276 L.N.McCartney where Vm is the volume fraction of the matrix.The suffices p and m will be used to refer properties k,u,and a to the particles and matrix,respectively. It can be shown from (7.2)to (7.7)that the effective bulk modulus, shear modulus,and thermal expansion coefficient are given by 4 3亚+3 p 1 kgkm km (7.9) ke V p+ +3 k。44m 15(l-Vnm4。-4m)P, 2(4-5ym)Wn4。+,m)+(7-5aunJ (7.10) 13 de =an+he 4Mm V (d-ap). 1,3 (7.11) k。44m It has been shown [5]that it is possible to express these relations as the sum of mixtures estimates plus correction terms so that 1 m (7.12) kmk。kn V ++3 kmk。44m (4-4m)P Le =Vplp +Vmklm- 9ka+8m (7.13) Vom +Valp+6(km +24m) (7.14) ++3 kmk。44where Vm is the volume fraction of the matrix. The suffices p and m will be used to refer properties k, µ, and α to the particles and matrix, respectively. It can be shown from (7.2) to (7.7) that the effective bulk modulus, shear modulus, and thermal expansion coefficient are given by m m p pm m p eff p m m mp m 4 3 3 1 , 3 4 4 V V kk k k k V V k k µ µ µ + + = ⎛ ⎞ ⎜ ⎟ + + ⎝ ⎠ (7.9) m p mp eff m m mp pm m m 15(1 )( ) 1 , 2(4 5 )( ) (7 5 ) V V V νµµ µ µ ν µ µ νµ ⎡ ⎤ − − = + ⎢ ⎥ − + +− ⎣ ⎦ eff m eff m p p m p m 1 3 4 ( ). 1 3 4 k V k µ α α α α µ + = + − + (7.11) 2 p m p m p m eff p m p m mp m 1 1 1 , 3 4 V V V V k k k kk V V k k µ ⎛ ⎞ ⎜ ⎟ − ⎝ ⎠ =+ − + + (7.12) 2 p m eff p p m m p m m m pm mp m m m ( ) , 9 8 6( 2 ) V V V V k V V k µ µ µ µµ µ µµ µ µ − =+ − + + + + (7.13) p m pm p m eff p p m m p m mp m 1 1 ( ) . 3 4 V V k k V V V V k k α α α αα µ ⎛ ⎞ ⎜ ⎟ − − ⎝ ⎠ =+ − + + (7.14) L.N. McCartney (7.10) It has been shown [5] that it is possible to express these relations as the sum of mixtures estimates plus correction terms so that 276