20.2 Shift Resistance
305
be represented as follows:
dγ nl = ϕ nl dω 0 dd,
(20.1)
where ϕ nl is the slip rate being a function of the directions n(α 0 , β 0 ) and l(ω 0 ).
Using the formulas of converting second-rank tensor components when switching
to new axes, we can record as follows based on (20.1):
dγ νλ = ϕ nl (n ν l λ + n λ l ν )dω 0 dd,
(20.2)
where each of the addends in the brackets is the product of cosines of angles between
the respective vectors.
When changing the direction where the shift from these slips is defined into the
opposite direction, a respective shift component will change only the sign, e.g.
dγ n(−l) = −dγ nl .
(20.3)
Hence,
ϕ(α 0 , β 0 , ω 0 + π) = −ϕ(α 0 , β 0 , ω 0 ).
(20.4)
Summing the respective strain tensor components from shifts (20.3) gives
γ ij =
1
2
R
ϕ nl (n i l j + n j l i )dω 0 dd, (i, j = x, y, z),
(20.5)
where l i , . . . , n j are the projections of the single vectors l and n onto the respective
axis of the Cartesian coordinate system, and R is the area of all positive and negative
slips.
If we fix a normal line ν(α, β) from a multitude of slip planes and directions
to an arbitrary slip plane and direction λ(ω) in this plane inside the slip direction
fan, the tangential stress component in this direction at the moment of slip is called
[3] the plastic shift resistance S νλ . In a common case, the shift resistance S νλ is [3]
some operator from the slip rate ϕ nl , which we will express using a symbolic form
S νλ ϕ nl .
If R means an area where slips occur at this point of time, and is the boundary
of that area, we have as follows according to the definition of shift resistance:
S νλ ϕ nl = τ νλ
when (ν, λ) ∈ R.
(20.6)
The area where no slips occur at this point in time is defined by the condition:
S νλ ϕ nl > τ νλ
when (ν, λ) /
∈ R.
(20.7)
305
be represented as follows:
dγ nl = ϕ nl dω 0 dd,
(20.1)
where ϕ nl is the slip rate being a function of the directions n(α 0 , β 0 ) and l(ω 0 ).
Using the formulas of converting second-rank tensor components when switching
to new axes, we can record as follows based on (20.1):
dγ νλ = ϕ nl (n ν l λ + n λ l ν )dω 0 dd,
(20.2)
where each of the addends in the brackets is the product of cosines of angles between
the respective vectors.
When changing the direction where the shift from these slips is defined into the
opposite direction, a respective shift component will change only the sign, e.g.
dγ n(−l) = −dγ nl .
(20.3)
Hence,
ϕ(α 0 , β 0 , ω 0 + π) = −ϕ(α 0 , β 0 , ω 0 ).
(20.4)
Summing the respective strain tensor components from shifts (20.3) gives
γ ij =
1
2
R
ϕ nl (n i l j + n j l i )dω 0 dd, (i, j = x, y, z),
(20.5)
where l i , . . . , n j are the projections of the single vectors l and n onto the respective
axis of the Cartesian coordinate system, and R is the area of all positive and negative
slips.
If we fix a normal line ν(α, β) from a multitude of slip planes and directions
to an arbitrary slip plane and direction λ(ω) in this plane inside the slip direction
fan, the tangential stress component in this direction at the moment of slip is called
[3] the plastic shift resistance S νλ . In a common case, the shift resistance S νλ is [3]
some operator from the slip rate ϕ nl , which we will express using a symbolic form
S νλ ϕ nl .
If R means an area where slips occur at this point of time, and is the boundary
of that area, we have as follows according to the definition of shift resistance:
S νλ ϕ nl = τ νλ
when (ν, λ) ∈ R.
(20.6)
The area where no slips occur at this point in time is defined by the condition:
S νλ ϕ nl > τ νλ
when (ν, λ) /
∈ R.
(20.7)