The Hashin-Shtrikman-Walpole lower and upper bounds for d-dimensional isotropic materials with n phases are the following (1): i=1

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1 Supplemental Document Mechanics of Partially Mineralized Collaen Yanxin Liu, Stavros Thomopoulos, Chanqin Chen, Victor Birman, Markus J. Buehler, & Guy M. Genin 1 Hashin-Shtrikman-Walpole bounds The Hashin-Shtrikman-Walpole lower and upper bounds for d-dimensional isotropic materials with n phases are the followin (1): n n φ i (Kmin + K i ) 1 Kmin K e φ i (Kmax + K i ) 1 Kmax (1) i=1 n n φ i (G min + G i ) 1 G min G e φ i (G max + G i ) 1 G max (2) i=1 where G i, K i, and φ i are the shear modulus, bulk modulus, and volume fraction of phase i, K min = 2d 1 (d 1)G min, K max = 2d 1 (d 1)G max, G min = min(h i), G max = max(h i ), and: i=1 i=1 h i = d2 K i + 2(d + 1)(d 2)G i G i. (3) 2d(K i + 2G i ) In all cases considered, Poisson s ratio was set to ν =.3 for every phase. To relate G and K to E and ν, the isotropic relations G i = E i / (2(1 + ν)) and Ki 3D = E i / (3(1 2ν)) were employed. 2 Linear bounds on lonitudinal moduli We present here the complete set of equations used to enerate bounds in Fiures 6 7. Fiure 6a is repeated here as Fiure S1, with encircled numerals added to simplify the mappin between equations and bounds. Note that E 1 in the vertical axis corresponds to a fiber-level modulus, which is ρ=.8 that of the fibril-level modulus. We further note that specific bounds were not enerated for Model D because this was a comparison case that is not as stronly supported by the literature as the other cases studied. 2.1 Hashin-Shtrikman bounds (Models A D) The yellow envelope in Fiure S1 (labels 1 & 2 ) correspond to the four phase (φ m <.21: mineralized ap reions, unmineralized ap reions, unmineralized overlap reions, and extrafibrillar matrix) and five phase Hashin-Shtrikman upper and lower bounds (φ m >.21: now includin extrafibrillar bioapatite), as described in the main text. Note that the lower bound is not reached on this raph. 2.2 Bounds for moduli durin intrafibrillar mineralization of ap channels (Models A, B, C, E) The bounds for intrafibrillar mineralization of ap channels ( φ m.21), labels 3 and 4 in Fiure S1) both result from series addition of unmineralized overlap reions, unmineralized ap 1

2 lonitudinal modulus E f iber 1 /Ebone model C model A model B bioapatite volume fraction, φ m Fiure 1: Bounds and estimates of fiber-level moduli, Models A C reions, and partially mineralized ap reions. The upper bound (label of bioapatite to all ap reions equally: ρ.4 E fibril o 3 ) corresponds to addition + L m/d E fibril (φ m ) + (.6 L m/d) E fibril (4),m where E,m fibril (φ m ) is estimated from a parallel addition of the stiffnesses of collaen and bioapatite in each ap reion: E,m fibril E fibril + α(φ m )E ha = E fibril + E ha φ m D/(ρL m ) (5) in which α(φ m ) is the area fraction of bioapatite platelets in ap channels of fibrils at tissue-level bioapatite volume fraction (φ m ), L m = 3nm is c-axis lenth of bioapatite platelets, parallelin the fibril direction, and D = 67nm is periodic spacin between ap reions in a fibril. The lower bound (label 4 ) corresponds to completely fillin one ap reion with bioapatite before beinnin to fill the next ap reion: ξ(φ m ) (φ m = φ max ) + 1 ξ(φ m) (6) (φ m = ) where ξ φ m /φ max, and φ max =.21 is the maximum volume fraction of bioapatite that can be accommodated by the ap channels. 2.3 Bounds for moduli associated with extrafibrillar mineralization followin complete intrafibrillar mineralization of ap channels (Models A C) Gap-nucleated extrafibrillar bioapatite (Model A) Model A involved nucleation of extrafibrillar bioapatite from the ap channels for φ m > φ max. An appropriate upper bound for the associated stiffenin effect (label 5 ) involved parallel addition 2

3 of bioapatite to overlap, mineralized ap, and unmineralized ap reions, and then a ssequent series combination of these three: +.4 ρeo fibril + E ha φ ex (φ m ) + L m /D ρ(e,m fibril ) max + E ha φ ex (φ m ) (.6 L m /D) ρe fibril (7) + E ha φ ex (φ m ) where φ ex (φ m φ max ) is the extrafibrillar bioapatite volume fraction, and (E,m fibril ) max = E,m fibril (φ m = φ max ) = 65 GPa is the estimated modulus of the fully mineralized portion of a ap channel. The associated lower bound (label 6 ) involved, as did Model A, addition of sheaths of extrafibrillar bioapatite emanatin from all mineralized ap channels simultaneously. The heiht of each sheath relative to the periodic spacin D between aps was: h m φ ex (φ m )/(1 ρ). (8) The expressions for effective moduli associated with label 6 differed as the sheaths rew to encompass (1) the mineralized ap reions, (2) the unmineralized portions of the ap reions, and (3) the unmineralized overlap reions. For (1), with extrafibrillar bioapatite extendin as a sheath over the mineralized ap reions (h m (φ m ) L m /D), the lower bound followed: h m (φ m ) + (L m/d h m (φ m )),m ) max + (1 ρ)e ha ρ(e,m fibril ) max +.4 ρeo fibril. (9a) ρ(e fibril + (.6 L m/d) ρe fibril For (2), with extrafibrillar bioapatite continuin extension as a sheath over the unmineralized portions of the ap reions (L m /D < h m (φ m ).6), L m /D ρ(e,m fibril + (h m(φ m ) L m /D) ) max + (1 ρ)e ha ρe fibril + (1 ρ)e ha + (.6 h m(φ m )) ρe fibril +.4 ρeo fibril. (9b) For (3), with extrafibrillar bioapatite completin the sheath as it extended over the unmineralized overlap reions, L m /D (.6 L m /D) ρ(e,m fibril + ) max + (1 ρ)e ha ρe fibril + (1 ρ)e ha + (h m (φ m ).6) ρeo fibril + (1 h m(φ m )) + (1 ρ)e ha ρeo fibril. (9c) 3

4 lonitudinal modulus E f iber 1 /Ebone model E bioapatite volume fraction, φ m Fiure 2: Bounds and estimates of fiber-level moduli, Models E. Nucleation-inhibited extrafibrillar mineralization (Model B) Model B involved a mineral sheath that spread over the lenth of a fiber from a sinle nucleation site when φ m > φ max. The upper bound for the associated stiffenin of a fiber (label 7 ) is a simple parallel addition of bioapatite and a mineralized fibril: Em fiber + φ ex (φ m )E ha, (1) where Em fiber is the modulus of a fiber predicted by Equations (4) and (6) for φ m > φ max intersection of the curves associated with labels 3 and 4 ). The associated lower bound (label 8 ), and in fact model B itself, involved rowth of an extrafibrillar bioapatite sheath within and over a fiber of uniform modulus Em fiber, beinnin with a sinle nucleation site. The moduli of the two reions were then added toether in series: 1 h m (φ m ) Em fiber + m h m (φ m ) + (1 ρ)e ha (the. (11) 2.4 Bounds for moduli of fibrils containin bioapatite in the overlap reions (Model E) Model E was analoous to Model A, except that bioapatite was allowed to accrue in the overlap reions followin complete mineralization of the ap channels and prior to accumulation of extrafibrillar bioapatite. Fiure 6c has been included as Fiure S2, aain with labels over the bounds. Hashin-Shtrikman bounds The yellow envelope in Fiure S2 (labels 9 and 1 ) aain correspond to Hashin-Shtrikman bounds, and are analoous to those for Model A D, with the only difference bein the addition of a phase associated with mineralized material in the overlap reions. As in Fiure S1, the ordinate rane does not extend down to the lower bound in this raph. 4

5 Bounds for moduli durin intrafibrillar mineralization of ap channels The bounds for intrafibrillar mineralization of ap channels in Model E ( 11 and 12 ) were identical to those of Models A-C ( 3 and 4 ), cf. Equations (4) (6). As above, the rane over which these bounds applied was φ m φ max. Bounds for moduli associated with intrafibrillar mineralization of the overlap reions followin complete intrafibrillar mineralization of ap channels In Model E, bioapatite accumulated within the overlap reions followin complete mineralization of the ap channels, over the rane φ max φ m (φ max + φ max o ), where φ max o =.12 is the maximum tissue-level volume fraction of bioapatite that miht be accommodated within the overlap reions. The modulus of the bioapatite in the overlap reions Eo,m fibril (φ m ) followed the Hashin-Shtrikman lower bound, as described in the main text. The maximum possible value of Eo,m fibril (φ m ) was that associated with φ m =.33: (Eo,m fibril ) max = 3.36 GPa. The upper bound associated with this intrafibrillar mineralization of the overlap reions (label 13 ) involved a series combination of mineralized ap reions, unmineralized portions of the ap reions, and mineralized overlap reions, analoous to Eq. (4): ( ] ) 1 (φ.4 m) = ρ E Eo,m fibril (φ m ) + L m/d (E,m fibril + (.6 L m/d) ) max E fibril. (12) The lower bound (label 14 ) was obtained by requirin that a sinle overlap reion be filled to the maximum level (φ max o =.12, correspondin to φ m =.33), before the next overlap reion bean to accrue bioapatite, analoous to Eq. (6): ( ) E (φ m) = ρ.4ζ(φ m ) (E fibril o,m where ζ (φ m φ max )/φ max o. +.4(1 ζ(φ m)) ) max Eo fibril + L m/d (E fibril + (.6 L m/d) ) max E fibril (13) Bounds for stiffenin associated with ssequent extrafibrillar mineralization Bounds for the stiffenin associated with extrafibrillar mineralization in Model E (labels 15 and 16 ) were analoous to those of Models A C (labels 5 and 6 ). The volume fraction rane over which these bounds applied was.33 φ m.53. The upper bound was identical to that of Eq. (15) except that the extrafibrillar volume fraction φ ex was replaced with: φ E ex = φ m (φ max and the overlap reion effective fibril-level modulus Eo fibril mineralized overlap reion, (Eo,m fibril ) max :,m + φ max o ), (14) was replaced with that of the fully 5

6 ( ) E +.4 ρ(eo,m fibril ) max + E ha φ E ex(φ m ) + L m /D ρ(e,m fibril ) max + E ha φ E ex(φ m ) (.6 L m /D) ρe fibril. (15) + E ha φ E ex(φ m ) The associated lower bound (label 16 ) as with that of label 6, involved addition of sheaths of extrafibrillar bioapatite emanatin from all mineralized ap channels simultaneously. The heiht of each sheath relative to the periodic spacin D between aps for model E was: h E m φ E ex(φ m )/(1 ρ). (16) The lower bounds (not shown here) were identical to those of Eq. (9) except that h E m(φ m ) replaces h m (φ m ), (Eo,m fibril ) max replaces Eo fibril. 3 Transverse modulus Transverse moduli of fibers were estimated by homoenizin isotropic estimates of the transverse moduli of mineralized and unmineralized portions of fibrils with either (1) unmineralized extrafibrillar matrix, includin non-collaenous proteins, with isotropic modulus E EF M =.287 MPa, or (2) extrafibrillar bioapatite, with isotropic modulus E HA = 11 GP a. The followin four elastic moduli were used for the transverse stiffnesses of the fibrils: 1. Unmineralized collaen fibril within an overlap reion, with transverse modulus (E 2 ) fibril o =14.3 MPa 2. Unmineralized collaen fibril within a ap reion,with transverse modulus (E 2 ) fibril =11.5 MPa 3. Fully mineralized collaen fibril within an overlap reion, with homoenized transverse modulus (E 2 ) fibril o,m (φ m ) (E2 max ) fibril o,m = 72.3 MPa 4. Fully mineralized collaen fibril within a ap reion, with homoenized transverse modulus (E 2 ) fibril,m (φ m ) (E2 max ) fibril,m = 22.9 GPa Althouh the homoenization scheme was somewhat involved, we emphasize that the followin are intended as simple, back-of-the-envelope estimates of transverse moduli. The above six transverse moduli were combined usin the two dimensional, two-phase Hashin-Shtrikman lower (E HS ) and upper (E HS ) bounds (Eq. 1 3) to obtain effective moduli of transverse slices containin (1) extrafibrillar material of volume fraction 1 ρ =.2, and (2) a fibril material with volume fraction ρ =.8. Eiht combinations were used, and the followin notation was introduced in which the first parameter in the parentheses represented the elastic modulus of a slice s extrafibrillar material, and the second the fibril s transverse elastic modulus: 1. E ) 2. E HS (E EF M, (E 2 ) fibril,m 3. E o ) (φ m )) E HS (E EF M, (E2 max ) fibril,m ) 6

7 4. E HS (E EF M, (E 2 ) fibril o,m 5. E HS (E HA, (E 2 ) fibril ) 6. E HS (E HA, (E max 2 ) fibril,m ) 7. E HS (E HA, (E 2 ) fibril o ) 8. E HS (E HA, (E max 2 ) fibril o,m ) (φ m )) E HS (E EF M, (E2 max ) fibril o,m ) Upper bounds were appropriate when the extrafibrillar material was stiffer than fibrils, and lower bounds when this material was more compliant than fibrils. Poisson s ratio was set to ν =.3 for all phases. Effective fiber transverse moduli were then calculated for models A, B, and E. 3.1 Intrafibrillar mineralization Bioapatite accumulatin in the ap reions (Models A, B, C, E) We considered the case with bioapatite completely fillin one ap reion before beinnin to fill the next ap reion, which correspondin to the lower bound of lonitudinal modulus (label 4 in Fi. 1 and label 12 in Fi. 2). This case is most potentially to delay the percolation threshold. 2 ξ(φ m )(L m /D)E HS (E EF M, (E max +.4E o ) 2 ) fibril,m ) + (.6 ξ(φ m )(L m /D))E ) (17) Bioapatite accumulatin ssequently in the overlap reions (Model E) Same as for intrafibrillar mineralization in the ap reion, we investiated the case which corresponds to the lower bound (label 14 ) of lonitudinal modulus, a sinle overlap reion filled to the maximum level, before the next overlap reion bean to accrue bioapatite: 2 (L m /D)E HS (E EF M, (E max +.4(1 ζ(φ m ))E HS 2 ) fibril,m (E EF M, (E 2 ) fibril o ) + (.6 (L m /D))E ) ) +.4ζ(φ m )E HS (E EF M, (E2 max ) fibril o,m ) (18) 3.2 Extrafibrillar mineralization Gap-nucleated extrafibrillar bioapatite (Models A and E) For φ max φ m, Model A involved extrafibrillar mineral emanatin as a sheath from the mineralized ap reions, and extendin over a fraction h m of the periodic spacin D of ap channels, where, as above, h m (φ m φ max )/(1 ρ). Note that the estimate for Model A at φ m = φ max was slihtly below the Hashin-Shtrikman bounds; rather, it matches with the estimate listed above for intrafibrillar mineralization of ap reions in Model E. As the sheath extended over the ap reions (h m L m /D), ( E fiber 2 )A (φ m) = h m (φ m )E HS (E HA, (E 2 ) fibril ) + ( L m /D h m (φ m ) ) E HS (E EF M, (E2 max ) fibril,m ) + (.6 L m /D ) E HS (E EF M, (E 2 ) fibril 7 ) +.4E o ) (19a)

8 For L m /D h m (φ m ).6, as the sheath encompassed the unmineralized portions of the ap channels, ( E fiber 2 )A (φ m) = ( L m /D ) E HS (Emax + (.6 h m (φ m ) ) E HS 2 ) fibril,m, E HA ) + ( h m (φ m ) L m /D ) E HS ((E 2) fibril, E HA ) (E EF M, (E 2 ) fibril ) +.4E o ) (19b) For h m (φ m ) >.6, as the sheath encompassed the overlap reion, ( E fiber 2 )A (φ m) = ( L m /D ) E HS (E HA, (E2 max ) fibril,m ) + (.6 L m /D ) E HS (E HA, (E 2 ) fibril ) + ( h m (φ m ).6 ) E HS (E HA, (E 2 ) fibril o ) + ( 1 h m (φ m ) ) E o ) (19c) The estimates for Model E were identical to those for Model A, except that (E 2 ) fibril o in Eq. 19 were sstituted with (E2 max ) fibril o,m. Nucleation-inhibited extrafibrillar bioapatite (Model B) Transverse moduli for Model B were estimated quasi-empirically from finite element estimates of E2 F E of tissue with fully mineralized ap reions (φ m = φ max =.21) and with fully mineralized extrafibrillar matrix (φ m =.41). Then, ( E fiber 2 )B (φ m) = ( 1 h m (φ m ) ) ( E2 F E (φ m =.21) ) + h m (φ m ) ( E2 F E (φ m =.41) ). (2) 4 Nonlinear model The stiffenin of a nonlinear collaen fiber by bioapatite was estimated for the sequence of mineralization of Model A. Estimates for the stiffenin associated with bioapatite accruin inside collaen fibers were presented in the main document. We present here estimates of the ssequent stiffenin by extrafibrillar bioapatite, derived usin procedures analoous to those used for the linear estimates of Model A, except now accountin for the material nonlinearity of the collaen. As before, the mechanics of three more buildin blocks had to be estimated: mineralized ap reions, ap reions, and overlap reions in parallel with 2% by volume extrafibrillar bioapatite (φ m =.2): 1. For a mineralized ap reion in parallel with 2% extrafibrillar bioapatite, the nonlinear function m,ex(ɛ fiber m,ex ) that predicted the fiber-level mechanical stress σ resultin from a strain ɛ m,ex was estimated by solvin the followin nonlinear equation, in which ρ =.8 is the area fraction of fibrils and is assumed to be independent of stretch: fiber m,ex(ɛ m,ex ) = σ = (1 ρ)e ha ɛ m,ex + ρ fibril,m (ɛ m,ex ). (21) 2. For a ap reion in parallel with 2% extrafibrillar bioapatite, the nonlinear function σ = fiber,ex (ɛ,ex ) was estimated by solvin: fiber,ex (ɛ,ex ) = σ = (1 ρ)e ha ɛ,ex + ρ fibril (ɛ,ex ). (22) 3. For an overlap reion in parallel with 2% extrafibrillar bioapatite, the nonlinear function σ = fiber o,ex (ɛ o,ex ) was estimated by solvin: fiber o,ex (ɛ o,ex ) = σ = (1 ρ)e ha ɛ o,ex + ρ fibril (ɛ o,ex ). (23) 8

9 Combinin in series different portions of mineralized ap reions, ap reions and overlap reions, surrounded either by EFM or extrafibrillar bioapatite, the nonlinear fiber-level stress-strain relationship could be estimated for partially mineralized collaen tissues. One of three expressions was used, dependin upon the extent to which the extrafibrillar sheath covered the fibrils, h m φ ex (φ m )/(1 ρ). For extrafibrillar bioapatite extendin as a sheath over the mineralized ap reions (h m (φ m ) L m /D): (f fiber ) 1 (σ, φ m ) = h m (φ m )(m,ex) fiber 1 (σ) + (L m /D h m (φ m )) (,m fibril ) 1 (σ/ρ) + (.6 L m /D)( fibril ) 1 (σ/(.8ρ)) +.4( fibril ) 1 (σ/ρ). (24a) For extrafibrillar bioapatite continuin extension as a sheath over the unmineralized portions of the ap reions (L m /D < h m (φ m ).6), (f fiber ) 1 (σ, φ m ) = (L m /D)( fiber m,ex) 1 (σ) + (h m (φ m ) L m /D) ( fiber,ex ) 1 (σ) + (.6 h m (φ m )) ( fibril ) 1 (σ/(.8ρ)) +.4( fibril ) 1 (σ/ρ). (24b) For extrafibrillar bioapatite completin the sheath as it extended over the unmineralized overlap reions, (f fiber ) 1 (σ, φ m ) = (L m /D)( fiber m,ex) 1 (σ) + (.6 L m /D)( fiber,ex ) 1 (σ) + (h m (φ m ).6) ( fiber o,ex ) 1 (σ) + (1 h m (φ m )) ( fibril ) 1 (σ/ρ). (24c) 5 Homoenized transverse modulus of a mineralized ap reion We treated the bioapatite and collaen within a mineralized ap reion a layered composite of two isotropic materials. The resultin composite was stronly orthotropic in plane, because the series combination of the two materials was much more compliant than the parallel combination. To minimize the effects of usin a small unit cell in the numerical simulations, effective isotropic properties were used, as obtained by averain over a uniform orientation distribution. The procedure for this is elementary and straihtforward, but we could not find the explicit expression elsewhere and thus present it in this section. Consider an orthotropic material with principal directions oriented an anle θ from the positive axes of the x 1 - x 2 plane. The stiffness tensor can be written in terms of that of the material in its principal axes as C ijkl (θ) = a mi a nj a rk a sl C mnrs () (25) where θ is a rotation around the x 3 axis measured relative to the x 1 direction, repeated indices imply summation, and cos(θ) sin(θ) a = sin(θ) cos(θ) 1 Notin that 1. a k3 = a 3k = for k 3, 2. C 1112 () = C 1121 () = C 1211 () = C 1222 () = C 2111 () = C 2122 () = C 2212 () = C 2221 () = C 2112 () = C 1221 () =, 9

10 3. C 1122 () = C 2211 (), C 1212 () = C 2121 (), and 4. a 11 = a 22, and a 12 = a 21 we can write the four independent constants of the stiffness tensor as: Notin that: C 1111 (θ) = a 4 11C 1111 () + 2a 2 11a 2 12(C 1122 () + C 1212 ()) + a 4 21C 2222 () C 1122 (θ) = a 2 11a 2 12(C 1111 () + C 2222 () 2C 1212 ()) + (a a 4 12)C 1122 () C 3333 (θ) = C 3333 () C 1133 (θ) = a 2 11C 1133 () + a 2 12C 2233 () (26) 1 π 1 π 1 π a 4 11dθ = 1 π a 2 11a 2 12dθ = 1 π a 2 11dθ = 1 π cos 4 θdθ = 1 π cos 2 θ sin 2 θdθ = 1 8, and cos 2 θdθ = 1 π sin 4 θdθ = 1 π sin 2 θdθ = 1 π the independent terms of the averaed stiffness tensor C, where () = 1 π C 1111 = 3 8 C 1111() (C 1122() + C 1212 ()) C 2222() C 1122 = 1 8 (C 1111() + C 2222 () 2C 1212 ()) C 1122() C 3333 = C 3333 () a 4 12dθ = 3 8, a 2 12dθ = 1 2, (27) ()dθ, could be written: C 1133 = 1 2 (C 1133() + C 2233 ()) (28) The components of C() can be written in terms of enineerin constants: C 1111 () = E1(1 ν23ν 32)Υ, C 2222 () = E2(1 ν13ν 31)Υ, C 3333 () = E 3 (1 ν12ν 21)Υ, C 1122 () = E1(ν 21 + ν31ν 23)Υ, C 1133 () = E1(ν 31 + ν21ν 32)Υ, C 2233 () = E2(ν 32 + ν12ν 31)Υ, and C 1212 () = 2G 12, (29) where Υ = 1/(1 ν 21 ν 12 ν 23 ν 32 ν 31 ν 13 2ν 21 ν 32 ν 13 ). Thus, C 1111 = 3 8 E 1(1 ν23ν 32)Υ (E 1(ν21 + ν31ν 23)Υ + 2G 12) E 2(1 ν13ν 31)Υ C 1122 = 1 8 (E 1(1 ν23ν 32)Υ + E2(1 ν13ν 31)Υ 4G 12()) E 1(ν21 + ν31ν 23)Υ C 3333 = E3(1 ν12ν 21)Υ C 1133 = 1 2 (E 1(ν21 + ν31ν 23)Υ + E2(ν 32 + ν12ν 31)Υ) (3) 1

11 The followin enineerin constants were assumed: ν 12 =.3 ν 32 = 1 ν 31 =.3 E1 = (1 α max )(E 2 ) fibril + α max E ha E2 (1 α max ) = (E 2 ) fibril + α max E ha E 3 = E fibril G 12 = + α max E ha (E 2 ) fibril o 2(1 + ν 12 )(1 α max) where α max =.58. The effective modulus was calculated by sstitutin these numbers into Eq. 3 and usin the well-known result: Ē 2 = (E 2 ) fibril,m = C C C C C C 1111 C C 3333 C 1111 C3333 C GPa. (31) 6 Fiber-level stiffness tensor C fiber Computin a tissue-level modulus required computation of the stiffness tensor of a fiber within a representative volume of mineralized or unmineralized extrafibrillar matrix, C fiber (φ m ). Transverse isotropy was assumed for this representative volume, meanin that C fiber (φ m ) contained five independent constants for each value of φ m. In addition to the symmetries in the previous section, the sscripts 2 and 3 are interchaneable, so that the only independent elements of the stiffness tensor (cf. Equation (32)) are: C fiber 1111 = 1 (1 ν fiber 23 ν fiber 32 )Υ fiber, C fiber 2222 = = 2 (1 ν fiber 13 ν fiber 31 )Υ fiber, C fiber 1122 = 1 (ν fiber 21 + ν fiber 31 ν fiber 23 )Υ fiber, C fiber 2233 = 2 (ν fiber 32 + ν fiber 12 ν fiber 31 )Υ fiber, and C fiber 1212 = 2G fiber 12, (32) where, as above, Υ fiber = 1/(1 ν fiber 21 ν fiber 12 ν fiber 23 ν fiber 32 ν fiber 31 ν fiber 13 2ν fiber 21 ν fiber 32 ν fiber 13 ). Thus, Usin the identity ν ij = ν ji E i /E j, in which repeated indices do not imply summation, the five independent enineerin constants needed for each value of φ m were 1, 2, G fiber 12, ν fiber 12, and ν fiber 23. The models in this article were used to estimate 1 (φ m ), 2 (φ m ); G fiber 12 (φ m ) was estimated accordin to the model used in (2): ( ) G fiber 12 (φ m ) 3β R 2 1 (φ m ) (33) L where R/L = 1 : 2 is a reasonable aspect ratio for fibers, and β = 1.7 was found throuh numerical simulations in (2). Followin (2), Poisson s ratio were taken to be constant throuhout the insertion site: ν fiber 23 =.3 and ν fiber 12 =.1. 11

12 References 1. Milton, G., 22. The theory of composites. Cambride: Cambride University Press. 2. Genin, G. M., A. Kent, V. Birman, B. Wopenka, J. D. Pasteris, P. J. Marquez, and S. Thomopoulos, 29. Functional radin of mineral and collaen in the attachment of tendon to bone. Biophys. J. 97: