32.6 Kröner Method for Plane Deformation
423
32.6 Kröner Method for Plane Deformation
For flat deformation (ε 33 = 0), the only component of the incompatibility tensor
different from zero S 33 will be a function of two variables. As already mentioned,
this function depicts the density of wedge dislocations continuously distributed in
the region of incompatible deformations.
Denote by ij (x, y, x 0 , y 0 ) the stress that occurs at an arbitrary point (x 1 =
x, x 2 = y) of the elastic body from the wedge dislocation introduced at some point
(x 0 , y 0 ) of the considered region with an angular opening equal to unity. In what
follows, we will call ij Green’s tensor function. Then the additional stress ˜
σ ij ,
which occurs in the body from discontinuous deformations, can be calculated by
the formula:
˜
σ ij = −
D
ij (x, y, x 0 , y 0 )S 33 (x 0 , y 0 )dx 0 dy 0 ,
(32.35)
where D is the area of incompatible deformations.
In accordance with the above, the stress σ ij in an inelastic body will be equal to
σ ij = σ
0
ij + ˜
σ ij ,
(32.36)
and σ 0
ij is the stress in the body from the given external influences, calculated under
the assumption of the ideal elasticity of the material.
To determine the stress σ ij , we use the elastic solution method. At the (k + 1)-th
loading stage, we determine the zero approximation for stress by formula (32.11)
by solving the corresponding elastic problem. Using the calculated stress, from the
condition (32.10) we refine the boundary of the region and use formulas (32.33)
and (32.11) to calculate the zero approximation for the incompatibility tensor. Then,
with the known function ij , it is easy to calculate the stress as a first approximation
from the dependencies (32.35) and (32.36). Next, the iteration process is repeated
until the required accuracy is obtained.
Thus, in the case of plane deformation, to solve the problem, it is necessary,
in addition to finding a solution to the corresponding elastic problem, to construct
Green’s tensor function. If this function is found, the solution is reduced to formal
calculations using the indicated algorithm.
The proposed method is especially effective in solving problems using the Kröner
theory of residual stresses. Already the first approximation should give a fairly good
result. If the region occupied by the body is rather simple (a circle, a plane with a
hole, etc.), the construction of Green’s function does not cause formal difficulties.
Examples of its calculation are given in the articles [9, 10].
423
32.6 Kröner Method for Plane Deformation
For flat deformation (ε 33 = 0), the only component of the incompatibility tensor
different from zero S 33 will be a function of two variables. As already mentioned,
this function depicts the density of wedge dislocations continuously distributed in
the region of incompatible deformations.
Denote by ij (x, y, x 0 , y 0 ) the stress that occurs at an arbitrary point (x 1 =
x, x 2 = y) of the elastic body from the wedge dislocation introduced at some point
(x 0 , y 0 ) of the considered region with an angular opening equal to unity. In what
follows, we will call ij Green’s tensor function. Then the additional stress ˜
σ ij ,
which occurs in the body from discontinuous deformations, can be calculated by
the formula:
˜
σ ij = −
D
ij (x, y, x 0 , y 0 )S 33 (x 0 , y 0 )dx 0 dy 0 ,
(32.35)
where D is the area of incompatible deformations.
In accordance with the above, the stress σ ij in an inelastic body will be equal to
σ ij = σ
0
ij + ˜
σ ij ,
(32.36)
and σ 0
ij is the stress in the body from the given external influences, calculated under
the assumption of the ideal elasticity of the material.
To determine the stress σ ij , we use the elastic solution method. At the (k + 1)-th
loading stage, we determine the zero approximation for stress by formula (32.11)
by solving the corresponding elastic problem. Using the calculated stress, from the
condition (32.10) we refine the boundary of the region and use formulas (32.33)
and (32.11) to calculate the zero approximation for the incompatibility tensor. Then,
with the known function ij , it is easy to calculate the stress as a first approximation
from the dependencies (32.35) and (32.36). Next, the iteration process is repeated
until the required accuracy is obtained.
Thus, in the case of plane deformation, to solve the problem, it is necessary,
in addition to finding a solution to the corresponding elastic problem, to construct
Green’s tensor function. If this function is found, the solution is reduced to formal
calculations using the indicated algorithm.
The proposed method is especially effective in solving problems using the Kröner
theory of residual stresses. Already the first approximation should give a fairly good
result. If the region occupied by the body is rather simple (a circle, a plane with a
hole, etc.), the construction of Green’s function does not cause formal difficulties.
Examples of its calculation are given in the articles [9, 10].