410
32 On Boundary Value Problems of Inelastic Body Mechanics
In Ilyushin’s method, as a zero approximation, we accept deformations from an
elastic solution, by which stresses are determined according to the deformation law
of the type (32.11). Stresses found by this method will not satisfy the equilibrium
equations. However, they can be considered as stresses that satisfy the equations of
the theory of elasticity in the presence of additional (fictitious) mass forces. Solving
these equations in displacements, deformations of the first approximation are found.
Then, according to the found strains, according to the law of the type (32.11), the
stresses of the first approximation are determined. Repeating this algorithm, the
stresses and strains of subsequent approximations are found. The process can be
completed as soon as the results of neighboring approximations coincide with the
desired accuracy.
We have already said that the convergence of Ilyushin’s elastic solution method
has not yet been proven. One can also specify a class of problems for which the
stress distribution in an inelastic body is close to the distribution of stresses in
the elastic state of the same body, while deformations can differ from elastic ones
by tens of times. Examples of such tasks are: a disk, compressible in diameter, a
rectangular plate, squeezed diagonally, etc. For such problems, it is advisable to
conduct the approximation process by stress. In Russian literature, this technique is
known as the Birger method of additional deformations [2].
With this in mind, we find at the end of the (k + 1)-th stage of loading stress
increment σ
(k+1)
ij 0 , considering the material to be perfectly elastic. Let us take this
increment as a zero approximation (the approximation number is denoted by the
last icon in the lower index) to increment the stress σ
(k+1)
ij
at the (k + 1)-th stage.
Further, by the stress in the zero approximation
σ
(k+1)
ij 0
= σ
(k)
ij + σ
(k+1)
ij 0 ,
(32.12)
let us define the boundary of the area where the inelasticity condition is met (32.10),
and calculate by the ratios (32.11), (32.6), and (32.9) fictitious forces in the
zero approximation f i0 , p iν0 . Then after solving the boundary value problem (32.5), (32.8), we calculate using formulas (32.3) and (32.1) the first approximation for the stress σ
(k+1)
ij 1
at the end of the (k + 1)-th loading stage. Given the
known σ
(k)
ij and σ
(k+1)
ij
, the loading path at the (k + 1)-th stage is specified and the
procedure is repeated. The iteration process at each stage can be completed as soon
as the difference between the stresses in the adjacent approximations is within the
desired accuracy. The algorithm diagram is shown in Fig. 32.1.
The described method has a wide generality due to the lack of any restrictions on
the type of relations (32.11). The only exception is a perfectly plastic material; this
case is specifically discussed in the next paragraph.
Note that for the above class of tasks, when the distribution of the stresses in
an inelastic body is close to the stress distribution at the elastic state of this body,
already the first approximations of the stresses may be good enough. In addition,
when solving problems, in which the stress field is close to uniform, the first
approximation also gives an almost exact result.
32 On Boundary Value Problems of Inelastic Body Mechanics
In Ilyushin’s method, as a zero approximation, we accept deformations from an
elastic solution, by which stresses are determined according to the deformation law
of the type (32.11). Stresses found by this method will not satisfy the equilibrium
equations. However, they can be considered as stresses that satisfy the equations of
the theory of elasticity in the presence of additional (fictitious) mass forces. Solving
these equations in displacements, deformations of the first approximation are found.
Then, according to the found strains, according to the law of the type (32.11), the
stresses of the first approximation are determined. Repeating this algorithm, the
stresses and strains of subsequent approximations are found. The process can be
completed as soon as the results of neighboring approximations coincide with the
desired accuracy.
We have already said that the convergence of Ilyushin’s elastic solution method
has not yet been proven. One can also specify a class of problems for which the
stress distribution in an inelastic body is close to the distribution of stresses in
the elastic state of the same body, while deformations can differ from elastic ones
by tens of times. Examples of such tasks are: a disk, compressible in diameter, a
rectangular plate, squeezed diagonally, etc. For such problems, it is advisable to
conduct the approximation process by stress. In Russian literature, this technique is
known as the Birger method of additional deformations [2].
With this in mind, we find at the end of the (k + 1)-th stage of loading stress
increment σ
(k+1)
ij 0 , considering the material to be perfectly elastic. Let us take this
increment as a zero approximation (the approximation number is denoted by the
last icon in the lower index) to increment the stress σ
(k+1)
ij
at the (k + 1)-th stage.
Further, by the stress in the zero approximation
σ
(k+1)
ij 0
= σ
(k)
ij + σ
(k+1)
ij 0 ,
(32.12)
let us define the boundary of the area where the inelasticity condition is met (32.10),
and calculate by the ratios (32.11), (32.6), and (32.9) fictitious forces in the
zero approximation f i0 , p iν0 . Then after solving the boundary value problem (32.5), (32.8), we calculate using formulas (32.3) and (32.1) the first approximation for the stress σ
(k+1)
ij 1
at the end of the (k + 1)-th loading stage. Given the
known σ
(k)
ij and σ
(k+1)
ij
, the loading path at the (k + 1)-th stage is specified and the
procedure is repeated. The iteration process at each stage can be completed as soon
as the difference between the stresses in the adjacent approximations is within the
desired accuracy. The algorithm diagram is shown in Fig. 32.1.
The described method has a wide generality due to the lack of any restrictions on
the type of relations (32.11). The only exception is a perfectly plastic material; this
case is specifically discussed in the next paragraph.
Note that for the above class of tasks, when the distribution of the stresses in
an inelastic body is close to the stress distribution at the elastic state of this body,
already the first approximations of the stresses may be good enough. In addition,
when solving problems, in which the stress field is close to uniform, the first
approximation also gives an almost exact result.