32.5 Using the Kröner Theory of Residual Stresses
421
T m = τ m ,
(32.31)
where T m is the maximum tangent macro-stress [6].
Under the load P s + P 1 , the body will slip along the planes and directions
of the idealized maximum tangential stresses [6]. In accordance with Axiom 20.1
(p. 303), plastic deformation is discontinuous in space and instantaneous in time.
Consequently, as a result of the slip in the body, a region of plastic deformation
is formed of a small thickness, extended in the direction of the minimum rate of
change of the maximum tangential stress, since the sections of the planes of this
region will be primarily involved in the slip process. Due to the small thickness, this
region can be considered a surface (in the flat case, a line), which in what follows
will be called a sliding surface (line).
The sliding system along the planes and directions of the maximum tangential
stress can be replaced by a kinematically equivalent discontinuity of displacements
distributed over the sliding surface. The density of this gap and the boundary of the
sliding surface can be determined from the condition that the idealized maximum
tangential stress on the indicated surface is equal to the yield strength of the material
τ m = τ s .
(32.32)
The latter condition is reduced to an integral equation with respect to the unknown
gap density of displacements on the sliding surface. With a known solution of
this equation, the determination of stresses and deformations does not present
fundamental difficulties.
Next, the second stage of loading is considered. Such an increment of the load
P 2 is given, at which again the condition (32.31) is met at any point of the body.
Then at the load P s + P 1 + P 2 , it is necessary to repeat the entire sequence of
calculations described for the first stage, etc.
Thus, for an ideal plastic body, the method of successive loading is in a sense
natural. Apparently, there is no other way to solve boundary value problems for a
perfectly plastic body.
32.5 Using the Kröner Theory of Residual Stresses
The inelastic deformation e ij is represented as a result of the continuous distribution
of discontinuous deformations (slides) in some area of the elastic body. Tension in
an inelastic body can be represented as a sum of two fields: the stress fields in an
elastic body from a given external influence and the stress fields in an elastic body
caused by slides. The second of these fields can be defined applying the Kröner
residual stress theory [11].
Inelastic deformation will not satisfy, generally speaking, the conditions of
compatibility [6]. To characterize the incompatibility of deformations, Kröner
introduced the [11] so-called tensor incompatibilities
421
T m = τ m ,
(32.31)
where T m is the maximum tangent macro-stress [6].
Under the load P s + P 1 , the body will slip along the planes and directions
of the idealized maximum tangential stresses [6]. In accordance with Axiom 20.1
(p. 303), plastic deformation is discontinuous in space and instantaneous in time.
Consequently, as a result of the slip in the body, a region of plastic deformation
is formed of a small thickness, extended in the direction of the minimum rate of
change of the maximum tangential stress, since the sections of the planes of this
region will be primarily involved in the slip process. Due to the small thickness, this
region can be considered a surface (in the flat case, a line), which in what follows
will be called a sliding surface (line).
The sliding system along the planes and directions of the maximum tangential
stress can be replaced by a kinematically equivalent discontinuity of displacements
distributed over the sliding surface. The density of this gap and the boundary of the
sliding surface can be determined from the condition that the idealized maximum
tangential stress on the indicated surface is equal to the yield strength of the material
τ m = τ s .
(32.32)
The latter condition is reduced to an integral equation with respect to the unknown
gap density of displacements on the sliding surface. With a known solution of
this equation, the determination of stresses and deformations does not present
fundamental difficulties.
Next, the second stage of loading is considered. Such an increment of the load
P 2 is given, at which again the condition (32.31) is met at any point of the body.
Then at the load P s + P 1 + P 2 , it is necessary to repeat the entire sequence of
calculations described for the first stage, etc.
Thus, for an ideal plastic body, the method of successive loading is in a sense
natural. Apparently, there is no other way to solve boundary value problems for a
perfectly plastic body.
32.5 Using the Kröner Theory of Residual Stresses
The inelastic deformation e ij is represented as a result of the continuous distribution
of discontinuous deformations (slides) in some area of the elastic body. Tension in
an inelastic body can be represented as a sum of two fields: the stress fields in an
elastic body from a given external influence and the stress fields in an elastic body
caused by slides. The second of these fields can be defined applying the Kröner
residual stress theory [11].
Inelastic deformation will not satisfy, generally speaking, the conditions of
compatibility [6]. To characterize the incompatibility of deformations, Kröner
introduced the [11] so-called tensor incompatibilities