Micro computed tomography studies of biomaterials and bone

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Appendix A

Appendix A: Elasticity

dx

dy

dz

τ

σ

x

y

z

xx

σ

xz zx

zz

zy

yz

yx xy

[image:1.595.146.486.281.583.2]

yy

Figure A.1:

A.1 Generalised Hooke’s Law for Linear Elastic Solids

If a linear elastic solid is subjected to a strain,ǫkl, the material will develop an internal

stress, denoted asσij. The stress tensor,σij, is a second rank symmetric tensor and is

defined in 3D cartesian coordinates by the relation

dFi = 3

X

j=1

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where dFi are the components of a resultant force vector acting over a differential

area, which is represented by the vectordAj. The stress tensor can be represented as

a matrix, where we have replaced the cartesian x, y, z basis with indices 1, 2, 3, as

  

σ11 τ12 τ13 τ21 σ22 τ23 τ31 τ32 σ33

 

. (A.2)

The first index specifies the direction in which the stress component acts, and the second index specifies the orientation of the surface upon which it acts (Figure A.1). For an infinitesimal volume element there can no torque, which implies τ12 = τ21,

τ13=τ31, andτ23=τ32.

The strain tensor, ǫkl, is also a symmetric second rank tensor. The tensor can be

constructed from the symmetric part of the Jacobian matrix of the displacement field,

u:

ǫkl=

1 2(

∂uk ∂xl

+ ∂ul

∂xk

). (A.3)

The stress and strain are related by a 4th rank tensor, Cijkl, which is written as the

generalised form of Hooke’s Law,

σij =Cijklǫkl, (A.4)

or alternatively as

ǫij =Sijklǫkl, (A.5)

where Cijkl is known as the stiffness or elasticity tensor, and Sijkl is known as the

compliance tensor.

Since bothǫij andσklhave 9 components, we can expect thatCijklto have 81

com-ponents. However, due to symmetries in the stress and strain tensors, and the thermo-dynamic consideration of reversible stress-strain, the number of independent compo-nents in the stiffness tensor can be reduced from 81 to 21[40]. The generalised stress-strain relationship can be written in matrix form, where the components of stress and strain are written as column vectors.

           σ1 σ2 σ3 σ4 σ5 σ6            =           

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§A.1 Generalised Hooke’s Law for Linear Elastic Solids 167

The stiffness in any direction can be obtained by expressing the stiffness tensor in rotated coordinates

Cpqrs=RipRjqRkrRlsCijkl, (A.7)

where R is a rotation matrix.

For isotropic linear elastic solids the stiffness matrix can be reduced to just 2 inde-pendent components [57],λandµ;

σij = 2µǫij +λtr(ǫij)δij, (A.8)

whereλ,µare the Lam´e parameters and tr(ǫij)is the trace of the strain tensor, andδij

is the Kronecker delta. We can also write the stress tensor as a linear combination of a constant tensor and symmetric traceless tensor. We can weight these respective tensor by introducing the bulk modulus, K, and the shear modulus, G:

σij = 3K(

1

3ǫkkδij) + 2G(ǫij− 1

3ǫkkδij). (A.9)

The bulk modulus of a substance, K, is thus defined as the hydrostatic pressure in-crease required to impart a relative dein-crease in the volume of the substance. The shear modulus, G, is defined as the ratio of the shear stress to shear strain.

K=−V∂P

∂V , (A.10)

G= F/A

∆x/h. (A.11)

The definitions of bulk and shear moduli are illustrated in Figure A.2.

Other related elastic moduli include Young’s modulus, E, which is the ratio of extensional stress to extensional strain. Young’s modulus is related to the bulk and shear moduli with the following equations:

K= E

3(1−2ν), (A.12)

G= E

2(1 +ν), (A.13)

whereν is the Poisson’s ratio, or the relative transverse strain divided by the relative axial strain

νyx=− ǫx ǫy

. (A.14)

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[image:4.595.76.478.100.302.2]

[a] [b]

Figure A.2:[a] Bulk modulus is a constant that relates the pressure required to impart a rela-tive change in a materials volume. [b] Shear modulus is the constant relating the shear stress (or surface force) to its shear strain.

isotropic materials Poisson’s ratio is the same in all directions. That is,νxy =νyx=νxy.

A.2 Finite Element Formulation

Consider an 8-noded cubic hexahedral element as shown in Figure A.3. Cubic ele-ments have the advantage of being easily applied to the coordinate system of tomo-graphic voxelated images. A displacement vectorU(x, y, z) at any point within the element can be found by interpolating the 8 nodal displacements, denoted asuij. The

displacement vector field inside the element has components U1(x, y, z),U2(x, y, z), U3(x, y, z).

U1(x, y, z) =N_ru1_r, (A.15)

U2(x, y, z) =Nru2r, (A.16)

U3(x, y, z) =Nru3r, (A.17)

where Nr is a 3×8 matrix linking the nodal displacements with the local

displace-ments within the voxel. This can be written in tensor notation:

Ui(x, y, z) =Nijkujk, (A.18)

whereNijk is a matrix of shape functions andujk is the k’th component of

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[image:5.595.269.367.112.242.2]

§A.2 Finite Element Formulation 169 r Nr 1 (1-x)(1-y)(1-z) 2 x(1-y)(1-z) 3 xy(1-z) 4 (1-x)y(1-z) 5 (1-x)(1-y)z 6 x(1-y)z 7 xyz 8 (1-x)yz

Table A.1: Shape functions for individual element

shape functions are quadratic in x,y,z and are given below in Table A.1.

To calculate the strain at a point within the voxel in terms of nodal displacements were operate with a matrix of derivatives. This matrix, L, a 6×3 matrix, links the displacement to the strain;

L=            ∂

∂x 0 0

0 _∂y∂ 0 0 0 _∂z∂

∂

∂z 0 ∂x∂

0 _∂z∂ _∂y∂

∂

∂y ∂x∂ 0

           . (A.19)

Using the L operator matrix and the definition of strain as the differential change in displacement,uas a function of the coordinate axis,x,

ǫ= ∂u

∂x, (A.20)

we can write

ǫα=LαiNijkujk (A.21)

ǫα=Sαjkujk (A.22)

whereSαjkis a matrix linking the nodal displacements to a strain 6-vector within each

voxel.

To evaluate the potential energy,E, stored within a differential element,dV, that is composed of an elastic solid with stiffness tensorCijkl, we can write:

E= 1 2

Z

ǫijCijklǫkldV. (A.23)

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[image:6.595.70.479.200.605.2]

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§A.2 Finite Element Formulation 171

Equation A.23:

E = 1 2

Z

[Sαjkujk]TCαβ[Sβsqusq]dV, (A.24)

by grouping theS andCmatrices together

E = 1 2u

T

rpDrpsqusq, (A.25)

where

Drpsq =

Z

[Sαjk]TCαβ[Sβsq. (A.26)

The stiffness matrix,D, is evaluated for each pixel by applying Simpson’s rule. The set of nodal displacements that minimise the strain energy is found by taking the partial derivative of the strain energy equation with respect to each nodal displacement. A conjugate gradient method is used to find the set of displacements which satisfy the following equation:

∂E

(8)

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Appendix B

Appendix B: Elasticity Tensors

B.1 Hip10R

Input C matrix:

0.8824 0.2402 0.2794 0.01286 -0.04285 0.02213 0.254 0.8018 0.3282 -0.005792 -0.1503 0.01408 0.2868 0.3219 1.08 -0.001149 -0.1802 0.001391 0.01017 -0.004411 0.001862 0.2688 0.007612 -0.058 -0.03181 -0.1336 -0.1756 0.006861 0.323 -0.003389 0.0253 0.01835 0.004187 -0.05865 -0.003467 0.2136

B.1.1 Orthotropic Approximation

Rotated C matrix:

0.8006 0.1473 0.4162 -0.03756 -0.001735 0.01097 0.1473 0.7917 0.2326 -0.05345 0.008713 0.02016 0.4162 0.2326 1.261 0.02614 0.002317 0.01114 -0.03756 -0.05345 0.02614 0.2784 0.009547 -0.004221 -0.001735 0.008713 0.002317 0.009547 0.3056 -0.008297 0.01097 0.02016 0.01114 -0.004221 -0.008297 0.1765 Orthotropic approximation:

0.8006 0.1473 0.4162 0 0 0 0.1473 0.7917 0.2326 0 0 0 0.4162 0.2326 1.261 0 0 0

0 0 0 0.2784 0 0

0 0 0 0 0.3056 0

0 0 0 0 0 0.1765

Deviation from Orthotropic = 7.572 percent

B.1.2 Transverse Isotropic Approximation

Rotated C matrix:

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TI approximation:

0.7184 0.2184 0.3288 0 0 0 0.2184 0.7184 0.3288 0 0 0 0.3288 0.3288 1.256 0 0 0

0 0 0 0.2954 0 0

0 0 0 0 0.2954 0

0 0 0 0 0 0.25

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§B.2 11R 175

B.2 11R

Input C matrix:

0.4495 0.1329 0.1555 -0.008464 -0.0141 0.01077 0.1359 0.4518 0.1828 -0.002898 -0.06472 0.008059 0.1586 0.1767 0.765 -0.02482 -0.09741 0.008499 -0.01294 -0.002887 -0.02014 0.1532 0.01083 -0.02322 -0.009585 -0.05099 -0.08988 0.01018 0.1831 -0.008721 0.01501 0.01311 0.008501 -0.02318 -0.00865 0.1051

Rotated C matrix:

0.4512 0.1068 0.1993 0.02875 -0.0004718 -0.005337 0.1068 0.4418 0.1318 0.002107 -0.006987 -0.009064 0.1993 0.1318 0.8278 -0.0086 -0.0002021 -0.006409 0.02875 0.002107 -0.0086 0.1587 -0.01368 0.0002324 -0.0004718 -0.006987 -0.0002021 -0.01368 0.1598 0.004928 -0.005337 -0.009064 -0.006409 0.0002324 0.004928 0.09562 Orthotropic approximation:

0.4512 0.1068 0.1993 0 0 0 0.1068 0.4418 0.1318 0 0 0 0.1993 0.1318 0.8278 0 0 0

0 0 0 0.1587 0 0

0 0 0 0 0.1598 0

0 0 0 0 0 0.09562

Distance from Orthotropic = 6.443 percent

Rotated C matrix:

0.4279 0.1267 0.1921 0.02658 0.007688 -0.03937 0.1267 0.431 0.1382 0.006987 -0.008648 0.02488 0.1921 0.1382 0.8276 -0.002793 -0.007124 -0.02478 0.02658 0.006987 -0.002793 0.1641 -0.01155 -0.0103 0.007688 -0.008648 -0.007124 -0.01155 0.1542 0.001471 -0.03937 0.02488 -0.02478 -0.0103 0.001471 0.1129 TI approximation:

0.4102 0.146 0.1651 0 0 0 0.146 0.4102 0.1651 0 0 0 0.1651 0.1651 0.8276 0 0 0

0 0 0 0.1592 0 0

0 0 0 0 0.1592 0

0 0 0 0 0 0.1321

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B.3 15R

Input C matrix:

0.9627 0.2737 0.3082 0.009797 -0.01364 0.008302 0.2927 1.021 0.3583 -0.01444 -0.07988 -0.005164 0.2867 0.316 1.265 -0.009204 -0.108 -0.002382 0.002626 -0.002682 -0.005403 0.3072 -0.0009517 -0.02439 -0.01029 -0.05257 -0.07873 -0.0003431 0.3067 -0.01303 0.0181 0.01036 0.003652 -0.024 -0.0137 0.2131

Rotated C matrix:

1.02 0.2552 0.3356 -0.04015 -0.007206 0.006666 0.2552 0.9467 0.2676 -0.02909 -0.0004952 0.01186 0.3356 0.2676 1.308 0.002877 0.005067 0.008662 -0.04015 -0.02909 0.002877 0.2941 0.002358 -0.01208 -0.007206 -0.0004952 0.005067 0.002358 0.3136 -0.003175 0.006666 0.01186 0.008662 -0.01208 -0.003175 0.2064 Orthotropic approximation:

1.02 0.2552 0.3356 0 0 0 0.2552 0.9467 0.2676 0 0 0 0.3356 0.2676 1.308 0 0 0

0 0 0 0.2941 0 0

0 0 0 0 0.3136 0

0 0 0 0 0 0.2064

Rotated C matrix:

0.9787 0.2977 0.3293 0.04703 -0.0193 0.06793 0.2977 0.9021 0.2718 0.01628 -0.0004479 -0.06861 0.3293 0.2718 1.303 0.006819 -0.02349 0.01591 0.04703 0.01628 0.006819 0.2936 0.007028 -0.004966 -0.0193 -0.0004479 -0.02349 0.007028 0.3104 0.002313 0.06793 -0.06861 0.01591 -0.004966 0.002313 0.2558 TI approximation:

0.9076 0.3305 0.3006 0 0 0 0.3305 0.9076 0.3006 0 0 0 0.3006 0.3006 1.303 0 0 0

0 0 0 0.302 0 0

0 0 0 0 0.302 0

0 0 0 0 0 0.2886

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§B.4 169R 177

B.4 169R

Input C matrix:

1.068 0.2983 0.3633 0.006548 -0.02199 0.05719 0.3734 1.113 0.468 -0.01151 -0.1207 0.04163 0.4406 0.4494 1.774 0.001031 -0.1485 0.01097 -0.004813 -0.004812 -0.01936 0.3645 0.02128 -0.0339 -0.01199 -0.082 -0.1186 0.02092 0.402 -0.01409 0.04516 0.03792 0.009899 -0.03751 -0.0139 0.2572

Rotated C matrix:

1.112 0.2742 0.478 -0.04554 -0.00651 0.02489 0.2742 1.059 0.404 0.01106 0.0004245 0.04362 0.478 0.404 1.83 0.01459 0.005668 0.01049 -0.04554 0.01106 0.01459 0.3771 0.02507 -0.006058 -0.00651 0.0004245 0.005668 0.02507 0.3764 -0.01342 0.02489 0.04362 0.01049 -0.006058 -0.01342 0.2467 Orthotropic approximation:

1.112 0.2742 0.478 0 0 0 0.2742 1.059 0.404 0 0 0 0.478 0.404 1.83 0 0 0

0 0 0 0.3771 0 0

0 0 0 0 0.3764 0

0 0 0 0 0 0.2467

Rotated C matrix:

1.122 0.2809 0.4749 0.05274 -0.01897 -0.01114 0.2809 1.046 0.4044 -0.01404 -0.002689 -0.05595 0.4749 0.4044 1.828 -0.003504 -0.03622 0.002019 0.05274 -0.01404 -0.003504 0.3735 -0.0196 -0.009373 -0.01897 -0.002689 -0.03622 -0.0196 0.3768 0.01574 -0.01114 -0.05595 0.002019 -0.009373 0.01574 0.2531 TI approximation:

1.01 0.3551 0.4397 0 0 0 0.3551 1.01 0.4397 0 0 0 0.4397 0.4397 1.828 0 0 0

0 0 0 0.3751 0 0

0 0 0 0 0.3751 0

0 0 0 0 0 0.3273

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B.5 177R

Input C matrix:

0.7036 0.201 0.2266 -0.006218 -0.02282 0.008248 0.2004 0.7293 0.2565 -0.008609 -0.104 -0.005156 0.2214 0.2524 0.9423 -0.02608 -0.1136 0.004009 -0.00583 -0.005474 -0.02176 0.2154 0.004889 -0.03127 -0.0144 -0.09114 -0.1052 0.004921 0.2621 -0.009929 0.008493 -0.001813 0.005833 -0.03234 -0.009883 0.1893

Rotated C matrix:

0.7289 0.1417 0.308 0.04107 -0.001346 -0.008224 0.1417 0.6727 0.1915 0.0197 -0.001609 -0.005529 0.308 0.1915 1.029 -0.02342 0.000871 0.009612 0.04107 0.0197 -0.02342 0.2384 0.009273 0.001073 -0.001346 -0.001609 0.000871 0.009273 0.2317 0.0106 -0.008224 -0.005529 0.009612 0.001073 0.0106 0.1689 Orthotropic approximation:

0.7289 0.1417 0.308 0 0 0 0.1417 0.6727 0.1915 0 0 0 0.308 0.1915 1.029 0 0 0

0 0 0 0.2384 0 0

0 0 0 0 0.2317 0

0 0 0 0 0 0.1689

Rotated C matrix:

0.6477 0.1994 0.2476 0 0 0 0.1994 0.6477 0.2476 0 0 0 0.2476 0.2476 1.031 0 0 0

0 0 0 0.2336 0 0

0 0 0 0 0.2336 0

0 0 0 0 0 0.2241

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§B.6 17R 179

B.6 17R

Input C matrix:

1.086 0.3224 0.3649 0.002533 -0.02698 -0.01932 0.327 1.033 0.3934 0.001088 -0.1016 -0.02227 0.3656 0.3864 1.484 -0.0075 -0.1294 -0.002912 -0.006747 0.005666 -0.005687 0.3001 -0.004083 -0.0314 -0.01424 -0.08388 -0.1231 -0.002716 0.3661 0.001779 -0.008447 -0.008201 -0.00481 -0.03489 0.001796 0.2623

Rotated C matrix:

1.032 0.2815 0.4271 0.05249 0.007399 0.007887 0.2815 1.058 0.3402 0.023 -0.006247 0.007927 0.4271 0.3402 1.552 -0.01794 -0.0001416 0.007197 0.05249 0.023 -0.01794 0.3459 0.003262 0.002066 0.007399 -0.006247 -0.0001416 0.003262 0.3151 0.01906 0.007887 0.007927 0.007197 0.002066 0.01906 0.2472 Orthotropic approximation:

1.032 0.2815 0.4271 0 0 0 0.2815 1.058 0.3402 0 0 0 0.4271 0.3402 1.552 0 0 0

0 0 0 0.3459 0 0

0 0 0 0 0.3151 0

0 0 0 0 0 0.2472

Rotated C matrix:

1.032 0.2834 0.4251 0.05519 0.00677 0.008498 0.2834 1.06 0.339 0.02249 -0.006562 0.006622 0.4251 0.339 1.553 -0.008107 -0.003089 0.007758 0.05519 0.02249 -0.008107 0.3445 0.003722 0.001795 0.00677 -0.006562 -0.003089 0.003722 0.3145 0.0203 0.008498 0.006622 0.007758 0.001795 0.0203 0.2479 TI approximation:

0.9795 0.3502 0.3821 0 0 0 0.3502 0.9795 0.3821 0 0 0 0.3821 0.3821 1.553 0 0 0

0 0 0 0.3295 0 0

0 0 0 0 0.3295 0

0 0 0 0 0 0.3147

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B.7 180R 2K

Input C matrix:

1.433 0.4783 0.5117 -0.002933 -0.01629 -0.01454 0.51 1.5 0.5703 0.0003395 -0.1177 -0.01913 0.5471 0.5737 1.755 -0.01905 -0.1319 0.003712 0.00391 -0.0009571 -0.009427 0.4571 0.002731 -0.04298 -0.02331 -0.1229 -0.136 0.002652 0.5031 -0.00614 -0.0144 -0.01961 0.001362 -0.04311 -0.005997 0.4056

Rotated C matrix:

1.502 0.4038 0.6478 -0.04876 0.009745 -0.01247 0.4038 1.411 0.5038 -0.00768 0.0106 -0.01788 0.6478 0.5038 1.858 0.03014 -0.007202 0.003493 -0.04876 -0.00768 0.03014 0.4615 -0.001909 0.0002716 0.009745 0.0106 -0.007202 -0.001909 0.48 -0.01035 -0.01247 -0.01788 0.003493 0.0002716 -0.01035 0.3825 Orthotropic approximation:

1.502 0.4038 0.6478 0 0 0 0.4038 1.411 0.5038 0 0 0 0.6478 0.5038 1.858 0 0 0

0 0 0 0.4615 0 0

0 0 0 0 0.48 0

0 0 0 0 0 0.3825

Rotated C matrix:

1.504 0.41 0.6428 0.05544 0.01583 -0.004208 0.41 1.407 0.5028 0.007156 0.01025 0.02984 0.6428 0.5028 1.861 -0.01506 -0.003214 -0.01421 0.05544 0.007156 -0.01506 0.4597 -0.001258 -0.002492 0.01583 0.01025 -0.003214 -0.001258 0.4785 0.01249 -0.004208 0.02984 -0.01421 -0.002492 0.01249 0.3856 TI approximation:

1.387 0.4786 0.5728 0 0 0 0.4786 1.387 0.5728 0 0 0 0.5728 0.5728 1.861 0 0 0

0 0 0 0.4691 0 0

0 0 0 0 0.4691 0

0 0 0 0 0 0.4542

Deviation from TI = 8.421 percent

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§B.8 180R 2K 222 181

B.8 180R 2K 222

Input C matrix:

0.9955 0.3275 0.3602 0.00289 -0.02607 -0.01348 0.3295 1.054 0.4066 -0.00308 -0.1255 -0.02141 0.3629 0.4071 1.388 -0.005914 -0.1405 0.0008195 0.003487 -0.005109 -0.007428 0.3613 0.0001235 -0.03898 -0.01979 -0.1223 -0.145 -6.144e-05 0.4137 -0.005453 -0.005815 -0.0204 0.00209 -0.03891 -0.005489 0.2997

Rotated C matrix:

1.057 0.2494 0.4849 -0.05089 0.00742 -0.01502 0.2494 0.995 0.3032 0.01438 0.005254 -0.01177 0.4849 0.3032 1.505 0.02475 -0.004775 0.007238 -0.05089 0.01438 0.02475 0.355 -0.00172 -0.003038 0.00742 0.005254 -0.004775 -0.00172 0.3799 -0.002395 -0.01502 -0.01177 0.007238 -0.003038 -0.002395 0.2806 Orthotropic approximation:

1.057 0.2494 0.4849 0 0 0 0.2494 0.995 0.3032 0 0 0 0.4849 0.3032 1.505 0 0 0

0 0 0 0.355 0 0

0 0 0 0 0.3799 0

0 0 0 0 0 0.2806

Rotated C matrix:

1.057 0.2528 0.4814 0.05769 0.00866 0.01268 0.2528 0.993 0.3029 -0.01393 0.004949 0.0138 0.4814 0.3029 1.507 -0.009817 -0.003507 -0.009431 0.05769 -0.01393 -0.009817 0.3545 0.0008431 -0.004073 0.00866 0.004949 -0.003507 0.0008431 0.3796 0.005195 0.01268 0.0138 -0.009431 -0.004073 0.005195 0.2809 TI approximation:

0.9725 0.3055 0.3921 0 0 0 0.3055 0.9725 0.3921 0 0 0 0.3921 0.3921 1.507 0 0 0

0 0 0 0.3671 0 0

0 0 0 0 0.3671 0

0 0 0 0 0 0.3335

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B.9 180R 2K 444

Input C matrix:

0.5062 0.1334 0.1774 0.01135 -0.008148 0.001146 0.1272 0.5547 0.2189 -0.001341 -0.09413 -0.00296 0.1916 0.238 0.9785 -0.0008432 -0.1308 0.005834 0.01958 -0.003241 0.0006309 0.2481 0.007618 -0.02172 0.0008535 -0.07066 -0.1135 0.009456 0.2917 -0.00519 0.004677 0.005463 0.01057 -0.02158 -0.005167 0.1839

Rotated C matrix:

0.5544 0.1024 0.2698 -0.02606 -0.00196 0.001071 0.1024 0.5374 0.1207 0.04749 0.02235 0.01261 0.2698 0.1207 1.068 0.003912 -0.001357 0.007785 -0.02606 0.04749 0.003912 0.2307 0.001115 0.001084 -0.00196 0.02235 -0.001357 0.001115 0.2548 0.002109 0.001071 0.01261 0.007785 0.001084 0.002109 0.1778 Orthotropic approximation:

0.5544 0.1024 0.2698 0 0 0 0.1024 0.5374 0.1207 0 0 0 0.2698 0.1207 1.068 0 0 0

0 0 0 0.2307 0 0

0 0 0 0 0.2548 0

0 0 0 0 0 0.1778

Distance from Orthotropic = 7.617 percent

Rotated C matrix:

0.5547 0.1043 0.2671 0.03364 -0.006024 -0.003159 0.1043 0.5305 0.1281 -0.0455 0.02124 -0.0094 0.2671 0.1281 1.065 0.01686 -0.006728 -0.009965 0.03364 -0.0455 0.01686 0.2359 -0.002004 -0.001234 -0.006024 0.02124 -0.006728 -0.002004 0.2548 0.001788 -0.003159 -0.0094 -0.009965 -0.001234 0.001788 0.1775 TI approximation:

0.5218 0.1251 0.1976 0 0 0 0.1251 0.5218 0.1976 0 0 0 0.1976 0.1976 1.065 0 0 0

0 0 0 0.2453 0 0

0 0 0 0 0.2453 0

0 0 0 0 0 0.1983

(19)

§B.10 180R 2K 888 183

B.10 180R 2K 888

Input C matrix:

0.3387 0.06879 0.1095 0.01504 -0.0006781 0.001039 0.06327 0.3589 0.148 -0.00701 -0.05848 0.00475 0.08743 0.1208 0.5621 0.002797 -0.0623 0.00867 0.01589 -0.007548 0.001783 0.1886 0.00657 -0.01707 0.006104 -0.04194 -0.05076 0.006583 0.2162 -0.001532 0.003062 0.004994 0.009058 -0.01707 -0.003097 0.142

Rotated C matrix:

0.3582 0.04848 0.1438 0.009387 -0.0003845 0.0009753 0.04848 0.3853 0.03107 -0.03735 -0.01655 0.01249 0.1438 0.03107 0.6175 -0.004882 0.00197 0.006568 0.009387 -0.03735 -0.004882 0.1648 -0.0008508 -0.002159 -0.0003845 -0.01655 0.00197 -0.0008508 0.1938 -0.006214 0.0009753 0.01249 0.006568 -0.002159 -0.006214 0.1375 Orthotropic approximation:

0.3582 0.04848 0.1438 0 0 0 0.04848 0.3853 0.03107 0 0 0 0.1438 0.03107 0.6175 0 0 0

0 0 0 0.1648 0 0

0 0 0 0 0.1938 0

0 0 0 0 0 0.1375

Rotated C matrix:

0.3595 0.05303 0.1368 0.02607 -0.01074 -0.002451 0.05303 0.3565 0.05352 -0.03414 0.01841 -0.006999 0.1368 0.05352 0.6051 0.03098 -0.00391 -0.007216 0.02607 -0.03414 0.03098 0.1852 -0.002833 0.001074 -0.01074 0.01841 -0.00391 -0.002833 0.1938 0.004468 -0.002451 -0.006999 -0.007216 0.001074 0.004468 0.1371 TI approximation:

0.3503 0.06073 0.09518 0 0 0 0.06073 0.3503 0.09518 0 0 0 0.09518 0.09518 0.6051 0 0 0

0 0 0 0.1895 0 0

0 0 0 0 0.1895 0

0 0 0 0 0 0.1448

(20)

B.11 180R 1K

Input C matrix:

0.795 0.2293 0.2507 -0.002195 -0.02393 -0.008778 0.2368 0.8276 0.288 -0.004045 -0.1049 -0.02024 0.2546 0.2835 1.065 -0.005396 -0.1222 -0.0005557 -0.003249 -0.00261 -0.006657 0.2437 -0.000505 -0.03266 -0.01442 -0.09039 -0.124 -0.000712 0.3067 -0.003193 -0.003076 -0.01093 0.003795 -0.03293 -0.002385 0.2051

Rotated C matrix:

0.8287 0.1718 0.3409 -0.04136 0.003487 -0.008722 0.1718 0.768 0.2167 -0.01374 -0.001283 -0.008263 0.3409 0.2167 1.161 0.02033 -0.001152 0.004315 -0.04136 -0.01374 0.02033 0.272 -0.0001663 0.00157 0.003487 -0.001283 -0.001152 -0.0001663 0.2613 -0.007962 -0.008722 -0.008263 0.004315 0.00157 -0.007962 0.1871 Orthotropic approximation:

0.8287 0.1718 0.3409 0 0 0 0.1718 0.768 0.2167 0 0 0 0.3409 0.2167 1.161 0 0 0

0 0 0 0.272 0 0

0 0 0 0 0.2613 0

0 0 0 0 0 0.1871

Rotated C matrix:

0.8283 0.1741 0.3387 0.04597 0.002644 0.008055 0.1741 0.7701 0.215 0.01358 -0.002226 0.007808 0.3387 0.215 1.163 -0.009554 -0.00579 -0.003767 0.04597 0.01358 -0.009554 0.2702 0.001417 -8.189e-05 0.002644 -0.002226 -0.00579 0.001417 0.261 0.01014 0.008055 0.007808 -0.003767 -8.189e-05 0.01014 0.1875 TI approximation:

0.7367 0.2366 0.2769 0 0 0 0.2366 0.7367 0.2769 0 0 0 0.2769 0.2769 1.163 0 0 0

0 0 0 0.2656 0 0

0 0 0 0 0.2656 0

0 0 0 0 0 0.2501

(21)

§B.12 19R 185

B.12 19R

Input C matrix:

0.4735 0.1325 0.156 0.01282 -0.0183 0.002054 0.1383 0.4906 0.1927 0.002911 -0.07842 -0.00457 0.1554 0.1851 0.7736 0.009426 -0.09968 0.0001561 0.01034 0.003393 0.009299 0.1627 -0.0004501 -0.02513 -0.005985 -0.06563 -0.09272 -0.0004355 0.1946 0.004522 -0.0006476 -0.004145 5.939e-05 -0.02521 0.004314 0.1245

Rotated C matrix:

0.4902 0.09658 0.2198 -0.03517 0.002075 0.002188 0.09658 0.4649 0.1244 -6.991e-05 0.007226 0.002841 0.2198 0.1244 0.8351 0.0147 -0.001038 0.0009265 -0.03517 -6.991e-05 0.0147 0.1687 -0.004442 8.328e-05 0.002075 0.007226 -0.001038 -0.004442 0.1742 -0.008123 0.002188 0.002841 0.0009265 8.328e-05 -0.008123 0.1127 Orthotropic approximation:

0.4902 0.09658 0.2198 0 0 0 0.09658 0.4649 0.1244 0 0 0 0.2198 0.1244 0.8351 0 0 0

0 0 0 0.1687 0 0

0 0 0 0 0.1742 0

0 0 0 0 0 0.1127

Rotated C matrix:

0.4836 0.1071 0.2159 0.03709 -0.006711 0.02202 0.1071 0.456 0.126 0.00232 0.01012 -0.02299 0.2159 0.126 0.8359 -0.005026 -0.0008391 0.01404 0.03709 0.00232 -0.005026 0.1695 0.004943 0.007281 -0.006711 0.01012 -0.0008391 0.004943 0.1722 0.007101 0.02202 -0.02299 0.01404 0.007281 0.007101 0.1212 TI approximation:

0.4397 0.1371 0.171 0 0 0 0.1371 0.4397 0.171 0 0 0 0.171 0.171 0.8359 0 0 0

0 0 0 0.1709 0 0

0 0 0 0 0.1709 0

0 0 0 0 0 0.1513

(22)

B.13 45R

Input C matrix:

1.164 0.3697 0.3965 -0.03202 -0.02027 -0.02704 0.3706 1.234 0.4354 -0.007053 -0.09989 -0.02099 0.3353 0.371 1.527 -0.0474 -0.1395 -0.005261 -0.009897 0.0087 -0.007838 0.3131 -0.0002889 -0.03105 -0.01163 -0.1008 -0.1407 0.0003878 0.3212 -0.004037 -0.01589 -0.01388 -0.00073 -0.03391 -0.01039 0.2692

Rotated C matrix:

1.233 0.3226 0.4151 -0.08922 0.008054 -0.01592 0.3226 1.129 0.3221 -0.06285 -0.005249 -0.02115 0.4151 0.3221 1.595 -0.003182 -2.136e-05 0.00139 -0.08922 -0.06285 -0.003182 0.306 0.003316 -0.004477 0.008054 -0.005249 -2.136e-05 0.003316 0.3246 -0.02021 -0.01592 -0.02115 0.00139 -0.004477 -0.02021 0.2566 Orthotropic approximation:

1.233 0.3226 0.4151 0 0 0 0.3226 1.129 0.3221 0 0 0 0.4151 0.3221 1.595 0 0 0

0 0 0 0.306 0 0

0 0 0 0 0.3246 0

0 0 0 0 0 0.2566

Rotated C matrix:

1.235 0.333 0.4119 0.09087 0.01676 -0.00916 0.333 1.126 0.317 0.05763 -0.00468 0.03814 0.4119 0.317 1.594 0.02055 -0.001153 -0.007007 0.09087 0.05763 0.02055 0.3031 -0.00382 -0.008291 0.01676 -0.00468 -0.001153 -0.00382 0.3236 0.0202 -0.00916 0.03814 -0.007007 -0.008291 0.0202 0.2613 TI approximation:

1.099 0.4143 0.3644 0 0 0 0.4143 1.099 0.3644 0 0 0 0.3644 0.3644 1.594 0 0 0

0 0 0 0.3133 0 0

0 0 0 0 0.3133 0

0 0 0 0 0 0.3426

(23)

§B.14 72R 187

B.14 72R

Input C matrix:

0.4519 0.1273 0.1478 0.02032 -0.00984 0.01237 0.1301 0.4158 0.172 0.004807 -0.04935 0.01639 0.142 0.1618 0.6828 0.02374 -0.072 0.002288 0.01121 0.005004 0.02528 0.09475 -0.003 -0.01104 -0.002744 -0.031 -0.05617 0.003623 0.151 0.003941 0.008483 0.008139 -0.001517 -0.01697 0.004096 0.1008

Rotated C matrix:

0.4447 0.1103 0.1273 -0.007052 0.008575 0.01201 0.1103 0.4145 0.1726 -0.006465 0.02042 0.01341 0.1273 0.1726 0.7164 -0.001597 -0.003431 0.0002339 -0.007052 -0.006465 -0.001597 0.09973 0.0004376 0.01643 0.008575 0.02042 -0.003431 0.0004376 0.1396 0.00183 0.01201 0.01341 0.0002339 0.01643 0.00183 0.09464 Orthotropic approximation:

0.4447 0.1103 0.1273 0 0 0 0.1103 0.4145 0.1726 0 0 0 0.1273 0.1726 0.7164 0 0 0

0 0 0 0.09973 0 0

0 0 0 0 0.1396 0

0 0 0 0 0 0.09464

Rotated C matrix:

0.4159 0.1375 0.171 0.01958 -0.005527 0.01501 0.1375 0.4048 0.1282 0.01258 0.009749 -0.04475 0.171 0.1282 0.7161 0.001259 0.004875 0.01241 0.01958 0.01258 0.001259 0.1364 -0.008968 0.008081 -0.005527 0.009749 0.004875 -0.008968 0.1025 0.007767 0.01501 -0.04475 0.01241 0.008081 0.007767 0.1145 TI approximation:

0.3994 0.1485 0.1496 0 0 0 0.1485 0.3994 0.1496 0 0 0 0.1496 0.1496 0.7161 0 0 0

0 0 0 0.1194 0 0

0 0 0 0 0.1194 0

0 0 0 0 0 0.1254

(24)

B.15 9R

Input C matrix:

0.5317 0.1478 0.1787 -0.009798 -0.01679 0.01048 0.1535 0.5031 0.194 -0.006151 -0.05949 0.004017 0.1772 0.1857 0.7669 -0.01821 -0.083 0.002232 -0.008323 -0.002527 -0.01374 0.1685 0.004785 -0.02018 -0.01322 -0.04908 -0.07403 0.004939 0.1814 -0.005809 0.00798 0.004341 0.003012 -0.02084 -0.005716 0.1325

Rotated C matrix:

0.5022 0.1226 0.2084 0.02947 0.0006607 0.0003617 0.1226 0.512 0.1622 0.01817 -0.004214 -0.001464 0.2084 0.1622 0.8146 -0.009763 0.0009604 -0.003396 0.02947 0.01817 -0.009763 0.1695 -0.007706 -0.001612 0.0006607 -0.004214 0.0009604 -0.007706 0.1759 0.00685 0.0003617 -0.001464 -0.003396 -0.001612 0.00685 0.1236 Orthotropic approximation:

0.5022 0.1226 0.2084 0 0 0 0.1226 0.512 0.1622 0 0 0 0.2084 0.1622 0.8146 0 0 0

0 0 0 0.1695 0 0

0 0 0 0 0.1759 0

0 0 0 0 0 0.1236

Rotated C matrix:

0.4733 0.1574 0.1849 0 0 0 0.1574 0.4733 0.1849 0 0 0 0.1849 0.1849 0.8145 0 0 0

0 0 0 0.1724 0 0

0 0 0 0 0.1724 0

0 0 0 0 0 0.1579

(25)

§B.16 A08R 189

B.16 A08R

Input C matrix:

0.4378 0.1374 0.1553 0.01434 -0.01203 0.006874 0.1405 0.4905 0.1876 0.001922 -0.07913 0.001362 0.1535 0.182 0.6686 0.01687 -0.09635 0.0001238 0.007175 0.001057 0.01361 0.1447 0.0005581 -0.02289 -0.006377 -0.06491 -0.08866 0.0003136 0.1805 0.002407 0.006367 0.003531 -0.002066 -0.02328 0.00208 0.1277

Rotated C matrix:

0.4908 0.09726 0.2196 0.03067 -0.003167 -0.006575 0.09726 0.432 0.1172 0.002339 0.003862 -0.00967 0.2196 0.1172 0.7412 -0.01455 0.0008014 -0.0001042 0.03067 0.002339 -0.01455 0.1479 0.0006981 -0.002329 -0.003167 0.003862 0.0008014 0.0006981 0.1573 0.01057 -0.006575 -0.00967 -0.0001042 -0.002329 0.01057 0.1142 Orthotropic approximation:

0.4908 0.09726 0.2196 0 0 0 0.09726 0.432 0.1172 0 0 0 0.2196 0.1172 0.7412 0 0 0

0 0 0 0.1479 0 0

0 0 0 0 0.1573 0

0 0 0 0 0 0.1142

Rotated C matrix:

0.488 0.1059 0.2168 0.03336 -0.01012 0.01441 0.1059 0.4229 0.1175 0.002955 0.007127 -0.02263 0.2168 0.1175 0.742 -0.004014 0.00323 0.01366 0.03336 0.002955 -0.004014 0.1477 0.001326 0.004908 -0.01012 0.007127 0.00323 0.001326 0.1563 0.009454 0.01441 -0.02263 0.01366 0.004908 0.009454 0.1211 TI approximation:

0.4286 0.1327 0.1672 0 0 0 0.1327 0.4286 0.1672 0 0 0 0.1672 0.1672 0.742 0 0 0

0 0 0 0.152 0 0

0 0 0 0 0.152 0

0 0 0 0 0 0.1479

(26)

B.17 A3R

Input C matrix:

0.701 0.1971 0.2388 -0.003313 -0.02831 -0.006959 0.2052 0.7071 0.2814 0.0001088 -0.1115 -0.01245 0.2398 0.2745 1.08 -0.01864 -0.1457 0.001547 -0.009354 -0.002068 -0.02254 0.24 0.001891 -0.03644 -0.01707 -0.09809 -0.142 0.001424 0.287 -0.002408 -0.004219 -0.008146 0.002308 -0.03669 -0.00264 0.1862

Rotated C matrix:

0.7091 0.1407 0.3296 0.04517 0.008924 0.009068 0.1407 0.6804 0.1927 0.006074 -0.004556 0.011 0.3296 0.1927 1.188 -0.01989 -0.0046 0.001515 0.04517 0.006074 -0.01989 0.2444 -0.001435 0.002619 0.008924 -0.004556 -0.0046 -0.001435 0.2559 0.008301 0.009068 0.011 0.001515 0.002619 0.008301 0.1686 Orthotropic approximation:

0.7091 0.1407 0.3296 0 0 0 0.1407 0.6804 0.1927 0 0 0 0.3296 0.1927 1.188 0 0 0

0 0 0 0.2444 0 0

0 0 0 0 0.2559 0

0 0 0 0 0 0.1686

Rotated C matrix:

0.6414 0.1964 0.26 0 0 0 0.1964 0.6414 0.26 0 0 0 0.26 0.26 1.189 0 0 0

0 0 0 0.2496 0 0

0 0 0 0 0.2496 0

0 0 0 0 0 0.2225

(27)

§B.18 Artibone256 191

B.18 Artibone256

Input C matrix:

0.7163 0.1087 0.107 -0.0008219 -0.00212 -0.0005858 0.1087 0.7234 0.1085 -0.0008288 -0.0006772 -0.001855 0.1069 0.1084 0.6904 -0.0002086 -0.0005007 -0.0002755 -0.0008257 -0.0008327 -0.0002079 0.2835 -0.000461 -0.0005085 -0.00213 -0.0006788 -0.0005052 -0.0004628 0.264 -0.0001352 -0.0005842 -0.001863 -0.0002805 -0.0005085 -0.0001356 0.2692

Rotated C matrix:

0.6903 0.1085 0.1069 -0.0002677 0.001247 0.0004401 0.1085 0.7235 0.1087 -0.001018 0.0006888 9.282e-05 0.1069 0.1087 0.7164 -0.00128 0.001092 0.0007961 -0.0002677 -0.001018 -0.00128 0.2692 5.537e-05 0.0007029 0.001247 0.0006888 0.001092 5.537e-05 0.264 -0.0002678 0.0004401 9.282e-05 0.0007961 0.0007029 -0.0002678 0.2835 Orthotropic approximation:

0.6903 0.1085 0.1069 0 0 0 0.1085 0.7235 0.1087 0 0 0 0.1069 0.1087 0.7164 0 0 0

0 0 0 0.2692 0 0

0 0 0 0 0.264 0

0 0 0 0 0 0.2835

Rotated C matrix:

0.7235 0.1084 0.1087 0.0006809 0.001207 -0.0008305 0.1084 0.6904 0.107 0.001063 0.0002682 -0.0002123 0.1087 0.107 0.7164 0.001322 0.00116 -0.000822 0.0006809 0.001063 0.001322 0.264 -0.0001337 0.0003029 0.001207 0.0002682 0.00116 -0.0001337 0.2692 0.0006512 -0.0008305 -0.0002123 -0.000822 0.0003029 0.0006512 0.2835 TI approximation:

0.6991 0.1163 0.1078 0 0 0 0.1163 0.6991 0.1078 0 0 0 0.1078 0.1078 0.7164 0 0 0

0 0 0 0.2666 0 0

0 0 0 0 0.2666 0

0 0 0 0 0 0.2914

(28)

(29)

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