14.3 Tresca Plasticity Condition
183
Fig. 14.2 Trace of the yield
surface on the octahedral
plane
σ
1 + σ
2 + σ
3 = 0.
(14.9)
Equations (14.9) result in that the components of the vector P satisfy Eq. (14.8) of
the plane π , e.g. P ∈ π .
Based on Eq. (14.5), the plasticity condition does not depend on Q and is fully
defined by the vector P. Consequently, the plasticity surface (14.5) is a cylinder
whose generatrix is perpendicular to the plane π . To build this surface, it is sufficient
to build its trace on the plane π .
This building is done in Fig. 14.2. Here the plane π is aligned with the figure
plane. The axes 1 , 2 , 3 are orthogonal projections of the coordinate axes 1, 2,
and 3 on the plane π . C is the trace of the plasticity surface on the plane π .
Let us show that to build the line C, it is sufficient to have experimental data for
1/12 of its part.
Indeed, first of all, the C trace must be symmetric relative to the axes 1 , 2 , 3
since the plasticity surface view cannot depend on axes numbering.
Second, the C line must not go through the origin of the coordinate system since
zero stresses in real solid bodies do not cause plastic deformations.
Third, any beam originating in the origin crosses C only once. Otherwise, there
would be 2 and more stress states corresponding to the occurrence of plasticity,
which is impossible.
Fourth, since initially isotropic materials are considered, which equally resist
elongation and compression, the C trace must be symmetric relative to the axes
perpendicular to the axes 1 , 2 , 3 . The latter are shown in Fig. 14.2 by dash lines.
It can be easily found that the Lode-Nadai parameter μ σ (p. 177) on the indicated
part 1/12 of the trace C changes from 0 to 1.
14.3 Tresca Plasticity Condition
We have already mentioned (p. 147) that historically, the first plasticity condition
was formulated by Tresca in 1864. Observing the extrusion of materials through
dies, Tresca found that the plasticity phenomenon occurred when the maximum
tangential stress (τ max ) reached a specific value:
183
Fig. 14.2 Trace of the yield
surface on the octahedral
plane
σ
1 + σ
2 + σ
3 = 0.
(14.9)
Equations (14.9) result in that the components of the vector P satisfy Eq. (14.8) of
the plane π , e.g. P ∈ π .
Based on Eq. (14.5), the plasticity condition does not depend on Q and is fully
defined by the vector P. Consequently, the plasticity surface (14.5) is a cylinder
whose generatrix is perpendicular to the plane π . To build this surface, it is sufficient
to build its trace on the plane π .
This building is done in Fig. 14.2. Here the plane π is aligned with the figure
plane. The axes 1 , 2 , 3 are orthogonal projections of the coordinate axes 1, 2,
and 3 on the plane π . C is the trace of the plasticity surface on the plane π .
Let us show that to build the line C, it is sufficient to have experimental data for
1/12 of its part.
Indeed, first of all, the C trace must be symmetric relative to the axes 1 , 2 , 3
since the plasticity surface view cannot depend on axes numbering.
Second, the C line must not go through the origin of the coordinate system since
zero stresses in real solid bodies do not cause plastic deformations.
Third, any beam originating in the origin crosses C only once. Otherwise, there
would be 2 and more stress states corresponding to the occurrence of plasticity,
which is impossible.
Fourth, since initially isotropic materials are considered, which equally resist
elongation and compression, the C trace must be symmetric relative to the axes
perpendicular to the axes 1 , 2 , 3 . The latter are shown in Fig. 14.2 by dash lines.
It can be easily found that the Lode-Nadai parameter μ σ (p. 177) on the indicated
part 1/12 of the trace C changes from 0 to 1.
14.3 Tresca Plasticity Condition
We have already mentioned (p. 147) that historically, the first plasticity condition
was formulated by Tresca in 1864. Observing the extrusion of materials through
dies, Tresca found that the plasticity phenomenon occurred when the maximum
tangential stress (τ max ) reached a specific value: