216
15 Plasticity Theory of Henky–Nadai–Ilyushin
σ x − ˜
σ x = λ(( − ˜
) + 2G(ε x − ˜
ε x ),
(15.66)
where λ is the Lame constant (p. 191).
If dependencies (15.66) are substituted to Eqs. (15.64), we obtain the equilibrium
equations in the Lame form [5]:
(λ + G)
∂
∂x
(( − ˜
) + 2GG(u − ˜
u) + ρ(X − ˜
X) = 0,
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ,
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
(15.67)
Equations (15.67) together with boundary conditions (15.65) have a single solution
that can be found by elasticity theory methods. This solution can be represented:
u 1 = u − ˜
u, v 1 = v − ˜
v, w 1 = w − ˜
w.
(15.68)
The solution (15.68) allows finding strains and displacements.
In this manner, the following is proved.
Theorem Displacements of a body point ( ˜
u, ˜
v, ˜
w) at some point of the unloading
stage differ from their values (u, v, w) at the beginning of unloading by the values
of elastic displacements (u 1 , v 1 , w 1 ) that would occur in the body if the forces
(X − ˜
X), (X ν − ˜
X ν ) are applied to it in a naturally non-stressed state [6].
In particular, if we assume ˜
X = ˜
X ν = 0, there is a case of full unloading and
residual displacements of body points will be
˜
u = u − u 1 , ˜
v = v − v 1 , ˜
w = w − w 1 .
The below follows from the proved theorem.
If the plasticity theory problem is solved for the body and the defined values of
the system of forces (X, X ν ) correspond to the true state (S), and if the fictitious
problem of elasticity theory is solved for the body, e.g. the same system of forces
(X, X ν ) is assigned with the fictitious state (S 1 ), as a result of the full unloading of
the body, it will still have displacements, strains, and stresses equal to the differences
of their values in the states (S) and (S 1 ).
It is suggested that residual stresses do not go beyond the elasticity limits.
The formulated conclusion is illustrated by the example of an elastic–plastic pure
bending of the beam depicted in Fig. 15.8. The epure S represents a distribution
of stresses at the elastic–plastic bending of a real beam. The epure S 1 represents
a fictitious solution to the elastic problem at the assumption there are no plastic
strains and the elastic properties of the material are preserved at any loads. The
epure (S) − (S 1 ) represents residual stresses after full unloading. The full solution
to the problem is given in the next chapter of the book.
15 Plasticity Theory of Henky–Nadai–Ilyushin
σ x − ˜
σ x = λ(( − ˜
) + 2G(ε x − ˜
ε x ),
(15.66)
where λ is the Lame constant (p. 191).
If dependencies (15.66) are substituted to Eqs. (15.64), we obtain the equilibrium
equations in the Lame form [5]:
(λ + G)
∂
∂x
(( − ˜
) + 2GG(u − ˜
u) + ρ(X − ˜
X) = 0,
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ,
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
(15.67)
Equations (15.67) together with boundary conditions (15.65) have a single solution
that can be found by elasticity theory methods. This solution can be represented:
u 1 = u − ˜
u, v 1 = v − ˜
v, w 1 = w − ˜
w.
(15.68)
The solution (15.68) allows finding strains and displacements.
In this manner, the following is proved.
Theorem Displacements of a body point ( ˜
u, ˜
v, ˜
w) at some point of the unloading
stage differ from their values (u, v, w) at the beginning of unloading by the values
of elastic displacements (u 1 , v 1 , w 1 ) that would occur in the body if the forces
(X − ˜
X), (X ν − ˜
X ν ) are applied to it in a naturally non-stressed state [6].
In particular, if we assume ˜
X = ˜
X ν = 0, there is a case of full unloading and
residual displacements of body points will be
˜
u = u − u 1 , ˜
v = v − v 1 , ˜
w = w − w 1 .
The below follows from the proved theorem.
If the plasticity theory problem is solved for the body and the defined values of
the system of forces (X, X ν ) correspond to the true state (S), and if the fictitious
problem of elasticity theory is solved for the body, e.g. the same system of forces
(X, X ν ) is assigned with the fictitious state (S 1 ), as a result of the full unloading of
the body, it will still have displacements, strains, and stresses equal to the differences
of their values in the states (S) and (S 1 ).
It is suggested that residual stresses do not go beyond the elasticity limits.
The formulated conclusion is illustrated by the example of an elastic–plastic pure
bending of the beam depicted in Fig. 15.8. The epure S represents a distribution
of stresses at the elastic–plastic bending of a real beam. The epure S 1 represents
a fictitious solution to the elastic problem at the assumption there are no plastic
strains and the elastic properties of the material are preserved at any loads. The
epure (S) − (S 1 ) represents residual stresses after full unloading. The full solution
to the problem is given in the next chapter of the book.