the frictional force between the femoral head and acetabulum is neglected since

# The frictional force between the femoral head and

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the frictional force between the femoral head and acetabulum is neglected since the fact of synovia in the hip joint [31]. The elastic modulus and Poisson ratios for the articular cartilages of the acetabulum and the femoral head are, respectively, El, it , and E2, A 2• The point m and n (see Fig.1) are two arbitrary points on the two hemispherical surfaces, which are very z2 = - 2R2 (4) Fig 1. The coordinate system for the ball-socket elastic contact modelling. -2- Biomed. Eng. Appl. Basis Commun. 2004.16:233-237. Downloaded from by 14.139.37.108 on 11/11/19. Re-use and distribution is strictly not permitted, except for Open Access articles.
BIOMEDICAL ENGINEERING- APPLICATIONS, BASIS & COMMUNICATIONS Or r2 1 1 R,-RZ 2 z2 -z, = 2 R2 R, (5) 2R,R2 r If the displacements of the points m and n at the t- axis direction are, respectively, wl and w2, then the shorten quantity between any two points at the z-axis, 4, can be written as following, 8=z2-z,+w1 +w2 Substituting Eq.(5) into (6), we have, w, +w2 =,5 -,8 r2 R, - R2 Q- (6) (7) where 2R,R2 . This is the displacement equation or geometric relationship of the ball-socket elastic contact modelling at the hip joint. 2.2 Physical Relationships and Equilibrium Equations on the Modelling It is assume that the displacement at any point M on the contact area is w due to the contact pressures. If the element area is taken as sdrpds , here the s is the distance from the point M to the element area, while the cp is the angle between the secant line through the point M and the oy axis (see Figure 2 ). Then, the contact pressure acting on the element area can be written as following, gsdrpds 235 Fig 2 . The coordinate system for calculating pressures on the contact area. where the A is the elastic contact area. Substituting Eq.(9) into (7), we have, (k,+k2)Jjgdsdcp=(5 -fire (10) A z z where k, _ = 1^ E' , kz 2 = 1)T E 2 It is because of the fact that the elastic contact area between the hip head and acetabulum is the circular one, therefore, at the boundary of the contact area, the following relationship must be held, w, + w2 =0 (11) Thus, from Eq.(7) or (10), the radii of the elastic contact area can be determined as following, rc = S/,fl (12) The integral equation ( 10) can be solved by the inverse method. The results are given as following: where the q is the contact pressures on the contact area. rc [3)rP(k, +k2)R,RZ 3 Thus, according to the Boussinesq Problem's 4(R, - R2 ) solutions [30], the displacement at any point M due to 3P the pressures gsdcpds can be written as following, qo = 27rr 2 c (13) 2 1- ' = dr ds (8) 2 1 2 W j jg p 3P r 2 z 2 7r E A q(,;, 77) = 2 z (1 _ ^ r - ^ +q If the two elastic bodies contact each other, then we have, r. 27rr 37rP(k, + k2 ) 4rc W, - 1- u' ffq drp ds 7zE A Where the r, is radii of the contact area ; the q0 is (9) the maximum contact stress; the q is the stress distributions of the contact area ; the P is the resultant W2-1-uz JJgdrpds 7r E2 A force acting on the hip joint.

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• Summer '20
• Rajendra Paramanik
• joints, Synovial joint, Hertzian contact stress

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