15.10 Theorem of Simple Loading
213
The existence and singularity of solving this problem have been proved.
15.10 Theorem of Simple Loading
Multiple experimental studies have shown that theory laws of small elastic–plastic
strains take place at least when the loading of a body element is simple or close to
simple.
There is a question: are there such loads applied to the body so that in each point
there is a process of simple loading, e.g. the director stress tensor is constant for
each point during the entire loading process (director tensor can change from point
to point)?
In the case of homogeneous strain, the answer to this question is trivial: the entire
body will endure simple loading if external forces change in proportion to the same
parameter, though all-around volumetric compression can change under an arbitrary
law.
In a general case, this question is rather complicated. So far, we have the
following theorem of simple loading proved by A.A. Ilyushin.
Theorem Assume that the external forces X(x, y, z), X ν change in proportion to
the same parameter
X(x, y, z) = λX
∗ (x, y, z),
X ν = λX
∗ (x, y, z),
(15.58)
where λ is the parameter defining the consecutive values of forces applied to the
body (such as time).
Simple loading in each body point will take place if the dependency σ i ∼ ε i can
be represented by the power law
σ i = Aε
χ
i , (A,χ − const),
(15.59)
and the condition of non-compressibility is met
= 3ε 0 = 0.
(15.60)
Conditions (15.58) – (15.60) are sufficient but not necessary.
The proof of this theorem in these strict conditions is elementary. Assume
that the plasticity theory problem is solved for a body at the fixed value of the
parameter λ. For certainty, let us adopt λ = 1. This means that in each body point,
stresses σ ∗
x , . . . , τ ∗
zx , strains ε ∗
x , . . . , γ ∗
zx , intensities σ ∗
? , ε ∗
? , and displacements
u ∗ , v ∗ , w ∗ are defined. The obtained solutions turn the equilibrium equations,
boundary conditions, and condition (15.60) into identical equations:
213
The existence and singularity of solving this problem have been proved.
15.10 Theorem of Simple Loading
Multiple experimental studies have shown that theory laws of small elastic–plastic
strains take place at least when the loading of a body element is simple or close to
simple.
There is a question: are there such loads applied to the body so that in each point
there is a process of simple loading, e.g. the director stress tensor is constant for
each point during the entire loading process (director tensor can change from point
to point)?
In the case of homogeneous strain, the answer to this question is trivial: the entire
body will endure simple loading if external forces change in proportion to the same
parameter, though all-around volumetric compression can change under an arbitrary
law.
In a general case, this question is rather complicated. So far, we have the
following theorem of simple loading proved by A.A. Ilyushin.
Theorem Assume that the external forces X(x, y, z), X ν change in proportion to
the same parameter
X(x, y, z) = λX
∗ (x, y, z),
X ν = λX
∗ (x, y, z),
(15.58)
where λ is the parameter defining the consecutive values of forces applied to the
body (such as time).
Simple loading in each body point will take place if the dependency σ i ∼ ε i can
be represented by the power law
σ i = Aε
χ
i , (A,χ − const),
(15.59)
and the condition of non-compressibility is met
= 3ε 0 = 0.
(15.60)
Conditions (15.58) – (15.60) are sufficient but not necessary.
The proof of this theorem in these strict conditions is elementary. Assume
that the plasticity theory problem is solved for a body at the fixed value of the
parameter λ. For certainty, let us adopt λ = 1. This means that in each body point,
stresses σ ∗
x , . . . , τ ∗
zx , strains ε ∗
x , . . . , γ ∗
zx , intensities σ ∗
? , ε ∗
? , and displacements
u ∗ , v ∗ , w ∗ are defined. The obtained solutions turn the equilibrium equations,
boundary conditions, and condition (15.60) into identical equations: