Chapter 32
On Boundary Value Problems of Inelastic
Body Mechanics
32.1 General Formulation of the Problem of Inelastic Solid
Mechanics
Recall that, from a physical point of view, the complete deformation ε ij is the result
of a change in the distances between particles of a solid body and the order of
location of these particles due to various structural disturbances in the body, changes
in structural bonds, etc.
If the order in the arrangement of particles in a loaded body and the distances
between them in an unloaded state are preserved, then the material experiences a
purely elastic deformation ε
y
ij . This deformation is connected with the stress σ ij by
Hooke’s law [8]
σ ij = λε
y
kk δ ij + 2G 0 ε
y
ij , (i,j = 1, 2, 3),
(32.1)
where the shear modulus G 0 and the Lame constant λ are expressed in terms of
Young’s modulus E and Poisson’s coefficient ν according to the formulas [8]
G 0 =
E
2(1 + ν)
, λ =
ν
(1 + ν)(1 − 2ν)
.
(32.2)
In formula (32.1) and later in this chapter, we use the notation introduced in
Chap. 12 and adopted in tensor analysis, namely, over repeated indexes (except for
the indexes i and j running through the values 1, 2, 3), summation is performed; the
character δ ij is 1 or 0, depending on whether i and j are equal or not.
As before (p. 309), the difference between full and elastic deformations will be
called inelastic deformation e ij :
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
V. Molotnikov, A. Molotnikova, Theory of Elasticity and Plasticity,
https://doi.org/10.1007/978-3-030-66622-4_32
407
On Boundary Value Problems of Inelastic
Body Mechanics
32.1 General Formulation of the Problem of Inelastic Solid
Mechanics
Recall that, from a physical point of view, the complete deformation ε ij is the result
of a change in the distances between particles of a solid body and the order of
location of these particles due to various structural disturbances in the body, changes
in structural bonds, etc.
If the order in the arrangement of particles in a loaded body and the distances
between them in an unloaded state are preserved, then the material experiences a
purely elastic deformation ε
y
ij . This deformation is connected with the stress σ ij by
Hooke’s law [8]
σ ij = λε
y
kk δ ij + 2G 0 ε
y
ij , (i,j = 1, 2, 3),
(32.1)
where the shear modulus G 0 and the Lame constant λ are expressed in terms of
Young’s modulus E and Poisson’s coefficient ν according to the formulas [8]
G 0 =
E
2(1 + ν)
, λ =
ν
(1 + ν)(1 − 2ν)
.
(32.2)
In formula (32.1) and later in this chapter, we use the notation introduced in
Chap. 12 and adopted in tensor analysis, namely, over repeated indexes (except for
the indexes i and j running through the values 1, 2, 3), summation is performed; the
character δ ij is 1 or 0, depending on whether i and j are equal or not.
As before (p. 309), the difference between full and elastic deformations will be
called inelastic deformation e ij :
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
V. Molotnikov, A. Molotnikova, Theory of Elasticity and Plasticity,
https://doi.org/10.1007/978-3-030-66622-4_32
407