286 L.N.McCartney Poisson's ratio The bounds are given by 1 (A- (w-) VV VA- V ≤v ≤VA 1 (7.50) where VA =VEVA+VRVA (7.51) These bounds are valid only if (-k-k号4-4)≥0， (7.52) and the bounds are reversed if (v-v)(-k)()0. Transverse bulk modulus The bounds are given by 11 1.1 1 S、 (7.53) k 1 m十 k m+ 1 m where (7.54) These bounds are valid only if 店≤， (7.55) and the bounds are reversed if
Poisson’s ratio The bounds are given by f m f m A A f m f m A A f m f m T T eff T T A A A m f m f fmm fmf TT T TT T 1 1 1 1 ( ) ( ) , 1 1 V V V V k k k k V V V V k k k k ν ν ν ν ν ν ν µ µ ⎛ ⎞ ⎛ ⎞ − − ⎜ ⎟ − − ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ − ≤ ≤ − + + + + (7.50) where f m A f A mA ν = + V V ν ν . (7.51) These bounds are valid only if f mf m f m A AT T T T ( )( )( ) 0, νν µµ − − −≥ k k (7.52) and the bounds are reversed if f mf m f m A AT T T T ( )( )( ) 0 νν µµ − k k − −≤ . Transverse bulk modulus The bounds are given by 2 2 f m f m f m f m T T T T eff T T m f T m f fmm fmf TT T TT T 1 1 1 1 1 1 1 , 1 1 V V V V k k k k k k V V k V V k k µ µ k k ⎛⎞ ⎛⎞ ⎜⎟ ⎜⎟ − − ⎝⎠ ⎝⎠ − ≤ ≤ − + + + + (7.53) where f m f m T TT 1 . V V kkk = + (7.54) These bounds are valid only if f m T T µ ≤ µ , (7.55) and the bounds are reversed if f m µ T T ≥ µ . 286 L.N. McCartney
Chapter 7:Multiscale Modeling of Composites 287 Transverse shear modulus The bounds are given by (4-)Wm ≤4s4- (-)2Wm 'm+'吧+ 吧 k+2 V+,4++2 (7.56) where u=Vihi +Vmu. (7.57) These bounds are valid for the practically important case 际≥k四，≥咒， (7.58) and the bounds are reversed if k≤k四，5≤四 Other cases can arise that are not considered here. Axial shear modulus The bounds are given by .-s≤A+以+ -(4-4VΨ (7.59) 4+吸+吸 where 4a='以+Vm吸 (7.60) These bounds are valid only if 咽≤， (7.61) and the bounds are reversed if
Transverse shear modulus The bounds are given by f m2 f m2 T T fm eff T T fm T T m m T f f f m T T f m T T mT f T m m mT f T f f T T T T ( ) ( ) , 2 2 V V V V k k V V V V k k µ µ µ µ µ µ µ µ µ µ µ µ µ µ µ − − − ≤ ≤ − + + + + + + (7.56) where f m T f T mT µ = + V V µ µ . Axial shear modulus The bounds are given by f m2 f m2 A A fm eff A A fm A A f mm A f mf mA f A A mA f A A ( ) ( ) , V V V V V V V V µ µ µ µ µ µ µ µ µ µ µ µµ − − − ≤ ≤ − + + + + (7.59) where f m A f A mA µ = + V V µ µ . (7.60) These bounds are valid only if m f A A µ ≤ µ , (7.61) and the bounds are reversed if m f µ A A ≥ µ . Chapter 7: Multiscale Modeling of Composites f m f m T T T T k k ≥ ≥ , f m f m T T T T k k ≤ ≤ , . These bounds are valid for the practically important case and the bounds are reversed if Other cases can arise that are not considered here. , (7.57) µ µ µ µ (7.58) 287