Chapter 25
Building a Shear Resistance Operator
25.1 General Form of the Shear Resistance Operator
Let us remind that due to Axiom 22.3 (see p. 323), the shear resistance S νλ in the
most general case is some integration–differentiation operator from slip intensity
(ϕ nl ). As per Axiom 22.2, any local slip in the model of a poly-crystalline body
changes mechanical properties (almost) in all directions. Hence, the operator S νλ ϕ nl
must contain the integration operation in the directions n and l. It must also contain
an integration–differentiation operation for the time (t), since time effects (aging,
recovery, and (based on Axiom 22.1) plastic strain rate) have a significant effect
on the dependency between stress and strain beyond the yield stress. Moreover, as
given above (Axiom 22.3), shear resistance must depend on elastic strains (but not
their rates!); for example, it must explicitly depend on the stress tensor components.
Based on the above, let us adopt that plastic shear resistance is a function (()
from the invariants (τ i , τ m ) of the stress deviator, the primary argument of isotropic
aging ((), the components of the aging tensor (( νλ ), and some operator (L νλ ϕ nl )
from slip intensity, as well as from the rate of this operator:
S νλ ϕ nl =
τ i , τ m , ,, , νλ , L νλ ϕ nl ,
∂
∂t
L νλ ϕ nl
.
(25.1)
For low plastic strains in the conditions of almost simple strain (p. 331), let us
represent the function from formula (25.1) as
S νλ ϕ nl = ψ(τ i , τ m ) + (τ i , . . .)
1 + ε
∂
∂t
L νλ ϕ nl
− U νλ ,
(25.2)
where ψ is the generalized function of elastic softening, ε is a small parameter, U νλ
is defined by formula (23.11), and is a function from arguments not depending
on the directions n and l.
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
V. Molotnikov, A. Molotnikova, Theory of Elasticity and Plasticity,
https://doi.org/10.1007/978-3-030-66622-4_25
341
Building a Shear Resistance Operator
25.1 General Form of the Shear Resistance Operator
Let us remind that due to Axiom 22.3 (see p. 323), the shear resistance S νλ in the
most general case is some integration–differentiation operator from slip intensity
(ϕ nl ). As per Axiom 22.2, any local slip in the model of a poly-crystalline body
changes mechanical properties (almost) in all directions. Hence, the operator S νλ ϕ nl
must contain the integration operation in the directions n and l. It must also contain
an integration–differentiation operation for the time (t), since time effects (aging,
recovery, and (based on Axiom 22.1) plastic strain rate) have a significant effect
on the dependency between stress and strain beyond the yield stress. Moreover, as
given above (Axiom 22.3), shear resistance must depend on elastic strains (but not
their rates!); for example, it must explicitly depend on the stress tensor components.
Based on the above, let us adopt that plastic shear resistance is a function (()
from the invariants (τ i , τ m ) of the stress deviator, the primary argument of isotropic
aging ((), the components of the aging tensor (( νλ ), and some operator (L νλ ϕ nl )
from slip intensity, as well as from the rate of this operator:
S νλ ϕ nl =
τ i , τ m , ,, , νλ , L νλ ϕ nl ,
∂
∂t
L νλ ϕ nl
.
(25.1)
For low plastic strains in the conditions of almost simple strain (p. 331), let us
represent the function from formula (25.1) as
S νλ ϕ nl = ψ(τ i , τ m ) + (τ i , . . .)
1 + ε
∂
∂t
L νλ ϕ nl
− U νλ ,
(25.2)
where ψ is the generalized function of elastic softening, ε is a small parameter, U νλ
is defined by formula (23.11), and is a function from arguments not depending
on the directions n and l.
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
V. Molotnikov, A. Molotnikova, Theory of Elasticity and Plasticity,
https://doi.org/10.1007/978-3-030-66622-4_25
341