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Does Temperature Effects the Growth of Microcracks in Tibia due to Volleyball?

M. Tsili , D. Zacharopoulos
Journal of Biomedical Engineering and Technology. 2017, 5(1), 6-11. DOI: 10.12691/jbet-5-1-2
Published online: April 08, 2017

Abstract

In present paper we considered if temperature plays role to the growth of microcracks in a tibia due to volleyball. We dealed with three particular points of bone and we based upon theories: of adaptive elasticity and upon energy density. We showed that both: neglecting or accounting temperature after a long time tibia at points of our interest will be strengthened (the mean length of their microcracks will be decreased). The result coincides with that of corresponding problem at macroscopic area. We concluded that temperature does not effects the growth of microcracks.

1. Introduction

It is well known that bone fracture due to many factors: as age 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, microstructure 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, bone density 2, 3, 8, 9, 10, 14, 17, 24, 25 and loading mode 14, 23, 24, 25, 26, 27, 28.

From the other hand very few known studies investigated the effect of temperature in bone disease 23, 25.

The purpose of this paper is to study if temperature plays role to the growth of microcracks in a tibia, due to volleyball. For that reason we will use the theories: of adaptive elasticity 29, 30 both neglecting and accounting temperature and energy density 31, 32, 33.

Macroscopically the bone has a volume V and a surface S. The volume V microscopically consists of microvolumes which generally are not homogenous 32, 33. Expanding the theory of adaptive elasticity [ 32, p. 322] at microscopic area we assume:

i) Every microvolume of the bone consists of a elastic microvolume (micromatrix bone) and of microcracks (pores) that is:

(1)

where is the volume of microcracks.

Sih [ 33, p.179] showed that every microvolume contains an homogenous microvolume. We suppose that the elastic microvolume given by (1) is an homogenous microvolume.

ii) The mechanical properties of microvolume of bone coincides with the mechanical properties of homogenous microvolume of micromatrix bone.

iii) The fraction of microvolume of the micromatrix bone is defined as [ 30, p. 322]:

(2)

where is the density of microvolume , while is the density of material (bone) and assume to be constant. From the above it follows .

iv) The porosity that is the mean length of microcraks of the microvolume alters with mass added /removal to /from micromatrix bone and linearly depends from the history of microstrain [ 29, p. 322]. The porosity is characterized by a parameter ê 30:

(3)

where is the initial fraction of the microvolume of micromatrix bone. With other words parameter is the change of the mean value of microcracks.

2. The Problem and the Physical Approximation

Assume that someone participates to a volleyball game or training and continues to be exersized, for a long time period. In previous paper we macroscopically study the internal remodeling of tibia due to volleyball 34. In present paper we want to predict the situation of his /her tibia after a long time, locally at three particularly points as indicated in Figure 1. The lasts are below.

2.1. The Mathematic Formulation of Our Problem

Assume that for the athlete was normal walking with constant velocity . Therefore the points Α, Β and of tibia were respectively under a compressive load , and due to ground reaction force at late stance phase during normal walking. The above are given by:

where is ground reaction force at late stance phase for foot and , are respectively the weights of foot and tibia 35. There is a statistical relationship between ground reaction force for foot G and velocity 36, 37, 38:

(5)

Substituting (5) into (4)1-2-3 and accounting 35:

it results:

Therefore the points Α, Β and of tibia were respectively under costant compressive loads , and .

Also the lasts had the same fraction of element volume and the same mean length of microcracks êο.

2.2. Biomechanical Analysis of Volleyball

At the athlete starts playing volleyball. During a volleyball game it is possible to observe the followings 34:

i) The players are running in order to successfully rebut the ball, before it comes to contact with the ground.

ii) The volleyball spiker is vertically jumping as high as possible, in order to hit the ball. Also the players of the opposite team, are simultaneously vertically jumping as high as possible, in order to rebut the hit of the volleyball spiker.

iii) Finally the player who serves the ball, is sometimes jumping as high as possible, but he (she) is not landing to his (her) initial location. This jump can be modeled as an oblique shoot in the plane Oxz (x, z are the horizontal and vertical axons respectively). Particularly the center of the mass of the player is launched from the origin, with a velocity uo. An angle α is formed between the direction of the vector of velocity uo and the horizontal axon, as it seems in Figure 2. The maximums high hM and horizontal displacement SM are respectively:

where g is the acceleration of the gravity

The server is mainly interested to jump as high as possible than as long as possible, in order to successfully send the ball towards the region of the opposite team. Also the international rules of volleyball forbides the server to leave his (her) area during the service. The last means that his (her) horizontal displacement is always under restriction. Therefore it holds from which it follows:

(9)

Since , the jump of the server can approximatelly be considered as a vertical. Therefore during the volleyball game, the tibia of the athlete is under an axial impact load, due as to running as to vertical jumps. Consequently the points Α, Β and of tibia were respectively under an axial load , and given by ( 34):

where , , and Fzi i= 1, 2, ....,x are respectively: the vertical component of ground reaction force at late stance, during the i time of running at points Α, Β and of tibia, given by respectively:

where is vertical component of ground reaction force at late stance, during phase running for foot. Assume that running velocity is constant.

There is a statistical relationship between ground reaction force and running velocity v 38, 39, 40, 41, 42, 43, 44:

(12)

Then (11) 1-2-3 of because of (6)1-2 and (12) take the forms:

Finally in (10)1-2-3 , and where , are respectively the vertical component of ground reaction force at landing phase, during the j time of jumping at points Α, Β and of tibia. We assume that that the above loads are constant. Therefore it follows that the total axial loads , and given by (10) are constant.

3. A Hollow Circular Cylinder Subjected to an Axial Load

Tibia is modeled as a hollow circular cylinder with lenght L, inner and outer radii a and b respectively, indicated in Figure 1.These radii correspond to endosteal and periosteal surface of bone and are constant due to internal remodeling 45. Since we deal with microscopic area, we will base upon the density energy theory 31, 32, 33.

The required equations of the above theory in cylindrical coordinates are:

i) the microstress equations [ 33, p.182]:

ii) the microstrain-microdisplacement relations [ 33, p.179]:

iii) the microstress - microstrain relations:

where:

where , and , are Young’s modulus and Poisson ratio in transverse and axial direction at macroscopic area.

The above hold because bone continues microscopically to be transversely isotropic material in microscopic area 46.

Finally rate remodeling equation 30 at microscopic area without /and accounting temperature are respectively:

where , are rate remodeling coefficients in transverse and axial direction respectively while is a rate remodeling coefficient depends from temperature.

The boundary conditions of our problem are:

i) at point A:

ii) at point B:

iii) at point :

where B is body’s weight of the runner in units of B.W. and assumed to be constant during training.

Our problem has a unique solution [ 33, p.186] and assume that microdisplacements are of the form:

where , , are unknowns. Then microstrains (15) are written as:

Therefore (16) because of (23) take the forms:

Applying (19)–(20)−(21) into (24) it is possible to obtain:

i) at point Α:

ii) at point Β:

iii) at point :

where

(28)

Employing (25), (26) and (27) into (22) it possible to obtain the microdisplacements at points Α, Β and respectively. Also employing (25), (26) and (27) into (23), it is possible to obtain the corresponding microstrains. Thus:

i) at point Α:

ii) at point Β:

iii) at point :

At continuity we distinguish the following cases:

i) Internal remodeling of tibia does depends upon temperature:

Then substituting (29), (30), (31) into (18)1 it follows:

i) at point Α:

(32)

ii) at point Β:

(33)

iii) at point :

(34)

Since the living bone is continually remodeling 47, 48 we assume that Young’s modulus and Poisson’s ratio depends upon tο 34, 49, 50, 51. Then from (17) it results that , , , depend also upon . At continuity we impose:

Therefore (32)-(33)-(34) conclude to the following form:

(36)

where:

i) at point Α:

ii) at point Β:

iii) at point :

where:

(40)

Initially the mean length of the pre-existing microcracks 52, 53 at points A, B and of tibia was

(41)

The solutions of (36) satisfying (41) are:

The acceptable solutions are given in Table 1. 34. Αccor-dingly to these results after a long time period, the tibia of athlete locally at points A, B and will be strenghted, that is the mean lenght of their microcracks will be decreased. The last can be explained as follows: i) the mean lenght of pre-existing microcracks 52, 53 wil be decreased or ii) some of preexisting microcracks 52, 53 will be closed or iii) a combination of both previous cases will be arised.

From the other hand we distinguish the following cases:

i) then from (37)3, (38)3 and (39)3 it is possible to obtain: and because of (42)2 it follows:

(43)

We observe that the decrease of porosity of tibia at points A, B, due only to mechanic loads. Particularly it analogically depends upon the magnitude of them.

ii) then from (37)3, (38)3 and (39)3 it is possible to obtain: and because of (42)1 it follows:

(44)

We observe that the decrease of porosity of tibia at points A, B , due only to mechanic loads. Particularly it revengelly depends upon the magnitude of them.

ii) The internal remodeling of tibia depends upon temperature:

Then substituting (29),(30),(31) into (18)2 it follows:

i) at point Α:

(45)

ii) at point B:

(46)

iii) at point :

(47)

Finally substituting (35) and

(48)

into (45)-(46) -(47) we again conclude to (36) where:

i) at point Α:

ii)at point Β:

iii) at point :

The solutions of (36) satisfying (41) are given by (42)1-2 and the same as at previous case are valid.

4. Discussion -Conclusion

Our model at both cases accounting and neglecting temperature coincide with the result of corresponding problem at macroscopic area 34, 54, 55, 56, 57, 58, 59, 60, 61, 62. From the above we conclude that temperature plays no one role to growth and propagation of microcracks in a tibia due to volley ball activity. In contrast with the above phenomenon exclusively due to mechanical loads.

References

[1]  Burstein, A., Reilly, D. and Martens, M. (1976). “Aging of bone tissue mechanical properties. J. Bone Joint Surg.” A58, 82-86.
In article      View Article
 
[2]  Thompson, D. (1980): “Age changes in bone mineralization, cortical thickness, and haversian canal area.” Calcif Tissue Int. 31, 5-11.
In article      View Article  PubMed
 
[3]  Grynpas, M. D.and Holmyard, D. (1988): “Changes inquality of bone mineral on aging and in disease.” Scan Microsc. 2, 1045-1054.
In article      
 
[4]  Hui, S. L., Slemenda, C. W. and Johnston, C.C. (1988): Age and bone mass as predictors of fracture in a prospective study. J. Clin. Invest. 81, 1804-1809.
In article      View Article  PubMed
 
[5]  Kiebzak G. M. (1991). “Age-related bone changes.” Exp. Gerontol. 26, 171-187.
In article      
 
[6]  Simmons, E. D., Pritzker, K. P. and Grynpas, M. D. (1991). “Age - related changes in the human femoral cortex.” J. Orthop. Res.9, 155-167.
In article      View Article  PubMed
 
[7]  Melvin JW. “Fracture mechanics of bone.” J. Biomech. Eng. 1993. Nov; 115(4B):549-554.
In article      
 
[8]  Currey, J. D., Brear, K. and Zioupos, P. (1996). “The effects of aging and changes in mineral content in degrading the toughness of human femora.” J. Biomech. 29, 257-260.
In article      View Article
 
[9]  Francis, R.M. (1996). “Low bone mineral content is common but osteoporotic fractures are rare in elderly rural Gambian women.” J. Bone Miner. Res.11, 1019-1025.
In article      
 
[10]  Aspray, T. J., Prentice, A., Cole, T. J., Sawo, Y., Reeve, J. and Francis RM. (1996). “Low bone mineral content is comon but osteoporotic fractures are rare in elderly rural Gambian women.” J. Bone Miner. Res.11, 1019-1025
In article      
 
[11]  Yeni, Y. N. and Norman, T. L. (2000). “Fracture toughness of human femoral neck: Effect of microstructure, composition and age.” Bone 26, 499-504.
In article      View Article
 
[12]  Wang, X., Shen, X., Li, X. and Agrawal C. M. (2002). “Age- related changes in the collagen network and toughness of bone.” Bone 31, 1-7.
In article      View Article
 
[13]  Akkus, O., Adar, F. and Schaffler, M. B. (2004). “Age-related changes in physicochemical properties of mineral crystals are related to impaired mechanical function of cortical bone.” Bone 34, 443-453.
In article      View Article  PubMed
 
[14]  Ritchie. R, Kinney H., Kruzic R., and Nalla R. (2005). “A fracture mechanics and mechanistic approach to the failure of cortical bone.” Fatigue Fract. Engng Mater Struct. Fatigue Fract. Engng Mater Struct 28, 345-37
In article      View Article
 
[15]  Behiri, JC. and Bonfield W. (1989): “Orientation depen-dence of the fracture mechanics of cortical bone.” J. Biomech,. 22, 863-872.
In article      View Article
 
[16]  Yeni, Y. N., Brown, C. U., Wang, Z. and Norman, T. L. (1997). “The influence of bone morphology on fracture toughness of the human femur and tibia.” Bone 21, 453-459.
In article      View Article
 
[17]  Yeni, Y. N., Brown, C.U. and Norman, T.L. (1998). “Influence of bone composition and apparent density on fracture toughness of the human femur and tibia.” Bone 22, 79-84.
In article      View Article
 
[18]  Feng, Z., Rho, J., Han, S. and Ziv, I. (2000). “Orientation and loading condition dependence of fracture toughness in cortical bone.” Mater. Sci. Engng CC11, 41-46.
In article      View Article
 
[19]  Brown, C. U., Yeni, Y. N. and Norman, TL.(2000). “Fracture toughness is dependent on bone location-A study of the femoral neck, femoral shaft and the tibial shaft.” J. Biomed. Mater. Res.49, 380-389.
In article      View Article
 
[20]  Phelps, J. B., Hubbard, G. B., Wang, X. and Agrawal C. M. (2000): “Microstructural heterogeneity and the fracture toughness of bone.” J. Biomed. Mater. Res.51, 735-471.
In article      View Article
 
[21]  Yeni, Y. N. and Norman, T. L. (2000). “Fracture toughness of human femoral neck: Effect of microstructure, composition and age.” Bone 26, 499-504.
In article      View Article
 
[22]  Seeman, E. (1999). “The structural basis of bone fragility in men.” Bone 25, 143-147.
In article      View Article
 
[23]  Rimnac, C. M., Petko, A. A., Santners, T. J. and Wright, T. M (1993). “The effect of temperature, stress and microstructure on the creep of compact bovine bone.” J. Biomech.26, 219-228.
In article      View Article
 
[24]  Ford C.M. and Keaveny, T.M. (1996). “The dependence of shear failure properties of trabecular bone on apparent density and trabecular orientation.” J. Biomech. 29, 1309-1317.
In article      View Article
 
[25]  Carter, D. R. and Hayes, W. C. (1976): “Fatigue life of compact bone-I. Effects of stress amplitude, temperature and density.” J. Biomech.9, 27-34, Biomech. 26, 219-228.
In article      
 
[26]  Norman, T.L., Nivargikar, S. V. and Burr, D. B. (1996). “Resistance to crack growth in human cortical bone is greater in shear than in tension.” J. Biomech.29, 1023-1031.
In article      View Article
 
[27]  Feng, Z., Rho, J., Han, S. and Ziv, I. (2000). “Orientation and loading condition dependence of fracture toughness in cortical bone.” Mater. Sci. Engng CC11, 41-46.
In article      View Article
 
[28]  Feng X. and McDonald J. (2011). “Disorders of Bone Remodeling,” Annu Rev Pathol.; 6: 121-145.
In article      
 
[29]  Cowin S. and Hegedus D. (1976). “Bone remodeling I: Theory of adaptive elasticity” J. Elastic. 6, pp. 313-326.
In article      View Article
 
[30]  Hegedus D. και Cowin S. (1976). “Bone remodeling II: Theory of adaptive elasticity” J. Elastic. 6, pp. 337-352.
In article      View Article
 
[31]  Sih G.C. (1985). “Mechanics and Physics of energy density theory”, Theoret., Appl., Fract. , Mech., 44, pp. 157-173.
In article      View Article
 
[32]  Sih G.C (1972-1982). “Mechanics of fracture, Introductory chapters”, Vol. I- VII, edited by G.C. Sih, Martinus Nijhoff, The Hague.
In article      
 
[33]  Sih GC(1988). “Thermomechanics of solids: nonequilibrium and irreversibility”, Theoretical and Applied Fracture Mechanics, 99, pp. 175-198.
In article      View Article
 
[34]  Τsili M. (2008). “The hyperthrophy of tibia induced by the volleyball” in: www.ispub.com/journal-of-internet journal of bioengineering , Volume .4. number 1.
In article      
 
[35]  Harless E. (1860). “The static moments of human limbs (in German). Treatsises of the Math-Phys. Class of the Royal Acad. of Sci. of Bavaria, 8, pp.69 και 257.
In article      
 
[36]  Andriachi T., Ogle J. and Galante J. (1977). “Walking speed as a base for normal and abnormal gait measurement.” J. Biomech., 10, pp. 261-268.
In article      View Article
 
[37]  Rohrle H., Scholten R., Sigolloto C. et., al., (1984). “Joint forces in the human pelvis - leg skeleton during walking.” J. Biomech., 17, pp. 409-424.
In article      View Article
 
[38]  Whalen R., Carter D. and Steele C. (1988). “Influence of the physical activity on the regulation of bone activity.” J. Biomech., 21., pp., 825-837.
In article      View Article
 
[39]  Αlexander N. and Jayes A. (1980). “Fourier analysis on forces exerted in walking and running.” J. Biomech. 13. pp. 383-390.
In article      
 
[40]  Bates B., Osterning L. and Sawhill J. (1983). “An assessment of subject variability, subject -shoe interaction and the evaluation of running shoes using ground reaction force data.” J. Biomech., 16., pp.181-191.
In article      View Article
 
[41]  Cavagna P. (1964). “Mechanical work in running” J. Appl. Physiol. 19., pp. 249-256
In article      
 
[42]  Cavagna P. and LaFortune M. (1980). “Ground reaction forces in distance -running.” J. Biomech., 13. pp. 397-406.
In article      
 
[43]  Fukanaga T., Matsuo A., Yuasa K., et., al., (1980). “Effect of running velocity on external mechanical output.” Ergonomics, 23, pp.,123-136
In article      View Article  PubMed
 
[44]  Winter D. (1983). “Moments of forces and mechanical power in jogging.” J.Biomech., 16, pp., 91-97.
In article      View Article
 
[45]  Frost H.M (1964). “Dynamics of bone remodeling in bone biodynamics.” (edited by Frost H.M) Little and Brown 316, Boston.
In article      
 
[46]  Reilly D.T. and Burstein A. (1975). J. Biomech., 8., p.393.
In article      
 
[47]  Cowin S. and Van -Burskirk W. (1978). “Internal bone remodeling induced by a medullary pin.” J. Biomech. 11, pp. 269-275.
In article      View Article
 
[48]  Wolff. J. (1884). “Das gesetz der transformation der inneren architecture knocken bei pathologism veranderungen der aussen knochenform.” Sitz Ber. Preuss Acad. d. Wiss 22, Sitz Physik- Math. K1.
In article      
 
[49]  Wolff J. (1892). Das gesetz der transformation knocken hirschald, Berlin.
In article      
 
[50]  Τsili M. (2000). “Theoretical solutions for internal bone remodeling of diaphyseal shafts using adaptive elasticity theory.” J. Biomech., 33 pp. 235-239.
In article      
 
[51]  Τsili M. (2008): “Internal bone remodeling induced by the distance - running and the unkown remodeling coefficients” in: www.ispub.com/journal- of- internet journal of bioengineering vol. 4. number 2.
In article      
 
[52]  Grifith A. (1921). “The phenomena of rupture and flow in solids” Philosophical Transactions of the Royal Society of London A 221, pp., 163-198
In article      View Article
 
[53]  Griffith A. (1924). “The theory of rupture.” In: Proc., Ist., Int., Congr., Appl., Mech. Biereno, C.B. Burgers, J.M(eds). Delft: Tech. Boekhandel en Drukkerij. J. Waltman Jr., pp. 54-63.
In article      
 
[54]  Calbet A., Diaz Herrera P and Rodrignex L. (1999). “High bone mineral density in male elite professional volleyball players.” Osteopor., Inter, 10, pp. 468-474.
In article      View Article  PubMed
 
[55]  Fehling P., Alekel L. and Classey J., et., al., (1995). “A comparison of bone mineral densities among female athletes in impact loading and active loading sports.” Bone 17., pp. 205-210.
In article      View Article
 
[56]  Ito M., Nakamura T., Ikesa S., et., al., (2001). “Effects of life time volleyball exersize on bone mineral densities in lum barspine, calcaneus and tibia for pro, peri- and postmenopausal women.” Ostepor. Intern., 12, pp.104-111.
In article      View Article  PubMed
 
[57]  Rittweger J., Beller G., Ehrig G., et., al., (2000). “Bone- muscle strenght indices for the human lower leg.” Bone , 27. pp. 319-326.
In article      View Article
 
[58]  Alfredson N., Nordsatorm P. and Lorentzon R., (1997). “Bone mass in female volleyball players. A comparison of total and regional bone mass in female volleyball players and nonactive females.” Calcif., Tissue, Int., 60, pp. 338-342.
In article      
 
[59]  Hara S., Yanagi H., Amagai H., et., al., (2001). “Effect of physical activity during teenage years, based on type of sport and duration of exersize on bone mineral density of young, premenopausal Japanese women.” Calcif., Tissue Int., 68., pp. 23-30.
In article      View Article
 
[60]  Nikander R., Sievanen H., Heinonen A. et., al., (2005). “Femoral neck structure in adult female athlete, subjected to different loading modalities.” J. Bone Miner., Res. 20., pp. 520-528.
In article      View Article
 
[61]  Nikander R., Sievanen H., Heinonen A. et., al., (2006). “Loading modalities and bone structures at nonweight-bearing upper extremity and weight - bearing lower extremity” A poct- study of adult female athletes.” Bone, 39, pp. 886-894.
In article      View Article  PubMed
 
[62]  Nichols L., Raugh M., Barrack M. (2007). “Bone mineral density in female high school athletes: Interactions of mentrual function and type of mechanical loading.” Bone 41., pp. 371-377.
In article      View Article  PubMed
 

Published with license by Science and Education Publishing, Copyright © 2017 M. Tsili and D. Zacharopoulos

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M. Tsili, D. Zacharopoulos. Does Temperature Effects the Growth of Microcracks in Tibia due to Volleyball?. Journal of Biomedical Engineering and Technology. Vol. 5, No. 1, 2017, pp 6-11. http://pubs.sciepub.com/jbet/5/1/2
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Tsili, M., and D. Zacharopoulos. "Does Temperature Effects the Growth of Microcracks in Tibia due to Volleyball?." Journal of Biomedical Engineering and Technology 5.1 (2017): 6-11.
APA Style
Tsili, M. , & Zacharopoulos, D. (2017). Does Temperature Effects the Growth of Microcracks in Tibia due to Volleyball?. Journal of Biomedical Engineering and Technology, 5(1), 6-11.
Chicago Style
Tsili, M., and D. Zacharopoulos. "Does Temperature Effects the Growth of Microcracks in Tibia due to Volleyball?." Journal of Biomedical Engineering and Technology 5, no. 1 (2017): 6-11.
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[1]  Burstein, A., Reilly, D. and Martens, M. (1976). “Aging of bone tissue mechanical properties. J. Bone Joint Surg.” A58, 82-86.
In article      View Article
 
[2]  Thompson, D. (1980): “Age changes in bone mineralization, cortical thickness, and haversian canal area.” Calcif Tissue Int. 31, 5-11.
In article      View Article  PubMed
 
[3]  Grynpas, M. D.and Holmyard, D. (1988): “Changes inquality of bone mineral on aging and in disease.” Scan Microsc. 2, 1045-1054.
In article      
 
[4]  Hui, S. L., Slemenda, C. W. and Johnston, C.C. (1988): Age and bone mass as predictors of fracture in a prospective study. J. Clin. Invest. 81, 1804-1809.
In article      View Article  PubMed
 
[5]  Kiebzak G. M. (1991). “Age-related bone changes.” Exp. Gerontol. 26, 171-187.
In article      
 
[6]  Simmons, E. D., Pritzker, K. P. and Grynpas, M. D. (1991). “Age - related changes in the human femoral cortex.” J. Orthop. Res.9, 155-167.
In article      View Article  PubMed
 
[7]  Melvin JW. “Fracture mechanics of bone.” J. Biomech. Eng. 1993. Nov; 115(4B):549-554.
In article      
 
[8]  Currey, J. D., Brear, K. and Zioupos, P. (1996). “The effects of aging and changes in mineral content in degrading the toughness of human femora.” J. Biomech. 29, 257-260.
In article      View Article
 
[9]  Francis, R.M. (1996). “Low bone mineral content is common but osteoporotic fractures are rare in elderly rural Gambian women.” J. Bone Miner. Res.11, 1019-1025.
In article      
 
[10]  Aspray, T. J., Prentice, A., Cole, T. J., Sawo, Y., Reeve, J. and Francis RM. (1996). “Low bone mineral content is comon but osteoporotic fractures are rare in elderly rural Gambian women.” J. Bone Miner. Res.11, 1019-1025
In article      
 
[11]  Yeni, Y. N. and Norman, T. L. (2000). “Fracture toughness of human femoral neck: Effect of microstructure, composition and age.” Bone 26, 499-504.
In article      View Article
 
[12]  Wang, X., Shen, X., Li, X. and Agrawal C. M. (2002). “Age- related changes in the collagen network and toughness of bone.” Bone 31, 1-7.
In article      View Article
 
[13]  Akkus, O., Adar, F. and Schaffler, M. B. (2004). “Age-related changes in physicochemical properties of mineral crystals are related to impaired mechanical function of cortical bone.” Bone 34, 443-453.
In article      View Article  PubMed
 
[14]  Ritchie. R, Kinney H., Kruzic R., and Nalla R. (2005). “A fracture mechanics and mechanistic approach to the failure of cortical bone.” Fatigue Fract. Engng Mater Struct. Fatigue Fract. Engng Mater Struct 28, 345-37
In article      View Article
 
[15]  Behiri, JC. and Bonfield W. (1989): “Orientation depen-dence of the fracture mechanics of cortical bone.” J. Biomech,. 22, 863-872.
In article      View Article
 
[16]  Yeni, Y. N., Brown, C. U., Wang, Z. and Norman, T. L. (1997). “The influence of bone morphology on fracture toughness of the human femur and tibia.” Bone 21, 453-459.
In article      View Article
 
[17]  Yeni, Y. N., Brown, C.U. and Norman, T.L. (1998). “Influence of bone composition and apparent density on fracture toughness of the human femur and tibia.” Bone 22, 79-84.
In article      View Article
 
[18]  Feng, Z., Rho, J., Han, S. and Ziv, I. (2000). “Orientation and loading condition dependence of fracture toughness in cortical bone.” Mater. Sci. Engng CC11, 41-46.
In article      View Article
 
[19]  Brown, C. U., Yeni, Y. N. and Norman, TL.(2000). “Fracture toughness is dependent on bone location-A study of the femoral neck, femoral shaft and the tibial shaft.” J. Biomed. Mater. Res.49, 380-389.
In article      View Article
 
[20]  Phelps, J. B., Hubbard, G. B., Wang, X. and Agrawal C. M. (2000): “Microstructural heterogeneity and the fracture toughness of bone.” J. Biomed. Mater. Res.51, 735-471.
In article      View Article
 
[21]  Yeni, Y. N. and Norman, T. L. (2000). “Fracture toughness of human femoral neck: Effect of microstructure, composition and age.” Bone 26, 499-504.
In article      View Article
 
[22]  Seeman, E. (1999). “The structural basis of bone fragility in men.” Bone 25, 143-147.
In article      View Article
 
[23]  Rimnac, C. M., Petko, A. A., Santners, T. J. and Wright, T. M (1993). “The effect of temperature, stress and microstructure on the creep of compact bovine bone.” J. Biomech.26, 219-228.
In article      View Article
 
[24]  Ford C.M. and Keaveny, T.M. (1996). “The dependence of shear failure properties of trabecular bone on apparent density and trabecular orientation.” J. Biomech. 29, 1309-1317.
In article      View Article
 
[25]  Carter, D. R. and Hayes, W. C. (1976): “Fatigue life of compact bone-I. Effects of stress amplitude, temperature and density.” J. Biomech.9, 27-34, Biomech. 26, 219-228.
In article      
 
[26]  Norman, T.L., Nivargikar, S. V. and Burr, D. B. (1996). “Resistance to crack growth in human cortical bone is greater in shear than in tension.” J. Biomech.29, 1023-1031.
In article      View Article
 
[27]  Feng, Z., Rho, J., Han, S. and Ziv, I. (2000). “Orientation and loading condition dependence of fracture toughness in cortical bone.” Mater. Sci. Engng CC11, 41-46.
In article      View Article
 
[28]  Feng X. and McDonald J. (2011). “Disorders of Bone Remodeling,” Annu Rev Pathol.; 6: 121-145.
In article      
 
[29]  Cowin S. and Hegedus D. (1976). “Bone remodeling I: Theory of adaptive elasticity” J. Elastic. 6, pp. 313-326.
In article      View Article
 
[30]  Hegedus D. και Cowin S. (1976). “Bone remodeling II: Theory of adaptive elasticity” J. Elastic. 6, pp. 337-352.
In article      View Article
 
[31]  Sih G.C. (1985). “Mechanics and Physics of energy density theory”, Theoret., Appl., Fract. , Mech., 44, pp. 157-173.
In article      View Article
 
[32]  Sih G.C (1972-1982). “Mechanics of fracture, Introductory chapters”, Vol. I- VII, edited by G.C. Sih, Martinus Nijhoff, The Hague.
In article      
 
[33]  Sih GC(1988). “Thermomechanics of solids: nonequilibrium and irreversibility”, Theoretical and Applied Fracture Mechanics, 99, pp. 175-198.
In article      View Article
 
[34]  Τsili M. (2008). “The hyperthrophy of tibia induced by the volleyball” in: www.ispub.com/journal-of-internet journal of bioengineering , Volume .4. number 1.
In article      
 
[35]  Harless E. (1860). “The static moments of human limbs (in German). Treatsises of the Math-Phys. Class of the Royal Acad. of Sci. of Bavaria, 8, pp.69 και 257.
In article      
 
[36]  Andriachi T., Ogle J. and Galante J. (1977). “Walking speed as a base for normal and abnormal gait measurement.” J. Biomech., 10, pp. 261-268.
In article      View Article
 
[37]  Rohrle H., Scholten R., Sigolloto C. et., al., (1984). “Joint forces in the human pelvis - leg skeleton during walking.” J. Biomech., 17, pp. 409-424.
In article      View Article
 
[38]  Whalen R., Carter D. and Steele C. (1988). “Influence of the physical activity on the regulation of bone activity.” J. Biomech., 21., pp., 825-837.
In article      View Article
 
[39]  Αlexander N. and Jayes A. (1980). “Fourier analysis on forces exerted in walking and running.” J. Biomech. 13. pp. 383-390.
In article      
 
[40]  Bates B., Osterning L. and Sawhill J. (1983). “An assessment of subject variability, subject -shoe interaction and the evaluation of running shoes using ground reaction force data.” J. Biomech., 16., pp.181-191.
In article      View Article
 
[41]  Cavagna P. (1964). “Mechanical work in running” J. Appl. Physiol. 19., pp. 249-256
In article      
 
[42]  Cavagna P. and LaFortune M. (1980). “Ground reaction forces in distance -running.” J. Biomech., 13. pp. 397-406.
In article      
 
[43]  Fukanaga T., Matsuo A., Yuasa K., et., al., (1980). “Effect of running velocity on external mechanical output.” Ergonomics, 23, pp.,123-136
In article      View Article  PubMed
 
[44]  Winter D. (1983). “Moments of forces and mechanical power in jogging.” J.Biomech., 16, pp., 91-97.
In article      View Article
 
[45]  Frost H.M (1964). “Dynamics of bone remodeling in bone biodynamics.” (edited by Frost H.M) Little and Brown 316, Boston.
In article      
 
[46]  Reilly D.T. and Burstein A. (1975). J. Biomech., 8., p.393.
In article      
 
[47]  Cowin S. and Van -Burskirk W. (1978). “Internal bone remodeling induced by a medullary pin.” J. Biomech. 11, pp. 269-275.
In article      View Article
 
[48]  Wolff. J. (1884). “Das gesetz der transformation der inneren architecture knocken bei pathologism veranderungen der aussen knochenform.” Sitz Ber. Preuss Acad. d. Wiss 22, Sitz Physik- Math. K1.
In article      
 
[49]  Wolff J. (1892). Das gesetz der transformation knocken hirschald, Berlin.
In article      
 
[50]  Τsili M. (2000). “Theoretical solutions for internal bone remodeling of diaphyseal shafts using adaptive elasticity theory.” J. Biomech., 33 pp. 235-239.
In article      
 
[51]  Τsili M. (2008): “Internal bone remodeling induced by the distance - running and the unkown remodeling coefficients” in: www.ispub.com/journal- of- internet journal of bioengineering vol. 4. number 2.
In article      
 
[52]  Grifith A. (1921). “The phenomena of rupture and flow in solids” Philosophical Transactions of the Royal Society of London A 221, pp., 163-198
In article      View Article
 
[53]  Griffith A. (1924). “The theory of rupture.” In: Proc., Ist., Int., Congr., Appl., Mech. Biereno, C.B. Burgers, J.M(eds). Delft: Tech. Boekhandel en Drukkerij. J. Waltman Jr., pp. 54-63.
In article      
 
[54]  Calbet A., Diaz Herrera P and Rodrignex L. (1999). “High bone mineral density in male elite professional volleyball players.” Osteopor., Inter, 10, pp. 468-474.
In article      View Article  PubMed
 
[55]  Fehling P., Alekel L. and Classey J., et., al., (1995). “A comparison of bone mineral densities among female athletes in impact loading and active loading sports.” Bone 17., pp. 205-210.
In article      View Article
 
[56]  Ito M., Nakamura T., Ikesa S., et., al., (2001). “Effects of life time volleyball exersize on bone mineral densities in lum barspine, calcaneus and tibia for pro, peri- and postmenopausal women.” Ostepor. Intern., 12, pp.104-111.
In article      View Article  PubMed
 
[57]  Rittweger J., Beller G., Ehrig G., et., al., (2000). “Bone- muscle strenght indices for the human lower leg.” Bone , 27. pp. 319-326.
In article      View Article
 
[58]  Alfredson N., Nordsatorm P. and Lorentzon R., (1997). “Bone mass in female volleyball players. A comparison of total and regional bone mass in female volleyball players and nonactive females.” Calcif., Tissue, Int., 60, pp. 338-342.
In article      
 
[59]  Hara S., Yanagi H., Amagai H., et., al., (2001). “Effect of physical activity during teenage years, based on type of sport and duration of exersize on bone mineral density of young, premenopausal Japanese women.” Calcif., Tissue Int., 68., pp. 23-30.
In article      View Article
 
[60]  Nikander R., Sievanen H., Heinonen A. et., al., (2005). “Femoral neck structure in adult female athlete, subjected to different loading modalities.” J. Bone Miner., Res. 20., pp. 520-528.
In article      View Article
 
[61]  Nikander R., Sievanen H., Heinonen A. et., al., (2006). “Loading modalities and bone structures at nonweight-bearing upper extremity and weight - bearing lower extremity” A poct- study of adult female athletes.” Bone, 39, pp. 886-894.
In article      View Article  PubMed
 
[62]  Nichols L., Raugh M., Barrack M. (2007). “Bone mineral density in female high school athletes: Interactions of mentrual function and type of mechanical loading.” Bone 41., pp. 371-377.
In article      View Article  PubMed