Page 18 - Mechanics of Asphalt Microstructure and Micromechanics

Introduction and Fundamentals for Mathematics and Continuum Mechanics 11

Where I 1 (first invariant, the trace), I 2 (second invariant), and I 3 (third invariant) are
the invariants of the tensor (matrix).

I = A = trA (1-33a)
1 ii
I = ( 1 2 2
1
tr A ⎤
A A − A A ji) = ⎡( trA − ( ) ⎦ (1-33b)
)
⎣
2 ii jj ij
2 2
I = ξ A A A = det A (1-33c)
3 ijk i 1 j 2 3 k
It can be proved that I 1 , I 2 , and I 3 are invariant when the tensor is transformed into
its representation into another rotated coordinate system.
Typically, it has three real roots.
⎡ ⎣ A − λδ ij ⎦ ⎤ n i q () = 0, (q 1,2,3) (1-34)
q ()
ij
and
q () ()
nn q = 1, (q 1,2,3)
i i
Replacing A with the stress tensor s or strain tensor e in the above operations, one
can find the principal stresses or principal strains. Due to the invariant nature of the
three invariants, they are often used in constitutive laws to avoid the violation of objec-
tivity principle.

1.5.6 Typical Operators
Operator (del) of vector calculus can be represented as:
∂ ∂ ∂ ∂
∇= e + e + e = e (1-35)
∂x 1 ∂x 2 ∂x 3 ∂x i
1 2 3 i
If f is a scalar function then f = gradf is the gradient of the function. It has applica-
tions in the plasticity theory for representing the flow rule. The gradient of a scalar
function is a vector.
The gradient of a vector field v is represented as:
∂v
∇= j e ⊗e (1-36)
v
∂x j i
i
It is a tensor.
The divergence of a vector field v is also scalar. It can be represented as:
∂v ∂v ∂v
∇• = j e • e = j δ = i (1-37)
v
x i j x ij ∂x
i i i
The curl of a vector is a vector and is defined as:
∂v
∇× = ξ j e (1-38)
v
ijk ∂x k
i