Modeling Material Microstructure - Continuum Mechanics of Anisotropic Materials - page 195

F 2 T ¼ a 1 1 tr E þ a 2 ð F ½ tr E þ 1 tr ½ E F Þ þ a 3 ð 1 tr ½ F E F þð tr E Þ Þ

F 2 F 2 tr

þ b 1 F tr ½ F E þ b 2 ð F tr ½ F E þ ð tr E F Þ Þþ b 3 ð ½ F E F Þ

þ 2 c 1 E þ 2 c 2 ðF E þ E FÞþ 2 c 3 ðF 2 E þ E F 2

Þ (7.37)

,tr F ,tr F 2 and tr F 3 .This
representation has been used to represent the elastic behavior of highly porous
cancellous bone tissue. The form of the functional dependence of the elasticity
tensor C ijkm (recall the stress-strain form of the anisotropic Hooke's law, T ij ¼
C ijkm E km ) upon fabric is given by

where a 1 , a 2 , a 3 , b 1 , b 2 , b 3 , c 1 , c 2 , and c 3 are functions of j

C ijkm ¼ a 1 d ij d km þ a 2 ð a 3 ðd ij F kq F qm þ d km F iq F qj Þ

F ij d km þ d ij F km Þþ

b 1 F ij F km þ b 2 ð b 3 F is F sj F kq F qm

þ F ij F kq F qm þ F km F iq F qj Þþ

(7.38)

c 1 ðd ki d mj þ d mi d kj Þþ c 2 ð

þ F ki d mj þ F kj d mi þ F im d kj þ F mj d ki Þ

c 3 ð þ F ir F rk d mj þ F kr F rj d mi þ F ir F rm d kj þ F mr F rj d ik Þ

where, as in ( 7.38 ) the a 1 , a 2 , a 3 , b 1 , b 2 , b 3 , c 1 , c 2 , and c 3 with the superscript c are
functions of ,tr F ,tr F 2 and tr F 3 . The fourth-rank elastic compliance tensor S ijkm j

(recall E ij ¼ S ijkm T km ) for the strain-stress relation is

a 1 d ij d km þ a 2 ð a 3 ðd ij F kq F qm þ d km F iq F qj Þ S ijkm ¼ F ij d km þ d ij F km Þþ

b 1 F ij F km þ b 2 ð b 3 F is F sj F kq F qm

þ F ij F kq F qm þ F km F iq F qj Þþ

(7.39)

c 1 ðd ki d mj þ d mi d kj Þþ c 2 ð

þ F ki d mj þ F kj d mi þ F im d kj þ F mj d ki Þ

c 3 ð

þ F ir F rk d mj þ F kr F rj d mi þ F ir F rm d kj þ F mr F rj d ik Þ

where a 1 , a 2 , a 3 , b 1 , b 2 , b 3 , c 1 , c 2 and c 3 with the superscript s are functions of f and
the invariants of F and porosity ) of the material.

The least elastic material symmetry for which the representation holds is
orthotropy. It therefore holds for transverse isotropy and isotropy as well as
orthotropy.

j or solid volume fraction (1- j

Problems

7.8.1 Specialize the representation ( 7.39 ) for the components of the fourth order
compliance tensor to the case of isotropic symmetry. Relate the 9 coefficients
in your result to the two elastic constants in equation (6.26), Young's
modulus and Poisson's ratio. Upon what parameter(s) do the Young's modu-
lus and Poisson's ratio obtained depend?
7.8.2 Specialize the representation ( 7.38 ) for the components of the fourth order
elasticity tensor to the case of isotropic symmetry. Relate the 9 coefficients in
your result to the two Lam ´ moduli in equation (6.23) or (6.24). The resulting
Lam ´ moduli are functions of which parameter(s)?