32.2 More About the Method of Elastic Solutions
409
F (σ ij ) = 0.
(32.10)
Let us also assume that the relations that agree with the experience are found
e ij = E ij (σ mn , I 1 , I 2 , . . .),
(32.11)
making it possible by some operator E ij from the stress at this point in time and
from some parameters (I 1 , I 2 , . . .) that characterize the entire previous process of
deformations to determine the inelastic deformation at the considered moment.
The problem can be formulated [4] as follows:
to find three functions u i that satisfy in the area occupied by the body the equilibrium
equations (32.5), on the border of —the conditions (32.8), if in the elastic
part of the region, the fictitious forces are zero, and in the region of inelastic
deformations defined by the condition (32.10), the relations (32.11), (32.11),
and (32.9) are true [7].
The formulated problem differs significantly from the classical problems of
mathematical physics in that the relations (32.11) depends on the history of deformation. Apparently, for the first time, this fact was noted by Khill [5]. He is also the
author of the idea of a method for solving inelastic problems by sequential step-bystep loading: “A process of plastic deformation has to be considered mathematically
as a succession of small increments of strain, even where the overall distortion is so
small that the change in external surfaces can be neglected” ([5], p. 90).
32.2 More About the Method of Elastic Solutions
In Chap. 16, we have already got acquainted with the essence and algorithm of the
elastic solution method. Here we turn again to the described algorithm in relation to
the formulation of the problem of plasticity theory, which is given in the previous
paragraph. For the ease of reading, here is a repetition of some of the information
set out earlier in Chap. 16.
Split the entire loading process in time into a series of sequential stages. The
increment of the external loads at each stage of loading will be considered small
enough that it is possible without significant errors to consider the trajectory of
loading to be linear at any point of the inelastic region at each loading stage. Such
splitting of the loading process into stages makes it easier to keep track of the history
of deformation.
Suppose that at the end of the k-th stage of loading (in particular, at the moment
of the occurrence of inelasticity at any point of the body), all the characteristics
of the deformation process: σ
(k)
ij , ε
(k)
ij , e
(k)
ij , etc. are known. To find a solution to
the problem, at the end of the (k + 1)-th stage, one can apply the iterative process
proposed by A. A. Ilyushin [4], which he called the method of elastic solutions.
409
F (σ ij ) = 0.
(32.10)
Let us also assume that the relations that agree with the experience are found
e ij = E ij (σ mn , I 1 , I 2 , . . .),
(32.11)
making it possible by some operator E ij from the stress at this point in time and
from some parameters (I 1 , I 2 , . . .) that characterize the entire previous process of
deformations to determine the inelastic deformation at the considered moment.
The problem can be formulated [4] as follows:
to find three functions u i that satisfy in the area occupied by the body the equilibrium
equations (32.5), on the border of —the conditions (32.8), if in the elastic
part of the region, the fictitious forces are zero, and in the region of inelastic
deformations defined by the condition (32.10), the relations (32.11), (32.11),
and (32.9) are true [7].
The formulated problem differs significantly from the classical problems of
mathematical physics in that the relations (32.11) depends on the history of deformation. Apparently, for the first time, this fact was noted by Khill [5]. He is also the
author of the idea of a method for solving inelastic problems by sequential step-bystep loading: “A process of plastic deformation has to be considered mathematically
as a succession of small increments of strain, even where the overall distortion is so
small that the change in external surfaces can be neglected” ([5], p. 90).
32.2 More About the Method of Elastic Solutions
In Chap. 16, we have already got acquainted with the essence and algorithm of the
elastic solution method. Here we turn again to the described algorithm in relation to
the formulation of the problem of plasticity theory, which is given in the previous
paragraph. For the ease of reading, here is a repetition of some of the information
set out earlier in Chap. 16.
Split the entire loading process in time into a series of sequential stages. The
increment of the external loads at each stage of loading will be considered small
enough that it is possible without significant errors to consider the trajectory of
loading to be linear at any point of the inelastic region at each loading stage. Such
splitting of the loading process into stages makes it easier to keep track of the history
of deformation.
Suppose that at the end of the k-th stage of loading (in particular, at the moment
of the occurrence of inelasticity at any point of the body), all the characteristics
of the deformation process: σ
(k)
ij , ε
(k)
ij , e
(k)
ij , etc. are known. To find a solution to
the problem, at the end of the (k + 1)-th stage, one can apply the iterative process
proposed by A. A. Ilyushin [4], which he called the method of elastic solutions.