CONTINUUM MECHANICS for ENGINEERS

A sum of dyads such as
is called a dyadic.
(g) vector-dyad products:
(2.2-18)
1. u vw e e e e
(2.2-19)
⋅( )= ⋅( uˆ v ˆ w ˆ )= uvw ˆ
2. uv w e e e e
(2.2-20)
( ) ⋅ =( uˆ v ˆ )⋅ w ˆ = uv w ˆ
3. u× ( vw)= ueˆ × v eˆ w eˆ ε uv w eˆ eˆ
(2.2-21)
4. ( uv)× w = ueˆ ( v eˆ × w eˆ )= uv w e e (2.2-22)
i i j j ε ˆ ˆ
k k jkq i j k i q
(Note that in products 3 and 4 the order of the base vectors is important.)
êi (h) dyad-dyad product:
(i) vector-tensor products:
(2.2-23)
1. vT ⋅ = veˆ ⋅ T ee ˆ ˆ = vT δ e ˆ = vT eˆ
(2.2-24)
2. T⋅ v= T ee ˆˆ ⋅ v eˆ = T e ˆδ v = T v eˆ
(2.2-25)
(Note that these products are also written as simply vT and Tv.)
(j) tensor-tensor product:
uv + uv + + u v
1 1 2 2
K N N
i i j j k k i i k k
i i j j k k i j j i
( ) =
( )⋅( )= ⋅
i i j j k k ijq i j k q k
( ) =
uv ws ueˆ v eˆ w eˆ s eˆ uv w s eˆ eˆ
i i j j k k q q i j j q i q
i i jk j k i jk ij k i ik k
ij i j k k ij i jk k ij j i
T⋅ S= T ee ˆˆ ⋅ S eˆ eˆ = T S ee ˆˆ
ij i j pq p q ij jq i q
(2.2-26)
Example 2.2-2
Let the vector v be given by v = ( a⋅nˆ) n+n ˆ ˆ × ( a× nˆ)
where a is an arbitrary
vector, and ˆn is a unit vector. Express v in terms of the base vectors ,
expand, and simplify. (Note that .)
êi nˆ⋅n= ˆ neˆ ⋅ neˆ = nnδ = nn = 1
i i j j i j ij i i