Multiscale Homogenization Theory: An Analysis Tool for Revealing Mechanical Design Principles in Bone and Bone Replacement Materials

Abstract

Biomimetics deals with the application of nature-made “design solutions” to the realm of engineering. In the quest to understand mechanical implications of structural hierarchies found in biological materials, multiscale mechanics may hold the key to understand “building plans” inherent to entire material classes, here bone and bone replacement materials. Analyzing a multitude of biophysical hierarchical and biomechanical experiments through homogenization theories for upscaling stiffness and strength properties reveals the following design principles: The elementary component “collagen” induces, right at the nanolevel, the mechanical anisotropy of bone materials, which is amplified by fibrillar collagen-based structures at the 100-nm scale, and by pores in the micrometer-to-millimeter regime. Hydroxyapatite minerals are poorly organized, and provide stiffness and strength in a quasi-brittle manner. Water layers between hydroxyapatite crystals govern the inelastic behavior of the nanocomposite, unless the “collagen reinforcement” breaks. Bone replacement materials should mimic these “microstructural mechanics” features as closely as possible if an imitation of the natural form of bone is desired (Gebeshuber et al., Adv Mater Res 74:265–268, 2009).

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

1 Nomenclature

5.1  

2 Introduction

Biomimetics deals with the application of nature-made “design solutions” to the realm of engineering. In this context, large efforts have aimed at imitating biological materials with interesting mechanical properties. However, biological materials are hierarchically organized and very complex, and frequently, the way they work is not easily comprehensible [1]. Hence, successful biomimetics solutions require a deep understanding of “universal” functioning principles of biological materials. It now appears that multiscale mechanics may hold the key to such an understanding of “building plans” inherent to entire material classes.

For relating the vision of hierarchical organization of materials to effective mechanical properties, we rely on continuum micromechanics, which is a well-established tool for structure–property investigations.

Based on various physical–chemical and mechanical experiments, our focus is the development of multiscale mechanical models, which mathematically and computationally quantify how the basic building blocks of biological materials (such as hydroxyapatite minerals, collagen, and water in all bones found throughout the vertebrate kingdom) govern the materials’ mechanical properties at different length scales, from a few nanometers up to the macroscopic level. Thereby, multiscale homogenization theory allows us, at each scale, to identify material representations which are as simple as possible, but as complex as necessary for reliable computational predictions of key material properties, such as poroelasticity, creep, and strength. This can be seen as “reverse” biomimetics engineering: (civil) engineering methods are used to understand biological systems, as described in more detail in Sect. 5.2.

This chapter is mainly devoted to the highly fascinating, hierarchically organized material class “bone” (see Sect. 5.3), and to the “universal” elementary building blocks inherent to this material class (Sect. 5.4). A multiscale micromechanics representation (Sect. 5.5) has opened, for the first time, a profound theoretical understanding of bone mechanics, which is consistent with virtually all major experimental observations, given in more detail in Sect. 5.6. One of our key findings is that bone’s mechanical properties are governed by porous polycrystals which the minerals build up as structural complement to the collagen fibrils found in all connective tissues (also in tendon, cartilage, skin). These polycrystals are central not only to the magnitude of elastic anisotropy of bone materials, but also to their tensile-to-compressive strength ratio resulting from universal failure characteristics of differently oriented submicron-sized mineral platelets (Sect. 5.7). Implications of these findings for bone biomaterial design conclude the chapter.

3 Fundamentals of Continuum Micromechanics

3.1 Representative Volume Element

In continuum micromechanics [2,  3,  4,  5,  6], a material is understood as a macrohomogeneous, but microheterogeneous body filling a representative volume element (RVE) with characteristic length ℓ, ℓ ≫ d, d standing for the characteristic length of inhomogeneities within the RVE (see Fig. 5.1), and $\mathcal{l}\,\ll\,\mathcal{L},\,\mathcal{L}$ standing for the cha- racteristic lengths of geometry or loading of a structure built up by the material defined on the RVE.

Fig. 5.1

Multistep homogenization: Properties of phases (with characteristic lengths of d and d ₂, respectively) inside RVEs with characteristic lengths of ℓ or ℓ ₂, respectively, are determined from homogenization over smaller RVEs with characteristic lengths of ℓ ₂ ≤ d and ℓ ₃ ≤ d ₂, respectively

In general, the microstructure within one RVE is so complicated that it cannot be described in complete detail. Therefore, quasi-homogeneous subdomains with known physical quantities (such as volume fractions or elastoplastic properties) are reasonably chosen. They are called material phases. The “homogenized” mechanical behavior of the overall material, i.e., the relation between homogeneous deformations acting on the boundary of the RVE and resulting (average) stresses, including the ultimate stresses sustainable by the RVE, can then be estimated from the mechanical behavior of the aforementioned homogeneous phases (representing the inhomogeneities within the RVE), their dosages within the RVE, their characteristic shapes, and their interactions. If a single phase exhibits a heterogeneous microstructure itself, its mechanical behavior can be estimated by introduction of an RVE within this phase, with dimensions ℓ ₂ ≤ d, comprising again smaller phases with characteristic length d ₂ ≪ ℓ ₂, and so on, leading to a multistep homogenization scheme (see Fig. 5.1).

3.2 Upscaling of Elasto-Brittle and Elastoplastic Material Properties

We consider an RVE consisting of n _r material phases, $r = 1,\ldots,{n}_{r}$, exhibiting elastoplastic or elasto-brittle material behavior. In case of ideal associated elastoplasticity, the RVE follows the constitutive laws:

$$\begin{array}{rcl}{ \sigma}_{r} {= }_{r} : ({\epsilon}_{r} - {\epsilon}_{r}^{\mathrm{p}}),& & \end{array}$$

(5.1)

$$\begin{array}{rcl}\dot{{\epsilon}_{r}^{\mathrm{p}}} =\dot{ {\lambda}}_{ r} \frac{\partial{\mathfrak{f}}_{r}} {\partial{\sigma}_{r}},\quad \dot{{\lambda}}_{r}{\mathfrak{f}}_{r}({\sigma}_{r}) = 0,\quad\dot{{\lambda}}_{r} \geq 0,\quad {\mathfrak{f}}_{r}({\sigma}_{r}) \leq 0.& & \end{array}$$

(5.2)

In (5.2), σ _r and ε _r are the stress and (linearized) strain tensors averaged over phase r with elasticity tensor ${}_{r}$; ε _r ^p are the average plastic strains in phase r, λ _r is the plastic multiplier of phase r, and ${\mathfrak{f}}_{r}({\sigma}_{r})$ is the yield function describing the (ideally) plastic characteristics of phase r.

In case of brittleness, the RVE follows the constitutive laws:

$$\begin{array}{rcl}{ \sigma}_{r} {= }_{r} :{ \epsilon}_{r}\quad ({\epsilon}_{r}^{\mathrm{p}} = 0)\quad \mathrm{if}\quad {\mathfrak{f}}_{ r}({\sigma}_{r}) < 0,& & \end{array}$$

(5.3)

${\mathfrak{f}}_{r}({\sigma}_{r})$ now being a failure function defining stress states σ _r related to brittle failure.

The RVE is subjected to Hashin boundary conditions, i.e., to “homogeneous” (“macroscopic”) strains E at its boundary, so that the kinematically compatible phase strains ε _r inside the RVE fulfill the average condition

$$\begin{array}{rcl} E ={ \sum \nolimits }_{r}{f}_{r}{\epsilon}_{r}& & \end{array}$$

(5.4)

with f _r as the volume fraction of phase r. In a similar way, the equilibrated phase stresses σ _r fulfill the stress average condition

$$\begin{array}{rcl} \Sigma ={ \sum\nolimits }_{r}{f}_{r}{\sigma}_{r}& & \end{array}$$

(5.5)

with Σ as the “macroscopic” stresses.

The superposition principle (following from linear elasticity and linearized strain) implies that the phase strains ε _r are linearly related to both the macroscopic strains E and the free strains ε _r ^p (which can be considered as independent loading parameters),

$$\begin{array}{rcl}{ \epsilon}_{r} = {\mathbb{A}}_{r} : E +{ \sum \nolimits }_{s}{G}_{rs} :{ \epsilon}_{s}^{\mathrm{p}}& & \end{array}$$

(5.6)

with ${\mathbb{A}}_{r}$ as the fourth-order concentration tensor [7], and ${G}_{rs}$ as the fourth-order influence tensors [8]. The latter quantify the phase strains ε _r resulting from plastic strains ε _s ^p, while the overall RVE is free from deformation, E = 0.

In the absence of plastic strains [${\mathfrak{f}}_{r} < 0$, ε _r ^p = 0 in (5.1)–(5.2)], the RVE behaves fully elastically, so that (5.6), (5.5), (5.4), and (5.1) yield a macroscopic elastic law of the form:

$$\begin{array}{rcl} \Sigma = {\mathbb{C}}^{\mathrm{hom}} : E\quad \mathrm{with}\quad {\mathbb{C}}^{\mathrm{hom}} ={ \sum \nolimits }_{r}{f{}_{r}}_{r} : {\mathbb{A}}_{r},& & \end{array}$$

(5.7)

as the homogenized elastic stiffness tensor characterizing the material within the RVE. In case of nonzero “free” plastic strains ε _r ^p, (5.7) can be extended to the form:

$$\begin{array}{rcl} \Sigma = {\mathbb{C}}^{\mathrm{hom}} : (E -{E}^{\mathrm{p}}),& & \end{array}$$

(5.8)

(5.8), together with (5.1), (5.5)–(5.7), gives access to the macroscopic plastic strains E ^p, reading as:

$$\begin{array}{rcl}{ E}^{\mathrm{p}}& =&-{\left [{\sum \nolimits }_{r}{f{}_{r}}_{r} : {\mathbb{A}}_{r}\right ]}^{-1} : \left \{{\sum \nolimits }_{r}{f{}_{r}}_{r} : \left [({\mathbb{A}}_{r} : E +{ \sum \nolimits }_{s}{G}_{rs} :{ \epsilon}_{s}^{\mathrm{p}})-{\epsilon}_{ r}^{\mathrm{p}}\right ]\right \} + E. \\ & & <EquationNumber>5.9</EquationNumber> \\ \end{array}$$

(5.9)

Matrix-inclusion problems [9,  10] allow for estimating concentration tensors ${\mathbb{A}}_{r}$ and influence tensors ${G}_{rs}$ [11], so that the estimate for the homogenized stiffness (5.7) can be written as [5]:

$$\begin{array}{rcl}{ \mathbb{C}}^{\mathrm{hom}} =\!{ \sum \nolimits }_{r}{f{}_{r}}_{r} :{ \left [\mathbb{I} + {\mathbb{P}}_{r}^{0} : {(}_{ r} - {\mathbb{C}}^{0})\right ]}^{-1} :{ \left \{{\sum \nolimits }_{s}{f}_{s}{\left [\mathbb{I} + {\mathbb{P}}_{s}^{0} : {(}_{ s} - {\mathbb{C}}^{0})\right ]}^{-1}\right \}}^{-1}\!\!,& & \\ && \\ \end{array}$$

(5.10)

where $\mathbb{I}$ is the fourth-order unity tensor, and the fourth-order Hill tensor ${\mathbb{P}}_{r}^{0}$ accounts for the characteristic shape of phase r in a matrix with stiffness ${\mathbb{C}}^{0}$. The two sums are taken over all phases of the heterogeneous material in the RVE. Choice of this stiffness describes the interactions between the phases: For ${\mathbb{C}}^{0}$ coinciding with one of the phase stiffnesses (Mori–Tanaka scheme [12,  13]), a composite material is represented (contiguous matrix with inclusions); for ${\mathbb{C}}^{0} = {\mathbb{C}}^{hom}$ (self-consistent scheme [14,  2]), a dispersed arrangement of the phases is considered (typical for polycrystals).

4 Bone’s Hierarchical Organization

Bone materials are characterized by an astonishing variability and diversity. Still, because of “architectural constraints” due to once chosen material constituents and their physical interaction, the fundamental hierarchical organization or basic building plans of bone materials remain largely unchanged during biological evolution. These building plans are expressed by typical morphological features which can be discerned across all bone materials. We here distinguish six levels of hierarchical organization in the line of Katz et al. [15], which have been quite generally accepted in the scientific community:

• At an observation scale of several 10 nm, the so-called elementary components of mineralized tissues can be distinguished. Long cylindrically shaped collagen molecules with a diameter of about 1.2 nm and a length of about 300 nm [16] are self-assembled in staggered organizational schemes with characteristic diameters of 50–500 nm [17,  18,  19,  20,  21,  22,  23,  24], and attached to each other at their ends by crosslinks. These cross-linked collagen molecules build up, together with the water-filled intermolecular space, the scale of wet collagen (Fig. 5.2a).

  Needle or plate-shaped mineral crystals, consisting of impure hydroxyapatite (HA; Ca₁₀[PO₄]₆[OH]₂) with typical 1–5 nm thickness, and 25–50 nm length [22], are penetrated by the water-filled intercrystalline space (Fig. 5.2c).

• Wet collagen is penetrated by intrafibrillar crystals, forming a collagen–HA network, called fibril (Fig. 5.2b), at an observation scale of about 10 nm.

• The extrafibrillar crystals form aggregates and build up, together with the intercrystalline space, a porous HA polycrystal called extrafibrillar space (Fig. 5.2c), at an observation scale of about 300 nm.

• At an observation scale of several micrometers, the ultrastructure comprises the fibrils (light areas in Fig. 5.2d) and the extrafibrillar space (dark areas in Fig. 5.2d).

• At an observation scale of 100–200 μm, the extravascular bone matrix comprises the ultrastructure and osteocyte-filled cavities called lacunae (Fig. 5.2e).

• At an observation scale of several 100 μm to several millimeters, the microstructure comprises the extravascular bone matrix and the vascular space (Haversian canals) (Fig. 5.2f).

Fig. 5.2

Multiscale view of bone structure, with key physical effects considered in micromechanics representation of Fig. 5.4: (a) wet collagen; reproduced from [25], Copyright National Academy of Sciences, USA; (b) mineralized collagen fibril; schematic sketch after [26]; (c) extrafibrillar porous polycrystal; (d) extracellular bone matrix; reproduced with kind permission from Springer Science+Business Media: [24], Fig. 5; (e) extravascular bone matrix [zoomed out of image (f)]; (f) cortical bone; reprinted from [27], with permission from American Institute of Physics, ©1979

5 Elastic and Strength Properties of the Elementary Components of Bone: Hydroxyapatite, Collagen, Water

As regards hydroxyapatite, tests with an ultrasonic interferometer coupled with a solid media pressure apparatus [28,  29] reveal isotropic elastic properties, which are in perfect agreement with recent ab initio calculations of elastic constants of ceramic crystals [30,  31] (Table 5.1); and quasi-static mechanical tests in uniaxial tension and compression on fairly dense samples of artificial hydroxyapatite biomaterials [32,  33] give access to the uniaxial tensile and shear strength of pure hydroxyapatite (Table 5.1). The latter is probably governed by detachment of one nonspherical nano- or microcrystal of hydroxyapatite from its neighbors [34]. This is mathematically expressed through [11]:

$$\begin{array}{rcl} & & \psi = 0,\ldots,2\pi : \\ & & {\mathfrak{f}}_{\mathrm{HA}\varphi\vartheta}({\sigma}_{\mathrm{HA}\varphi\vartheta}) ={ \frac{{\sigma}_{\mathrm{HA}}^{\mathrm{ult,t}}} {{\sigma}_{\mathrm{HA}}^{\mathrm{ult,s}}}\,\max }_{\psi}\vert{\sigma}_{\mathrm{HA}}^{Nn}\vert + {\sigma}_{\mathrm{ HA}}^{NN} - {\sigma}_{ HA}^{ult,t} = 0 \\ \end{array}$$

(5.11)

with spherical coordinates φ and $\vartheta$ defining the crystal needle orientation vector $\underline{N} ={ \underline{e}}_{r}$ in the reference frame (${\underline{e}}_{1}$, ${\underline{e}}_{2}$, ${\underline{e}}_{3}$), and with ψ defining the orientation of vector $\underline{n}$ related to shear stresses (see Fig. 5.3), with σ_HA ^ult,t as the uniaxial tensile strength and σ_HA ^ult,s as the shear strength of pure hydroxyapatite; and with ${\sigma}_{\mathrm{HA}}^{Nn} = \underline{N} \cdot{\sigma}_{\mathrm{HA}\varphi\vartheta} \cdot\underline{n}$ and ${\sigma}_{\mathrm{HA}}^{NN} = \underline{N} \cdot{\sigma}_{\mathrm{HA}\varphi\vartheta} \cdot\underline{N}$ as the normal and shear stress components related to a surface with normal $\underline{N}(\varphi,\vartheta)$.

Table 5.2 “Universal” (tissue and location-independent) elasticity and strength values of elementary constituents of bone

Fig. 5.3

Cylindrical (needle-like) HA inclusion oriented along vector $\underline{N}$ and inclined by angles $\vartheta$ and φ with respect to the reference frame (${\underline{e}}_{1}$, ${\underline{e}}_{2}$, ${\underline{e}}_{3}$); local base frame (${\underline{e}}_{r}$, ${\underline{e}}_{\vartheta}$, ${\underline{e}}_{\varphi}$) is attached to the needle

As regards (molecular) collagen, its (transversely isotropic) elastic properties are well accessible through those of dry rat tail tendon, a tissue consisting almost exclusively of collagen (i.e., the intermolecular porous space, which is increasing with the water content of “wet collagen,” is minimal for “dry collagen” [37,  16]). Corresponding components of the elasticity tensor of (molecular) collagen (see Table 5.1), derived from Brillouin light scattering tests [17], are in perfect agreement with Young’s modulus of a collagen I monomer, when measured through an optical tweezer-interferometer system [38]. Strength tests, also on rat tail tendon [36], reveal the crosslink-failure-induced quasi-brittle strength of collagen [39], when related to the molecular collagen portion of the tissue [11] (see Table 5.1). The latter is mathematically expressed through [11]:

$$\begin{array}{rcl}{ \mathfrak{f}}_{\mathrm{col}}({\sigma}_{\mathrm{col}}) = \vert{\underline{e}}_{3} \cdot{\sigma}_{\mathrm{col}} \cdot{\underline{e}}_{3}\vert- {\sigma}_{\mathrm{col}}^{\mathrm{ult}} \leq 0,& & \\ \end{array}$$

(5.12)

where the direction three coincides with the principal orientation direction of collagen and where σ_col ^ult is the uniaxial (tensile and compressive) strength of molecular collagen (see Fig. 5.4).

We assign the ultrasonics-derived bulk modulus of water [35] (and no strength value) to phases comprising water with mechanically insignificant noncollagenous organic matter (Table 5.1).

Fig. 5.4

Micromechanical representation of bone material by means of a six-step homogenization procedure. Reproduced from [11]

6 Multiscale Micromechanical Representation of Bone

Across the hierarchical organization of cortical bone material, the following “universal” microstructural patterns are considered in the framework of a multistep homogenization scheme (Fig. 5.4, see [40,  41,  11] for details): The first homogenization step refers to an observation scale of several nanometers, where cross-linked collagen molecules form a (linear elastic-perfectly brittle) contiguous matrix, which is “perforated” by intermolecular, water-filled spaces. A Mori–Tanaka scheme relates the stiffnesses and volume fractions of molecular collagen and intermolecular space to the homogenized elasticity tensor of wet collagen [specification of (5.10) for RVE of Fig. 5.4a]. The second homogenization step refers to an observation scale of several hundreds of nanometers, where wet collagen and mineral crystal agglomerations penetrate each other, building up the mineralized collagen fibril. A self-consistent scheme relates the stiffnesses and volume fractions of wet collagen and (intrafibrillar) hydroxyapatite crystal agglomerations to the homogenized elasticity tensor of the fibril [specification of (5.10) for RVE of Fig. 5.4b]. The third homogenization step also refers to an observation scale of several hundreds of nanometers, where (extrafibrillar) hydroxyapatite crystal agglomerations and intercrystalline space build up a porous polycrystal (extrafibrillar space). A self-consistent scheme relates the stiffnesses and volume fractions of hydroxyapatite crystals and intercrystalline space to the homogenized elasticity tensor of the porous hydroxyapatite crystal [specification of (5.10) for RVE of Fig. 5.4c]. The fourth homogenization step refers to an observation scale of some micrometers, where mineralized fibrils are embedded as inclusions into the extrafibrillar mineral foam, forming together the extracellular bone matrix (ultrastructure). A Mori–Tanaka scheme relates the stiffnesses and volume fractions of the extrafibrillar space and the fibril to the homogenized elasticity tensor of the extracellular bone matrix [specification of (5.10) for RVE of Fig. 5.4d]. The fifth homogenization step refers to an observation scale of about 100 μm, where lacunar pores are embedded in an ultrastructural matrix building up the extravascular bone material. A Mori–Tanaka scheme relates the stiffnesses and volume fractions of the ultrastructure and the lacunae to the homogenized elasticity tensor of the extravascular bone material [specification of (5.10) for RVE of Fig. 5.4e]. The sixth homogenization step refers to an observation scale of about 1 mm, where the extravascular matrix is perforated by Haversian canals, building up the microstructure. A Mori–Tanaka scheme relates the stiffnesses and volume fractions of the extravascular bone material and the Haversian canals to the homogenized elasticity tensor of the microstructure [specification of (5.10) for RVE of Fig. 5.4f].

In addition, repeated specification of (5.6) for all RVEs in Fig. 5.4 allows for upscaling strains from the level of the elementary components, all the way up to the macroscopic bone material of Fig. 5.4f. Considering, in addition, the elastoplastic behavior of sliding hydroxyapatite minerals [11,  42], according to (5.11), (5.1), and (5.2), the elastoplastic material behavior at all scales is defined, inclusive of quasi-brittle failure once the collagen failure surface, (5.12), is reached.

7 Experimental Validation of Multiscale Micromechanics Theory for Bone

The micromechanical model of Sect. 5.5 predicts, for each set of tissue-specific volume fractions, the corresponding tissue-specific elasticity and strength properties at all observation scales of Fig. 5.4. Thus, a strict experimental validation of this model is realized as follows: (1) different sets of volume fractions [variables f _r in (5.1)–(5.10)] are determined from composition experiments on different bone samples with different ages, from different species and different anatomical locations; (2) these volume fractions are used as model input, and (3) corresponding model-predicted stiffness and strength values [model output, (5.10)] are compared to results from stiffness and strength experiments on the same or very similar bone samples.

The volume fractions within each of the considered RVEs (Fig. 5.4) are accessible through the following experiments: At the microstructural and the extravascular observation scales (Fig. 5.4e,f), polarized light microscopy (Fig. 5.2f) visualizes, in cortical bone, Haversian canals and lacunae [27,  43,  44], from which the vascular and lacunar porosities can be derived, see [41] for details. For determination of the volume fractions in all other RVEs, the pioneering volume measurements and weighing experiments on wet, dehydrated, and demineralized cortical bone specimens, performed by Lees [45,  46,  16], are of central importance; they give access to the mineral, (molecular) collagen, and water content at the extracellular scale (Fig. 5.2d and RVE Fig. 5.4d). The collagen volume fraction in a piece of extracellular bone material (Fig. 5.2d and Fig. 5.4d), together with the spatial organization of collagen molecules within the extracellular matrix, characterized by a staggered longitudinal arrangement quantified by the Hodge–Petruska scheme [47] and by a quasi-hexagonal transverse arrangement as elucidated by the pioneering neutron diffraction experiments of Lees and coworkers [37,  16], gives access to the volume fraction of collagen fibrils in an extracellular RVE (Fig. 5.4d, see [41] for details). Concerning the RVEs below the extracellular scale (Fig. 5.4b,c), their mineral volume fraction can be determined from Hellmich and Ulm’s finding [48] that the hydroxyapatite concentration in the extra-collagenous space is the same inside and outside the fibrils, see [41] for details. The class of porous hydroxyapatite biomaterials [49,  32,  33,  50,  51,  52] directly mimics the porous polycrystal of Fig. 5.4c. For such materials, referred to in Fig. 5.5a–d, the mineral volume fraction is directly related to porosity or apparent mass density, accessible from mass and volume measurements on biomaterial samples. Finally, the ultrastructural collagen volume and that of wet collagen in fibrils (Fig. 5.4b) give access to the intermolecular porosity within an RVE of wet collagen (see Fig. 5.4a and [41] for details).

Stiffness and strength values are gained from ultrasonics experiments on cortical [45,  46,  43,  44,  53] and on trabecular bone samples [54], and from quasi-static uniaxial, compressive or tensile, mechanical tests [55,  56,  57,  58,  42,  59,  60,  61,  62,  63,  64,  65,  66,  67,  68,  69,  70,  71] (see [34,  41,  11] for details).

Throughout all different observation scales, and across a great variety of bone tissues (from different anatomical locations of different mammals including humans, cows, elephants, deers, rabbits, and horses), micromechanics model predictions for stiffness and strength agree very well with corresponding experimental data (see Fig. 5.5). This unparalleled quantification of bone’s mechanical behavior allows us to draw basic conclusions on how bone works mechanically – described in the next chapter.

Fig. 5.5

Comparison between micromechanical model predictions and corresponding experiments: (a, b) elasticity in terms of Young’s modulus E and Poisson’s ratio ν of hydroxyapatite biomaterials (see Fig. 5.4c, [34]); (c, d) uniaxial tensile and compressive strength of hydroxyapatite biomaterials (see Fig. 5.4c, [34]); (e) radial and axial normal stiffness (C ₁₁₁₁ and C ₃₃₃₃) of cortical and trabecular bone at ultrastructural (ultra) and extravascular (exvas) scales (see Fig. 5.4d,e, [41]); (f) uniaxial strength of cortical bone at the macroscopic scale (see Fig. 5.4f, [11]), with mean and standard deviation being depicted for experimental tensile strength (dark color) and experimental compressive strength (light color)

8 How Bone Works: Mechanical Design Characteristics of Bone Revealed Through Multiscale Micromechanics

Bone materials are built up in a remarkably anisotropic fashion, with respect to both stiffness and strength.

As regards elasticity, the experimentally verified micromechanics theory of Sects. 5.5 and 5.6 reveals the following sources of anisotropy:

• 1. The intrinsic anisotropy of molecular collagen, quantified through Brillouin light scattering tests of Cusack and Miller [17] and given at the bottom of Table 5.1;

• 2. The axially oriented arrangement of molecular collagen in the mineralized collagen fibril (Fig. 5.4b); of the mineralized collagen fibrils in the extracellular bone matrix (Fig. 5.4d); and of the vascular porosity (Haversian canals) in cortical bone (Fig. 5.4f).

In contrast, bone mineral is poorly organized [20,  24,  72,  73,  74,  75], so that bone materials without collagen and vascular pores, such as hyperpycnotic tissues [16,  76] or artificial hydroxyapatite biomaterials [32,  77,  29,  78,  51,  52], represented basically through the extrafibrillar RVE of Fig. 5.4c, are virtually isotropic. At the same time, the high connectivity of the individual crystals, forming a porous polycrystal in the extrafibrillar space of bone materials, is important for providing sufficient transverse stiffness.

As regards strength, the micromechanics model of Sects.  5.5 and 5.6 shows an additional feature beyond anisotropy, namely the sensitivity of bone to tensile versus compressive loading type (not existing in the elastic regime): The extrafibrillar space (see Fig. 5.2c and RVE in Fig. 5.4c) is ten times weaker under tensile loading than under compressive loading, a characteristic which it shares with a lot of other quasi-brittle materials, such as sandstone or concrete [79]. The reason for that lies in the localization or concentration of homogeneous tensile or compressive strains from the boundary of the extrafibrillar RVE into its individual needle phases, leading eventually to failure of the most unfavorably stressed needle according to criterion (5.11) (see Fig. 5.6). In overall uniaxial tension, this local failure is dominated by tensile normal stresses in needle phases more or less aligned with the loading direction (for more or less porous materials; see Fig. 5.6a,b). In overall uniaxial compression, however, the aforementioned local failure is still dominated by tensile normal stresses in one needle phase – this phase now being aligned more or less perpendicular to the loading direction (for more or less porous materials; see Fig. 5.6c,d). Accordingly, it takes much more overall compressive stresses to produce as much normal tensile stresses in needles perpendicular to the loading direction than tensile stresses in needles aligned with the loading direction, stemming from overall tensile loads. This explains the ratio of 1:10 between tensile and compressive strength in hydroxyapatite biomaterials.

Fig. 5.6

Orientation of crystal needle initiating overall failure of hydroxyapatite polycrystals, under tensile (a, b) and compressive (c, d) uniaxial macroscopic loading: (a, c) local tensile failure in highly porous polycrystals; (b, d) local combined tensile-shear failure in polycrystals with low porosity

In real bones, however, two differences with respect to this situation are significant:

• 1. The hydroxyapatite crystals are nanometer-sized instead of micrometer-sized, so that they are intimately bond to the structured water layers between them [80,  81,  82,  83]. This evokes ideal plastic sliding rather than brittle failure, once the elastic regime of the crystals is left.

• 2. Molecular collagen (shown to be quasi-brittle, both experimentally [36] and computationally [84]) reinforces the multiporous hierarchical nanocomposite “bone” both in axial tension and in axial compression, up to the same extent. Accordingly, bone materials show a tensile-to-compressive strength ratio of 2:3, instead of 1:10.

Bone replacement materials should mimic these “microstructural mechanics”, features as closely as possible if an imitation of the natural form of bone is desired [105]. Hence, the design characteristics of natural bone imply the following design rules for creating bone replacement materials close to their natural “prototypes”:

• Use of an anisotropic organic template

• Growth of nanometer-sized mineral crystals

• Realization of a connected network of crystals

In this context, the following recent developments in the bone biomaterials community seem noteworthy: As regards anisotropic organic templates, recent bone biomaterials rely on materials such as starch [85], chitosan [86], or collagen itself [87,  88,  89,  90,  91,  92]. While micron-sized hydroxyapatite crystals defined the state of the art in biomaterial production in the 1980s and 1990s [32,  77,  33,  50], modern processing techniques, such as sol–gel combustion [93] or precipitation from an aqueous solution [87,  88,  94,  95,  96,  97,  92], allow for production of nanometer-sized hydroxyapatite crystals. Finally, also interconnected networks of hydroxyapatite crystals could be recently produced, by means of a layer-by-layer deposition method [92].

9 Some Conclusions from a Biological Viewpoint

To the knowledge of the authors, the aforementioned design rules concern all different kinds of bones, across the entire subphylum vertebrata, in the sense that they satisfactorily explain, in a unified fashion, the large variety of mechanical properties depicted in Fig. 5.5e,f. In this sense, they are “universal” for the vertebrates, but not beyond this subphylum. The aforementioned design principles may result from “architectural constraints” [98,  99] merely due to once chosen material constituents and their physical interactions that imply the fundamental hierarchical organization patterns or basic building plans of bone materials (see Fig. 5.2), which remain largely unchanged during biological evolution. In this sense, it seems highly probable that the same principles hold for extinct species like dinosaurs, although this cannot be tested in full completeness, as fossils are chemically altered postmortem [100].

Mineralized skeletal structures built up of bone typically span three orders of magnitude (Fig. 5.7), whereby the mammals currently define the upper size limit. The concept of an inner skeleton (“endoskeleton”) is also followed out of the phylum chordata, namely with sponges (using either silica or calcium carbonates as mineral, Fig. 5.7) and with phylum echinodermata, based on mineralized tissues containing calcium carbonates (Fig. 5.7). The same holds for certain members of phylum mollusca, namely cephalopods (Fig. 5.7). Alternatively, load-carrying organs may be realized as outer skeletons (exoskeletons), at tiny formats containing silica (e.g., in diatoms), and at larger formats containing calcium carbonates and chitin as organic component (with phylum arthropoda and with all classes of phylum mollusca except for certain cephalopods, see Fig. 5.7).

Fig. 5.7

Overview over elementary components of mineralized tissues and size of inner and outer skeletons built thereof, across the animal kingdom (see also [101,  102,  103,  104])