponding whole-organ Finite Element (FE) analyses of a partially edentulous human mandible characterized by atrophy of the alveolar ridge show that volumetric strain concentrations/peaks within the organ are decreased when considering material anisotropy, and increased when considering material inhomogeneity.

Micromechanical modeling of solid-type and plate-type deformation patterns within softwood materials. A review and an improved approach

Abstract Wood exhibits a highly diverse microstructure. It appears as a solid-type composite material at a length scale of some micrometers, while it resembles an assembly of plate-like elements arranged in a honeycomb fashion at the length scale of some hundreds of micrometers. These structural features result in different load-carrying mechanisms at different observation scales and under different loading conditions. In this paper, we elucidate the main load-carrying mechanisms by means of a micromechanical model for softwood materials. Representing remarkable progress with respect to earlier models reported in the literature, this model is valid across various species. The model is based on tissue-independent stiffness properties of cellulose, lignin, hemicellulose, and water obtained from direct testing and lattice-dynamics analyses. Sample-specific characteristics are considered in terms of porosity and the contents of cellulose, lignin, hemicelluloses and water, which are obtained from mass density measurements, environmental scanning micrographs, analytical chemistry, and NMR spectroscopy. The model comprises three homogenization steps, two based on continuum micromechanics and one on the unit cell method. The latter represents plate-like bending and shear of the cell walls due to transverse shear loading and axial straining in the tangential stem direction. Accurate representation of these deformation modes results in accurate (orthotropic) stiffness estimates across a variety of softwood species. These stiffness predictions deviate, on average, by less than 10% from corresponding experimental results obtained from ultrasonic or quasi-static testing. Thus, the proposed model can reliably predict microscopic and macroscopic mechanical properties from internal structure and composition, and is therefore expected to significantly support wood production technology (such as drying techniques) and mechanical analyses of timber structures.

Characterization of the deformation behavior of intermediate porosity interconnected Ti foams using micro-computed tomography and direct finite element modeling.

Under load-bearing conditions metal-based foam scaffolds are currently the preferred choice as bone/cartilage implants. In this study X-ray micro-computed tomography was used to discretize the three-dimensional structure of a commercial titanium foam used in spinal fusion devices. Direct finite element modeling, continuum micromechanics and analytical models of the foam were employed to characterize the elasto-plastic deformation behavior. These results were validated against experimental measurements, including ultrasound and monotonic and interrupted compression testing. Interrupted compression tests demonstrated localized collapse of pores unfavorably oriented with respect to the loading direction at many isolated locations, unlike the Ashby model, in which pores collapse row by row. A principal component analysis technique was developed to quantify the pore anisotropy which was then related to the yield stress anisotropy, indicating which isolated pores will collapse first. The Gibson-Ashby model was extended to incorporate this anisotropy by considering an orthorhombic, rather than a tetragonal, unit cell. It is worth noting that the natural bone is highly anisotropic and there is a need to develop and characterize anisotropic implants that mimic bone characteristics.

Micromechanics of bone tissue-engineering scaffolds, based on resolution error-cleared computer tomography.

Synchrotron radiation micro-computed tomography (SRmuCT) revealed the microstructure of a CEL2 glass-ceramic scaffold with macropores of several hundred microns characteristic length, in terms of the voxel-by-voxel 3D distribution of the attenuation coefficients throughout the scanned space. The probability density function of all attenuation coefficients related to the macroporous space inside the scaffold gives access to the tomograph-specific machine error included in the SRmuCT measurements (also referred to as instrumental resolution function). After Lorentz function-based clearing of the measured CT data from the systematic resolution error, the voxel-specific attenuation information of the voxels representing the solid skeleton is translated into the composition of the material inside one voxel, in terms of the nanoporosity embedded in a dense CEL2 glass-ceramic matrix. Based on voxel-invariant elastic properties of dense CEL2 glass-ceramic, continuum micromechanics allows for translation of the voxel-specific nanoporosity into voxel-specific elastic properties. They serve as input for Finite Element analyses of the scaffold structure. Youngs modulus of a specific CT-scanned macroporous scaffold sample, predicted from a Finite Element simulation of a uniaxial compression test, agrees well with the experimental value obtained from an ultrasonic test on the same sample. This highlights the satisfactory predictive capabilities of the presented approach.

Hybrid method for quantification of stress states in shotcrete tunnel shells: combination of 3D in situ displacement measurements and thermochemoplastic material law

Abstract A hybrid method for quantification of the loading of shotcrete tunnel shells, combining thermochemomechanical material modeling of shotcrete with 3D in situ displacement measurements in the framework of the non-linear finite element method, is presented. Histories of displacement fields, determined from in situ measurements by suitable interpolation functions, are prescribed on the outer boundary of a part of the tunnel, discretized in 2D or 3D. The main goal is the determination of fields of safety degrees, amounting to 0% for the unloaded shell and to 100% for the material loaded up (locally) to the compressive strength. A comprehensive investigation of the significance of the individual material characteristics of shotcrete as well as that of the third dimension in space on the structural behavior is performed. The Sieberg tunnel in Lower Austria, constituting a part of the high capacity railway line between Vienna and Salzburg, is used as the vehicle for this investigation. This tunnel was very recently completed.

Micromechanical Explanation of Elasticity and Strength of Gypsum: From Elongated Anisotropic Crystals to Isotropic Porous Polycrystals

Gypsum is made up of interlocked and elongated crystals. The random nature of its morphology suggests to resort to homogenization of random media to investigate its mechanical properties from the scale of the single crystals upwards. Unfortunately, the usual homogenization schemes fail to quantitatively predict the influence of the porosity on the effective Youngs modulus of gypsum. This is clearly due to the inability of such approaches to take into account the elongated nature of the crystals. A modification of the classical self-consistent scheme is proposed. It is validated against elastic characteristics computed by finite element analyses, and also against experiments on real dried gypsum samples (with empty pores). Finally, a strength model based on brittle failure is presented. The whole strength domain in the space of macroscopic principal stresses is derived. The comparison to experimental data in both simple tension and simple compression is remarkably good.

Intravoxel bone micromechanics for microCT-based finite element simulations

While micro-FE simulations have become a standard tool in computational biomechanics, the choice of appropriate material properties is still a relevant topic, typically involving empirical grey value-to-elastic modulus relations. We here derive the voxel-specific volume fractions of mineral, collagen, and water, from tissue-independent bilinear relations between mineral and collagen content in extracellular bone tissue (J. Theor. Biol. 287: 115, 2011), and from the measured X-ray attenuation information quantified in terms of grey values. The aforementioned volume fractions enter a micromechanics representation of bone tissue, as to deliver voxel-specific stiffness tensors. In order to check the relevance of this strategy, we convert a micro Computer Tomograph of a mouse femur into a regular Finite Element mesh, apply forces related to the dead load of a standing mouse, and then compare simulation results based on voxel-specific heterogeneous elastic properties to results based on homogeneous elastic properties related to the spatial average over the solid bone matrix compartment, of the X-ray attenuation coefficients. The element-specific strain energy density illustrates that use of homogeneous elastic properties implies overestimation of the organ stiffness. Moreover, the simulation reveals large tensile normal stresses throughout the femur neck, which may explain the mouse femur necks trabecular morphology being quite different from the human case, where the femur neck bears compressive forces and bending moments.

Layered water in crystal interfaces as source for bone viscoelasticity: arguments from a multiscale approach

Extracellular bone material can be characterised as a nanocomposite where, in a liquid environment, nanometre-sized hydroxyapatite crystals precipitate within as well as between long fibre-like collagen fibrils (with diameters in the 100 nm range), as evidenced from neutron diffraction and transmission electron microscopy. Accordingly, these crystals are referred to as ‘interfibrillar mineral’ and ‘extrafibrillar mineral’, respectively. From a topological viewpoint, it is probable that the mineralisations start on the surfaces of the collagen fibrils (‘mineral-encrusted fibrils’), from where the crystals grow both into the fibril and into the extrafibrillar space. Since the mineral concentration depends on the pore spaces within the fibrils and between the fibrils (there is more space between them), the majority of the crystals (but clearly not all of them) typically lie in the extrafibrillar space. There, larger crystal agglomerations or clusters, spanning tens to hundreds of nanometers, develop in the course of mineralisation, and the micromechanics community has identified the pivotal role, which this extrafibrillar mineral plays for tissue elasticity. In such extrafibrillar crystal agglomerates, single crystals are stuck together, their surfaces being covered with very thin water layers. Recently, the latter have caught our interest regarding strength properties (Fritsch et al. 2009 J Theor Biol. 260(2): 230–252) – we have identified these water layers as weak interfaces in the extrafibrillar mineral of bone. Rate-independent gliding effects of crystals along the aforementioned interfaces, once an elastic threshold is surpassed, can be related to overall elastoplastic material behaviour of the hierarchical material ‘bone’. Extending this idea, the present paper is devoted to viscous gliding along these interfaces, expressing itself, at the macroscale, in the well-known experimentally evidenced phenomenon of bone viscoelasticity. In this context, a multiscale homogenisation scheme is extended to viscoelasticity, mineral-cluster-specific creep parameters are identified from three-point bending tests on hydrated bone samples, and the model is validated by statistically and physically independent experiments on partially dried samples. We expect this model to be relevant when it comes to prediction of time-dependent phenomena, e.g. in the context of bone remodelling.

A multiscale poromicromechanical approach to wave propagation and attenuation in bone

Ultrasonics is an important diagnostic tool for bone diseases, as it allows for non-invasive assessment of bone tissue quality through mass density-elasticity relationships. The latter are, however, quite complex for fluid-filled porous media, which motivates us to develop a rigorous multiscale poromicrodynamics approach valid across the great variety of different bone tissues. Multiscale momentum and mass balance, as well as kinematics of a hierarchical double porous medium, together with Darcys law for fluid flow and micro-poro-elasticity for the solid phase of bone, give access to the so-called dispersion relation, linking the complex wave numbers to corresponding wave frequencies. Experimentally validated results show that 2.25 MHz acoustical signals transmit healthy cortical bone (exhibiting a low vascular porosity) only in the form of fast waves, agreeing very well with experimental data, while both fast and slow waves transmit highly osteoporotic as well as trabecular bone (exhibiting a large vascular porosity). While velocities and wavelengths of both fast and slow waves, as well as attenuation lengths of slow waves, are always monotonously increasing with the permeability of the bone sample, the attenuation length of fast waves shows a minimum when considered as function of the permeability.