Matrices and Tensors - Continuum Mechanics of Anisotropic Materials - page 384

equal to C kmij , in general. The results of interest are for fourth order tensors in three
dimensions with these particular symmetries because it is these fourth order tensors
that linearly relate two symmetric second order tensors in three dimensions. Due to
the special indicial symmetries just described, the change of basis (A.149) may be
introduced in (A.153) and may be rewritten as

C ¼ C ij e i e j ¼ C ab e a e b ;

(A.154)

where the 36 components of c ijkm , the fourth order tensor in three dimensions (with
the symmetries c ijkm ¼ c jikm and c ijkm ¼ c ijmk ) are related to the 36 components of

^

c ij , the second order tensor in six dimensions by

c 11 ¼ ^ c 1111 ; ^ c 22 ¼ c 2222 ; ^ c 33 ¼ c 3333 ; ^ c 23 ¼ c 2233 ; ^ c 32 ¼ c 3322 ;

c 13 ¼ ^ c 1133 ; ^ c 31 ¼ c 3311 ; ^ c 12 ¼ c 1122 ; ^ c 21 ¼ c 2211 ;

c 44 ¼ ^ 2 c 2323 ; ^ c 55 ¼ 2 c 1313 ; ^ c 66 ¼ 2 c 1212 ; ^ c 45 ¼ 2 c 2313 ; ^ c 54 ¼ 2 c 1323 ;

c 46 ¼ ^ 2 c 2312 ; ^ c 64 ¼ 2 c 1223 ; ^ c 56 ¼ 2 c 1312 ; ^ c 65 ¼ 2 c 1213 ;

p c 2311 ; ^ p c 1123 ; ^

p c 1311 ; ^

p c 1113 ;

c 41 ¼ ^ c 14 ¼ c 51 ¼ c 15 ¼

p c 1211 ; ^ p c 1112 ; ^

p c 2322 ; ^

p c 2223 ;

c 61 ¼ ^ c 16 ¼ c 42 ¼ c 24 ¼

p c 1322 ; ^ p c 2213 ; ^

p c 1222 ; ^

p c 2212 ;

c 52 ¼ ^ c 25 ¼ c 62 ¼ c 26 ¼

p c 2333 ; ^ p c 3323 ; ^

p c 1333 ; ^

p c 3313 ;:

c 43 ¼ ^ c 34 ¼ c 53 ¼ c 35 ¼

p c 1233 ; c 36 ¼ p c 3312 :

c 63 ¼

(A.155)

J T , as well as
the two indicial symmetries of c ijkm , the linear relationship between T and J ,

T T and J

Using the symmetry of the second order tensors, T ¼ ¼

T ij ¼ C ijkm J km ; (A.156)

may be expanded to read

T 11 ¼ c 1111 J 11 þ c 1122 J 22 þ c 1133 J 33 þ 2 c 1123 J 23 þ 2 c 1113 J 13 þ 2 c 1112 J 12 ;

T 22 ¼ c 2211 J 11 þ c 2222 J 22 þ c 2233 J 33 þ 2 c 2223 J 23 þ 2 c 2213 J 13 þ 2 c 2212 J 12 ;

T 33 ¼ c 3311 J 11 þ c 3322 J 22 þ c 3333 J 33 þ 2 c 3323 J 23 þ 2 c 3313 J 13 þ 2 c 3312 J 12 ;

T 23 ¼ c 2311 J 11 þ c 2322 J 22 þ c 2333 J 33 þ 2 c 2323 J 23 þ 2 c 2313 J 13 þ 2 c 2312 J 12

T 13 ¼ c 1311 J 11 þ c 1322 J 22 þ c 1333 J 33 þ 2 c 1323 J 23 þ 2 c 1313 J 13 þ 2 c 1312 J 12 ;

T 12 ¼ c 1211 J 11 þ c 1222 J 22 þ c 1233 J 33 þ 2 c 1223 J 23 þ 2 c 1213 J 13 þ 2 c 1212 J 12 :

(A.157)