302 Random Composites the coordinates of the hemisphere with the radius a constructed on a contact surface.If go is taken as the pressure at the point C,then one can show that qds=10A a (6.18) where A--sin) (6.19) which gives k+血6：-r产n'pp=a-图-） (6.20) a 2RR2 Finally,the parameters a and a can be determined for this problem as 3 P(k +k2 RR2 a=3 (6.21) 4 R-R2 9n2 P2(k+k2)(R-R2) 16 RR which gives maximal pressure on the contact surface equal to 3P (6.22) 9= 2a2 Then,the normal stress can be defined as o:=-J3grdrz (6.23) -1*石4同 Let us note that the shear stresses are equal to 0,which result from the spherical symmetry of the reinforcing particle.However in the case of ellipsoidal reinforcement the shear stresses differ from 0. The main purpose of further analysis is to determine the probabilistic characteristics of maximal contact stresses as well as contact surface geometrical parameters.Since the spherical particle surrounding the matrix is considered,let us assume that the difference R-R2=e is smaller than R2.This parameter is treated as302 Random Composites the coordinates of the hemisphere with the radius a constructed on a contact surface. If q0 is taken as the pressure at the point C, then one can show that A a q qds 0 ∫ = (6.18) where ( ) ϕ π 2 2 2 sin 2 A = a − r (6.19) which gives ( ) ( ) ( ) 1 2 1 2 2 0 1 2 0 2 2 2 2 sin 2 R R r R R a r d a k k q − − = − + ∫ ϕ ϕ α π π (6.20) Finally, the parameters a and α can be determined for this problem as ( ) ( )( ) ⎪ ⎪ ⎩ ⎪ ⎪ ⎨ ⎧ + − = − + = 3 1 2 1 2 2 1 2 2 2 3 1 2 1 2 1 2 16 9 4 3 R R P k k R R R R P k k R R a π α π (6.21) which gives maximal pressure on the contact surface equal to 2 2 3 a P q π = (6.22) Then, the normal stress can be defined as ( ) ( ) ( )⎥ ⎦ ⎤ ⎢ ⎣ ⎡ + = + = − + = − + − − ∫ 2 2 3 0 3 / 2 3 2 2 0 5 / 2 3 2 2 1 3 a z z qz r z q qrdrz r z a a σ z (6.23) Let us note that the shear stresses are equal to 0, which result from the spherical symmetry of the reinforcing particle. However in the case of ellipsoidal reinforcement the shear stresses differ from 0. The main purpose of further analysis is to determine the probabilistic characteristics of maximal contact stresses as well as contact surface geometrical parameters. Since the spherical particle surrounding the matrix is considered, let us assume that the difference R1-R2=ε is smaller than R2. This parameter is treated as