Research Paper
A micropolar anisotropic constitutive model of cancellous bone from discrete homogenization

https://doi.org/10.1016/j.jmbbm.2012.07.012Get rights and content

Abstract

Cosserat models of cancellous bone are constructed, relying on micromechanical approaches in order to investigate microstructure-related scale effects on the macroscopic properties of bone. The derivation of the effective mechanical properties of cancellous bone considered as a cellular solid modeled as two-dimensional lattices of articulated beams is presently investigated. The cell walls of the bone microstructure are modeled as Timoshenko thick beams. The asymptotic homogenization technique is involved to get closed form expressions of the equivalent properties versus the geometrical and mechanical microparameters, accounting for the effects of bending, axial, and transverse shear deformations. Considering lattice microrotations as additional degrees of freedom at both the microscopic and macroscopic scales, an anisotropic micropolar equivalent continuum model is constructed, the effective mechanical properties of which are identified. The effective elastic moduli of various periodic cell structures are computed in situations of low and high effective densities to assess the impact of the transverse shear deformation. The stress distribution in a cracked bone sample is computed based on the effective micropolar model, highlighting the regularizing effect of the Cosserat continuum in comparison to a classical elasticity continuum model.

Highlights

► Micromechanical model of trabecular bone viewed as an anisotropic cellular solid material. ► Effective mechanical properties versus lattice geometry and microscopic moduli. ► Micropolar continuum model for trabecular bone. ► Linear Cosserat finite element. ► Regularization of stresses around a sharp crack in a bone sample.

Introduction

There are two major forms of bone tissue at the macroscopic level, compact or cortical bone, and cancellous or trabecular bone. The location of these bone types in the femur is illustrated in Fig. 1; cortical or compact bone is a dense material with a specific gravity of almost two in humans and a little over two in cattle; it forms most of the outer shell of a whole bone, a shell of variable thickness.

Cancellous bone generally exists only within the confines of the cortical bone coverings; it is also called trabecular bone because it is composed of short struts of bone material called trabeculae. The connected trabeculae give cancellous bone a spongy appearance, and it is consequently often called spongy bone (Fig. 2).

In mechanical terms, bone is a complex hierarchical material in which 'different geometrical features occur over several length scales that can be classified as follows: (a) The nanoscale represents a single fiber and crystals, b) the microscale represents the random network of trabecular bone composed of random struts or plates, and (c) the macroscale represents the whole bone, which includes both trabecular (porous) and cortical (solid) bone types. These structural levels are illustrated in Fig. 3.

Trabecular bone is very complex due to its randomness and spatial heterogeneity. In order to simplify the forthcoming analysis, one can choose to represent bone as having an idealized periodic microstructure: in this contribution, we model bone with a periodic prismatic structure in a two dimensional context. This work addresses the modeling of trabecular bone as hexagonal periodic structures.

We first present a review of the literature related to the modeling approaches and geometric models of trabecular bone at both the structural and continuum levels, focusing on generalized continuum models.

Considering first microstructural modeling approaches, Gibson (1985) proposed to model cancellous bone by a periodic network of cubic or hexagonal cells; this model was used to estimate the compressive behavior of cancellous bone as a function of the mechanical properties of the cell material and dimensions of the cell (thickness t, length l). Guo et al. (1994) studied the compressive fatigue of trabecular bone modeled by a 2-D hexagonal honeycomb structure. Silva and Gibson (1997) developed a two-dimensional finite element model of the human vertebral trabecular bone using a technique based on Voronoï diagrams to investigate its mechanical behavior. Silva and Gibson (1997) also showed that the variability in the arrangement of cell walls creates a small amount of variability in the elastic constants of honeycombs, but that overall the relationship between the microstructure and the elastic properties for isotropic and anisotropic non-periodic honeycombs are not different from periodic honeycombs. This result lends some evidence in utilizing periodic unit cells in modeling the random geometry of trabecular bone. Analytical models were also developed to investigate the strength asymmetry observed in trabecular bone (Keaveny et al., 1994, Kaplan et al., 1985). The geometrical formulations include a two dimensional honeycomb. The models were loaded under a remote uniaxial stress and the unit cells were aligned with the direction of loading (Ford and Gibson, 1998). Visual inspection of several representative cross sections suggested that a choice of 60° for the parameter θ was reasonable to represent, on the average, the relatively steep angle at which the oblique trabeculae lie relative to the horizontal direction. Also, a value resulting in a degree of material anisotropy that is consistent with previous measurements for this type of trabecular bone was chosen, namely h/l=0.5 for the geometry of that model (h is the vertical cell length) see Fig. 6.

Although there have been many continuum models of bone developed over the last two decades on the basis of classical elasticity (Taylor et al., 2002, Bowman et al., 1998), those models ignore microstructure-related scale effects on the macroscopic mechanical properties. Consequently, they do not provide a complete description of the mechanical behavior when the microstructural size of bone approaches the macroscopic length scale. The structural hierarchy of materials with microstructures such as bones plays an important role in determining their macroscopic mechanical behavior as well as the stress and strain distributions. Such microstructural effects are most pronounced near bone–implant interfaces and in areas of high strain gradients. This issue is presently investigated by studying generalized continuum mechanics theories which account for the influence of microstructure-related scale effects on the macroscopic properties of bone. Bone is a strongly heterogeneous material with microstructural features, requiring generalized continuum mechanics theories when the macroscopic length scale (identified as the smallest length scale of the deformation pattern) becomes comparable or smaller than the typical microstructural length scale, such as the size of trabeculae in cancellous bone. Especially, the classical assumption inherent to classical elasticity is no more valid under such conditions, due to which occur in zones of high stress and strain gradients, for instance in the vicinity of cracks or within a bone prosthesis region.

The incorporation of the microstructural scale within the continuum framework involves the relaxation of the local action hypothesis, which states that the constitutive model at a given material points can be elaborated from field variables defined in a local neighborhood of that point. Continuum phenomenological models that abandon the local action principle introduce a spatial interaction and a typical length scale of this interaction to account for the influence of neighboring points in the formulation of constitutive equations. Such enhanced continuum formulations aim at incorporating information related to the microstructure, and they follow three possible main strategies, namely integral nonlocal models, higher-order gradient models and micropolar (otherwise coined Cosserat) theories. All three models share the fact that a characteristic length inherent to the material is introduced in the field equations. Second order models for bones have emerged recently in the literature (Gitman et al., 2010), in which the authors evidence the stress reduction close to the crack line in contrast to classical continuum elasticity which shows a stress peak.

In the class of micropolar models developed in this paper, independent rotational degrees of freedom are considered in addition to the translational degrees of freedom (the displacement vector), in the form of a microrotation vector field. This entails that the material can transmit couple stresses in addition to tractions; those couple stresses develop internal work in the variation of microcurvatures, defined as the spatial gradients of the microrotation, a second order tensor. From a historical perspective, the Cosserat Brothers, who introduced the concept bearing their name (Cosserat and Cosserat, 1909), developed the theory of non-symmetric elasticity, and further developments emerged in the 1960s (Eringen, 1968, Mindlin, 1964, Mindlin and Tiersten, 1962). Based on the generalized continuum theory established by the Cosserat brothers about 100 years ago, Eringen, 1966, Eringen, 1968, Eringen, 1976, Eringen, 1999 formulated a general theory of Cosserat continuum coined the micropolar theory, adequate for materials possessing microstructures, such as bones, but also for granular composites, amorphous metals/ceramics, polymers. This theory can well explain the discrepancies between experiments and the classical theory of elasticity in cases when the effects of material microstructures are known to contribute significantly to the body's overall deformation, for example, in materials with a granular microstructure such as human bones (Lakes et al., 1990, Lakes, 1995, Shmoylova et al., 2007). A special case of Cosserat theory is the couple–stress theory, in which the microrotation and macrorotation coincide (Koiter, 1964). Micropolar theory assumes that the interaction between continuum particles through a surface element dA occurs not only through a force vector (FidA) but also through a moment vector (MidA). This establishes the “force–stress” tensor expressed as force per unit area, σij, and the “couple–stress” tensor expressed as moment per unit area, mij, from the Euler-Cauchy equilibrium principle. In terms of kinematics, material particles have additional rotational degrees of freedom ϕi allowing to better capture the behavior of heterogeneous materials like bone, which have microstructural dimensions comparable to the size of specimen.

The existence of couple stress has been first evidenced by Yang and Lakes (1981), who measured the effect of the size of a bone specimen on the apparent stiffness of cortical bone in quasi-static torsion; they further obtained the characteristic length scales for torsion and bending for cortical bone in the context of couple stress theory.

As an example, bone trabeculae are modeled as isotropic micropolar materials in Lakes (1993). Lakes and co-workers conducted a series of experiments on bone and other cellular materials, in which they observed a stiffening effect in such materials in bending and in torsion (Park and Lakes, 1987) and a tougher notched bone Lakes et al. (1990) than expected from classical Cauchy-type elasticity. Those last authors found that Cosserat elasticity provides better predictions of the response of bone than classical elasticity theory. Tanaka and Adachi (1999) represented the trabeculae as beam elements in a lattice structure in which the lattice elements are rigidly interconnected to each other, for the purpose of bone remodeling. The microstructure was then embedded in the continuum in the context of the couple stress theory. The issue of application of higher-order continuum theories to mechanical analysis of bone (both cortical and cancellous) has further been addressed by Fatemi et al. (2002), who analyzed a simplified two-dimensional bone-prosthesis configuration using a micropolar-based FE formulation. The stress and strain intensities they calculated at the bone-prosthesis interface are different from those predicted by classical elasticity. In addition, Fatemi et al. (2003) identified the micropolar elastic constants of cancellous bone in the context of micromechanical analyses. In this approach, it is assumed that the bone tissue is an isotropic, Cauchy-type elastic material at the microscopic level, whereas cancellous bone behaves as a homogeneous, anisotropic micropolar-type continuum at the macroscopic level. The effective elastic constants for the micropolar continuum were determined from the response of a bone specimen, whose microstructure was obtained from micro-CT scans.

The main goal of this work is to construct a continuum micropolar model for trabecular bones with a high density, accounting for the transverse shear of the microbeams building the lattice, and reflecting the overall anisotropy of the initial lattice structure. We shall describe the cell walls as Timoshenko beams, and consider a lattice with a regular enough topology so that it can be considered as quasi periodic and thus a repetitive representative unit cell can be identified. The micropolar constants for a range of periodic two-dimensional cell topologies will be derived, relying on the asymptotic homogenization technique to get closed form expressions of the equivalent mechanical properties versus the geometrical and mechanical microparameters. This approach is novel since most of the works considering a Cosserat model for bone adopt it as a postulate, whereas we construct such a micropolar constitutive model from a micromechanical analysis of the underlying bone microstructure. Cell wall bending, transverse shear and axial stretching are taken as the deformation mechanisms in the analysis. Such mechanisms are quite essential for typical cellular structures having a high relative density as the cell walls then have large aspect ratios, or if the cell wall material has small shear rigidity.

This work is organized as follows: in Section 2, a two-dimensional micropolar continuum model equivalent to the initial lattice is constructed, and its mechanical properties are identified versus the microbeams geometry and mechanical behavior. The specific case of hexagonal lattices is considered in Section 3, and the effect of transverse shear is analyzed, comparing the homogenized values with Gibson and Ashby model. The homogenized micropolar anisotropic constitutive model leads to the development of a specific four node finite element for planar situations. Based on this, finite element simulations of bone samples including cracks are performed (Section 4), showing the regularizing effect of the micropolar degrees of freedom. Finally, a summary of the main results and perspectives are exposed in Section 5.

Section snippets

A 2D micropolar effective continuum model of bone with transverse shear

We adopt the viewpoint of trabecular bone as a cellular solid consisting of a quasi periodical lattice of cells having an hexagonal topology, the cell walls being modeled as thick beams. The characteristics of many periodic cellular structures are discussed in detail in the well-known contribution of Gibson and Ashby (1997), wherein the equivalent mechanical properties of honeycomb lattices are derived by analyzing strain and stress states in unit cells through the application (in most cases)

Application: effective micropolar properties of bone modeled as a hexagonal lattice

The general anisotropic hexagonal unit cell of the two-dimensional honeycomb under consideration is pictured in Fig. 6; it consists of three beams, a vertical beam of length h and two inclined beams of length l. The representative unit cell is easily identified for the hexagonal lattice (Fig. 7). The vertical beams have elastic and shear moduli Es1 and Gs1, respectively, while the two inclined beams have elastic and shear moduli Es and Gs, respectively. The dimensionless parameters θ and α=h/l

Application to bone fracture

Simulations of a square bone sample including a crack are performed, using the previously obtained homogenized anisotropic micropolar model. The results are compared to those obtained by a classical four node finite element called CPS4 within the Abaqus environment.

Conclusion

This paper is a micromechanical approach of microstructural effects in the macroscopic continuum mechanical properties of trabecular bone. As a main novelty, a Cosserat anisotropic continuum model has been developed from the discrete homogenization of a quasi periodical lattice model of the cancellous bone microstructure, whereby the effective mechanical properties of bone are here directly related to the lattice micro-geometry and micromechanical elastic properties. The cell walls of this

References (49)

  • A. Mourad et al.

    A nonlinearly elastic homogenized constitutive law for the myocardium

    Computational Fluid and Solid Mechanics

    (2003)
  • H.B. Muhlhaus et al.

    Dispersion and wave propagation in discrete and continuous models for granular materials

    International Journal of Solids and Structures

    (1996)
  • H.C. Park et al.

    Torsion of a micropolar elastic prism of square cross section

    International Journal of Solids Structures

    (1987)
  • M.J. Silva et al.

    The effects of non-periodic microstructure on the elastic properties of two-dimensional cellular solids

    International Journal of Mechanical Sciences

    (1995)
  • M.J. Silva et al.

    Modeling the mechanical behavior of vertebral trabecular bone: effects of age-related changes in microstructure

    Bone

    (1997)
  • W.E. Warren et al.

    Three-fold symmetry restrictions on two-dimensional micropolar materials

    European Journal of Mechanics A/Solids

    (2002)
  • W.E. Warren et al.

    Foam mechanics: the linear elastic response of two dimensional spatially periodic cellular materials

    Mechanics of Materials

    (1987)
  • H.X. Zhu et al.

    Effects of cell irregularity on the elastic properties of 2D Voronoi honeycombs

    Journal of the Mechanics and Physics of Solids

    (2001)
  • N. Bakhvalov et al.

    Homogenisation: Averaging Processes in Periodic Media

    (1984)
  • S.M. Bowman

    Creep contributes to the fatigue behavior of bovine trabecular bone

    Journal of Biomechanical Engineering

    (1998)
  • Broek, D., 1974. Elementary Fracture Mechanics....
  • Bucalem, M.L., Bathe, K.J., 2011 The Mechanics of Solids and Structures- Hierarchical Modeling and Finite Element...
  • Cosserat, E., Cosserat, F., 1909. Théorie des Corps Déformables. A. Hermann et Fils,...
  • S.C. Cowin et al.

    Tissue Mechanics

    (2007)
  • Cited by (115)

    View all citing articles on Scopus
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