22.3 Function of Elastic Softening
323
Based on this result, we can suggest that for a poly-crystalline body, the strain rate
( ˙
γ nl ) in the slip plane n in the direction l at time t depends only on the tangential
stress component (τ nl ) in this direction at this time, but it does not depend on the
rate of this component.
Since plastic strain is a consequence of local slips characterized by the slip
intensity ϕ nl , the following results from the above.
Axiom 22.3 In the case of deformation beyond the yield stress, the shear resistance
is an integration–differentiation operator of slip intensity and does not depend on
the rate of elastic strains (stresses).
This axiom is a generalization of the anisotropy postulate of M. Ya. Leonov [2].
22.3 Function of Elastic Softening
Due to the elastic anisotropy of crystalline grains and their chaotic arrangement,
internal forces in solid bodies are characterized by some irregularity even in the
case of macro-ordinary strain. Due to this irregularity of stressed state, there are
always crystals in a poly-crystalline body whose tangential stress components in the
shear plane and direction exceed the nominal value, and shears may occur at some
loading in these crystals. However, for these slips to cause a macroscopic effect, it
is insufficient to have shears occurred in a single grain only; in the case of loading,
elastically deformed neighboring elements may return this grain to the initial state.
During the macro-uniform loading of a poly-crystalline body, the maximum
number of grains (their relative volume) where shears may occur depends on the
multitude of directions where high tangential stresses occur, and the latter depends
on the type of the stressed state. An increased number of planes and directions
where high stresses occur at the same maximum tangential stress increases the
relative volume of grains where plastic strains may occur. This results in the
following.
Axiom 22.4 In case any components of tangential stresses rise in all planes, the
increment of the shear resistance from elastic strains only is negative.
Let us represent that the proportional loading of a body element occurs beyond
the elastic limit at an infinitely low speed. A respective maximum tangential stress
at the yield stress is designated as S ∞
m . The value S ∞
m being the highest yield stress
will be called instantaneous yield stress. The following is assumed.
Axiom 22.5 In the case of deformation of a plastic material with an infinitely
high speed, first slips occur in planes and directions including the planes with the
maximum tangential stress in the direction of its action.
Let us suggest that the considered materials have the same instantaneous
yield stresses at uniaxial elongation and compression. For such materials, normal
hydrostatic stresses do not have effects on plastic strains, so the yield condition in
323
Based on this result, we can suggest that for a poly-crystalline body, the strain rate
( ˙
γ nl ) in the slip plane n in the direction l at time t depends only on the tangential
stress component (τ nl ) in this direction at this time, but it does not depend on the
rate of this component.
Since plastic strain is a consequence of local slips characterized by the slip
intensity ϕ nl , the following results from the above.
Axiom 22.3 In the case of deformation beyond the yield stress, the shear resistance
is an integration–differentiation operator of slip intensity and does not depend on
the rate of elastic strains (stresses).
This axiom is a generalization of the anisotropy postulate of M. Ya. Leonov [2].
22.3 Function of Elastic Softening
Due to the elastic anisotropy of crystalline grains and their chaotic arrangement,
internal forces in solid bodies are characterized by some irregularity even in the
case of macro-ordinary strain. Due to this irregularity of stressed state, there are
always crystals in a poly-crystalline body whose tangential stress components in the
shear plane and direction exceed the nominal value, and shears may occur at some
loading in these crystals. However, for these slips to cause a macroscopic effect, it
is insufficient to have shears occurred in a single grain only; in the case of loading,
elastically deformed neighboring elements may return this grain to the initial state.
During the macro-uniform loading of a poly-crystalline body, the maximum
number of grains (their relative volume) where shears may occur depends on the
multitude of directions where high tangential stresses occur, and the latter depends
on the type of the stressed state. An increased number of planes and directions
where high stresses occur at the same maximum tangential stress increases the
relative volume of grains where plastic strains may occur. This results in the
following.
Axiom 22.4 In case any components of tangential stresses rise in all planes, the
increment of the shear resistance from elastic strains only is negative.
Let us represent that the proportional loading of a body element occurs beyond
the elastic limit at an infinitely low speed. A respective maximum tangential stress
at the yield stress is designated as S ∞
m . The value S ∞
m being the highest yield stress
will be called instantaneous yield stress. The following is assumed.
Axiom 22.5 In the case of deformation of a plastic material with an infinitely
high speed, first slips occur in planes and directions including the planes with the
maximum tangential stress in the direction of its action.
Let us suggest that the considered materials have the same instantaneous
yield stresses at uniaxial elongation and compression. For such materials, normal
hydrostatic stresses do not have effects on plastic strains, so the yield condition in