342
25 Building a Shear Resistance Operator
Using the results of slip synthesis (p. 306), let us choose the operator L νλ ϕ nl as
follows:
L νλ ϕ nl = aϕ νλ + bb νλ + gr νλ + cγ νλ , (a, b, c, g − const).
(25.3)
The first addend here is proportional to the displacement intensity of dislocations
ϕ νλ in those planes and the directions where the shear resistance S νλ is defined. The
second addend is proportional to the vector slip intensity and changes as a projection
to the direction λ of shear in the direction l for n = ν, e.g. as a shear component
in the plane with the normal line ν from slip in the same plane. The third addend
is proportional to the tensor intensity of slip, e.g. to the shear component (γ νλ ) in
the plane with the normal line ν from all slips in planes parallel to the direction
where the shear resistance is sought. The last addend is proportional to the strain
component γ νλ from slips in all planes and directions.
To describe the strain of hardening materials in the studies [2, 5], the operator
L νλ ϕ nl contained only two addends (either aϕ νλ and cγ νλ or bb νλ and cγ νλ ),
since setting the operator L νλ ϕ nl in a more general form (for example, as (25.3)
significantly simplifies the solution of some partial problems. Furthermore, the
papers given in this paragraph in the expression of shear resistance (25.2) did not
take into account aging and the effects of the slip rate (ε = A = B = 0) and
also assumed ψ ≡ . The shear resistance dependency on the rate of the operator
L νλ ϕ nl was first introduced in the article [3].
25.2 Boundary Condition
The operator (25.3) can also be written as follows:
L νλ ϕ nl =
{L}
ϕ νl F 1 (ω lλ )dω lλ + gr νλ + cγ νλ , (ω lλ =
lλ),
(25.4)
where
F 1 (ω) =
a
2
[δ(ω) − δ(|ω| − π)] + b cos ω.
In the latter formula, δ is the Dirac function, and the function F 1 (ω) being the core
of the integral operator (25.4) characterizes the sensitivity of the material to strain
anisotropy.
If we adopt the continuity postulate [1], the function F 1 must satisfy the condition
F 1 (0) = ∞
at a = 0.
(25.5)
25 Building a Shear Resistance Operator
Using the results of slip synthesis (p. 306), let us choose the operator L νλ ϕ nl as
follows:
L νλ ϕ nl = aϕ νλ + bb νλ + gr νλ + cγ νλ , (a, b, c, g − const).
(25.3)
The first addend here is proportional to the displacement intensity of dislocations
ϕ νλ in those planes and the directions where the shear resistance S νλ is defined. The
second addend is proportional to the vector slip intensity and changes as a projection
to the direction λ of shear in the direction l for n = ν, e.g. as a shear component
in the plane with the normal line ν from slip in the same plane. The third addend
is proportional to the tensor intensity of slip, e.g. to the shear component (γ νλ ) in
the plane with the normal line ν from all slips in planes parallel to the direction
where the shear resistance is sought. The last addend is proportional to the strain
component γ νλ from slips in all planes and directions.
To describe the strain of hardening materials in the studies [2, 5], the operator
L νλ ϕ nl contained only two addends (either aϕ νλ and cγ νλ or bb νλ and cγ νλ ),
since setting the operator L νλ ϕ nl in a more general form (for example, as (25.3)
significantly simplifies the solution of some partial problems. Furthermore, the
papers given in this paragraph in the expression of shear resistance (25.2) did not
take into account aging and the effects of the slip rate (ε = A = B = 0) and
also assumed ψ ≡ . The shear resistance dependency on the rate of the operator
L νλ ϕ nl was first introduced in the article [3].
25.2 Boundary Condition
The operator (25.3) can also be written as follows:
L νλ ϕ nl =
{L}
ϕ νl F 1 (ω lλ )dω lλ + gr νλ + cγ νλ , (ω lλ =
lλ),
(25.4)
where
F 1 (ω) =
a
2
[δ(ω) − δ(|ω| − π)] + b cos ω.
In the latter formula, δ is the Dirac function, and the function F 1 (ω) being the core
of the integral operator (25.4) characterizes the sensitivity of the material to strain
anisotropy.
If we adopt the continuity postulate [1], the function F 1 must satisfy the condition
F 1 (0) = ∞
at a = 0.
(25.5)