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.

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.

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 u_o. An angle α is formed between the direction of the vector of velocity u_o and the horizontal axon, as it seems in Figure 2. The maximums high h_M and horizontal displacement S_M 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 F_zi 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