Bone Mechanics Bone is a hard connective tissue Forms rigid skeleton Yield strain is small < 0.01 Elastic modulus is high (18 GPa) compared with normal working stresses Stress-strain relation is linear in elastic range Strain-rate dependence of stress is minor in normal conditions Bone is frequently approximated as a linear (Hookean) elastic material 2011: This is too long to cover the boundary value solution 19 January 1999 This fits comfortably in one lecture (80 mins). Not time for Bone remodeling which should be a separate lecture. Rather than cram in more stuff, add some explanatory detail to existing material such as: 1. index and/or direct notation form for orthotropic or transversely isotropic strain energy or stress-strain relation (Decided against it jan 2000) 2. Stiffness or compliance matrix for transverse isotropy (added for TI jan 2000) 3. more explanation of boundary conditions in torsion example (looks ok, jan 2000) It would be worth adding simple beam problem as an alternative example that could either be (a) shown if time or (b) shown instead of torsion problem in years when torsion is given as homework or design assignment.

Bone Anisotropy Bone is a composite Bone has organized microstructure mineral matrix collagen fibers Bone has organized microstructure lamellar (layered) Haversian (tubular) trabecular (spongy, fabric-like) Elastic moduli vary with type of loading: tension – compression bending – shear Elastic moduli vary with orientation transverse vs. axial Bone is anisotropic requires more than two elastic constants It is hard to allow enough time for the example but topic 6 is shorter so there is plenty of time in the two lectures combined. Plan to revisit the example the next time before getting into bone remodeling. That worked well this time. However additions are needed: Generic introduction to boundary conditions after Navier’s equations More explanation of torsion problem including a separate slide showing boundary conditions. A more direct derivation? Maybe mainly in polar coords A more detailed discussion of GJ and the moment twist relation plus an analogy with the moment curvature relation and I. Equations for J (and I) for other cross-sections. A more stepwise approach to the derivation with each step on a separate slide.

Constitutive Law for Linear (Hookean) Elasticity Elasticity tensor, Cijkl, a 4th-order tensor of material constants 34 = 81 components symmetry conditions: Tij = Tji εkl = εlk → 6x6 = 36 independent constants Increment of work (strain energy) dW =Tijdεij =Cijklεkldεij ⇒ W = ∫Cijklεkldεij = ½Cijklεklεij = ½Cklijεijεkl ⇒ Cijkl = Cklij → leaves 21 independent constants simplest special case – Isotropy: λ and μ are the Lamé constants

Isotropic Hookean Solids: Technical Constants Measured from standard tests Uniaxial test: Young’s modulus, E = slope of the stress-strain curve Poisson ratio, ν = (-) ratio transverse:axial strain T11 = Eε11=λ(ε11 + ε22 + ε33) + 2με11 = λ(ε11 – νε11 – νε11) + 2με11 → E = λ(1 – 2ν) + 2μ T22 = 0 = λ(ε11+ε22+ε33) + 2με22 = λ(–ε22/ν + ε22 + ε22) + 2με22

Isotropic Hookean Solids: Technical Constants Shear modulus, G = half slope of the shear stress vs. shear strain curve For i ≠ j, Tij = 2μεij ⇒ G = μ Bulk modulus, K = mean stress σ0 divided by volume change, Δ (dilatation)

Stiffness Matrix [cij] Represent the stress and strain tensors as column matrices [σi] = [T11, T22, T33, T23, T13, T12]T [ei] = [ε11, ε22, ε33, 2ε23, 2ε13, 2ε12]T [σi] = [cij][ej] [cij] is the (6x6) stiffness matrix e.g. for isotropic Hookean materials

Compliance Matrix [sij] The inverse of the stiffness matrix [ei] = [sij] [σj] [sij] is the (6x6) compliance matrix e.g. for isotropic Hookean solids, in terms of the technical constants:

Orthotropy bone often assumed to be orthotropic different properties in the three mutually perpendicular directions: 3 Young's moduli; 3 shear moduli; 3 independent Poisson ratios →3 uniaxial tests and 3 plane shear tests structural axes of orthotropic symmetry are defined by bone microstructure Long bone structural axes (1) radial (2) circumferential (3) longitudinal As for isotropy, stiffness matrix has 12 non-zero components, but 9 independent values instead of 2

Orthotropy: Stiffness Matrix Technical constants 3 Young's moduli for uniaxial strain along each axis, Ei 6 Poisson ratios, νij for strain in the j-direction when loaded in the i-direction (i ≠ j) νijEj = νjiEi (no sum) leaving 3 independent Poisson ratios 3 shear moduli, Gij = Gji for shear in the i-j plane

Orthotropy: Compliance Matrix

Transverse Isotropy E1 and E2 are similar compared with E3 Similarly, ν31 and ν32 are close compared with ν21 ⇒ greater differences between axial and transverse directions than between radial and circumferential Transversely isotropic materials one preferred (“fiber”) axis, i.e. long axis of the bone in long bones, the "fibers" are the osteons isotropic properties in plane transverse to fibers stiffness matrix simplifies from 9 to 5 independent constants Et, Ef, 𝜈f, 𝜈t, Gf: c11=c22 c13=c23 c44=c55 c66=0.5(c11-c12)

Technical Constants for Human Bone From SC Cowin, Chapter 2 in Handbook of Bioengineering, 1987

Bone Growth and Remodeling Bone continually remodels growth, reinforcement, resorption depends on stress and strain There is an optimal range of stress for maximum strength understressed or overstressed bone can weaken stresses on fractured bone affect healing stress-dependent remodeling affects surgical implant and prosthesis design, e.g. fracture fixation plates, surgical screws, artificial joints 1978: radiographic evidence of bone resorption seen in 70% of total hip replacement patients 14 January 1999 This fits comfortably in one lecture (80 mins). 5 mins left for a feew more nice pictures but probably enough numbers and facts already. This should be a whole lecture supplemented only with more nice pictures aas they become available.

Stress-Dependent Remodeling Osteoclasts - cells responsible for resorption Osteoblasts - cells responsible for growth compressive stress stimulates formation of new bone and is important for fracture healing loss of normal stress → loss of calcium and reduced bone density Time scales: fastest remodeling is due to change in mineral content healing - weeks remodeling - months/years growth/maturation - years 14 January 1999 This fits comfortably in one lecture (80 mins). 5 mins left for a feew more nice pictures but probably enough numbers and facts already. This should be a whole lecture supplemented only with more nice pictures aas they become available.

Types of Bone Remodeling Two types of remodeling in bone: 1. surface (external) remodeling change in bone shape and dimensions deposition on to or resorption of bone material from inner or outer surfaces 2. internal remodeling change in: bulk density trabecular size orientation osteon size, etc. 14 January 1999 This fits comfortably in one lecture (80 mins). 5 mins left for a feew more nice pictures but probably enough numbers and facts already. This should be a whole lecture supplemented only with more nice pictures aas they become available.

Functional Adaptation and Optimal Design Principal of Functional Adaptation, Roux (1895): “the ability of organs (and cells, tissues and organisms) to adapt their capacity to function in response to altered demands by practice” Functional adaptation in bone is remodeling of structure, shape & mechanical properties in response to altered loading Related to the engineering concept of optimal design, e.g. Theory of Uniform Strength — attempts to produce the same maximum normal stress (brittle material) or shear stress (ductile material) throughout the body for a specific loading Theory of Trajectorial Architecture — concentrates material in the paths of force transmission, such as principal stress lines, e.g. fiber reinforcing of composite (kevlar-mylar) sails Principle of Maximum-Minimum Design — maximize strength for minimum weight or cost 14 January 1999 This fits comfortably in one lecture (80 mins). 5 mins left for a feew more nice pictures but probably enough numbers and facts already. This should be a whole lecture supplemented only with more nice pictures aas they become available.

Stress Adaptation of Trabecular Bone G.H. von Meyer’s trabecular bone architecture in human femur (1867) Principal stress trajectories of Culmann’s crane 14 January 1999 This fits comfortably in one lecture (80 mins). 5 mins left for a few more nice pictures but probably enough numbers and facts already. This should be a whole lecture supplemented only with more nice pictures as they become available.

Remodeling of Trabecular Bone: Wolff's Law Wolff (1872): when loads are changed by trauma or change in activity, functional remodeling reorients bone trabeculae so they align with the new principal stress axes Wolff never actually proved this Wolff's “law of bone transformation” (1884): “there is a perfect mathematical correspondence between the structure of cancellous bone in proximal femur and Culmann’s trajectories” Culmann’s trajectories and other of Wolff’s assertions were suspect, but photoelastic studies (Pauwels,1954) confirmed Wolff's law 14 January 1999 This fits comfortably in one lecture (80 mins). 5 mins left for a feew more nice pictures but probably enough numbers and facts already. This should be a whole lecture supplemented only with more nice pictures aas they become available.

Bone Mechanics: Key Points Under physiological loads, bone can be assumed Hookean elastic with a high elastic modulus (10-20 GPa) The microstructure of the bone composite makes the material response anisotropic. Compared with an isotropic Hookean elastic solid which has two independent technical constants, transversely isotropic linearly elastic solids have five independent elastic constants and orthotropic Hookean solids have nine. For human cortical bone orthotropy is a somewhat better assumption than transverse isotropy, but transverse isotropy is a much better approximation than isotropy. Internal remodeling results in altered bone properties External remodeling results in altered bone size or shape Bone growth and remodeling is stress-adaptive Wolff’s law described how trabecular bone reorients when principal stress axes change