284 L.N.McCartney These bounds are valid only if 4。≤lm, (7.40) and the bounds are reversed ifp Shear modulus Bounds for the shear modulus of a particulate composite may be written in the following form (4。-4m)2V'n π- ≤ler V,m+'m4,+ (9km +8um)um 6(km+24m） (7.41) (4。-4m)V'm Volm +Vmkp+ 9k。+8,), 6(k。+24,) where 正=V。+'mlm (7.42) These bounds are valid for the practically important case kp≥km,lp≥lm (7.43) and the bounds are reversed if kp≤km,lp≤lm Other cases can arise that are not considered here. Thermal expansion Bounds for the thermal expansion coefficient of a particulate composite may be written in the following formp m µ ≤ µ , (7.40) and the bounds are reversed if µp ≥ µ m . Shear modulus Bounds for the shear modulus of a particulate composite may be written in the following form 2 p m pm eff m mm pm mp m m 2 p m pm p pp pm mp p p ( ) (9 8 ) 6( 2 ) ( ) , (9 8 ) 6( 2 ) V V k V V k V V k V V k µ µ µ µ µ µ µ µ µ µ µ µ µ µ µ µ µ − − ≤ + + + + − ≤ − + + + + where p p mm µ =V V µ µ + . (7.42) Thermal expansion Bounds for the thermal expansion coefficient of a particulate composite may be written in the following form L.N. McCartney These bounds are valid only if (7.41) p m p m k k ≥ , k k p m p m ≤ ≤ , . These bounds are valid for the practically important case and the bounds are reversed if Other cases can arise that are not considered here. ≥ µµ , (7.43) µ µ 284