Chapter 17
Additions and Generalizations to the
Strain Theory of Plasticity
17.1 Generalizations of Goldenblatt and Prager
The simplicity of the ratios of the strain theory of plasticity
σ
ij = 2G s ε
ij , σ i = (ε i ), σ 0 = 3Kε 0
with dJ
2 > 0,
σ
ij = 2Gε
ij , σ 0 = 3Kε 0
with dJ
2 < 0
(17.1)
and their formal affinity with the respective ratios of elasticity theory attracted wide
attention of engineers and researchers. Another important advantage of the theory
is an opportunity to solve multiple applied elastic–plastic problems. Some of these
solutions were given in the previous chapter. It was also shown that a simple process
of approximate solution was built based on the ratios (17.1) to solve marginal
problems—a method of elastic solutions.
After 1928, when the fundamental study of Lode [15] was published, multiple
experimental works were done to find the applicability limits of the strain theory of
plasticity. It was found and generally admitted that we must know to what extent the
conditions of proportional loading are implemented in each element of the body
volume to judge about the possibility of using the strain theory. In the case of
proportional loading, the laws (17.1) of the strain theory have been experimentally
proved.
However, for a non-hardening material, the term of proportional loading has no
sense, since when the plasticity condition is met, which binds components of the
stress tensor, the stressed state may change only by changing the ratios between
stresses. Nevertheless, strain theory is used in the case of ideal plastic material
taking the obtained results as approximation.
Experimental studies show that the laws of the strain theory of plasticity remain
rather accurate even when loading differs from proportional. Slight divergence from
© 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_17
245
Additions and Generalizations to the
Strain Theory of Plasticity
17.1 Generalizations of Goldenblatt and Prager
The simplicity of the ratios of the strain theory of plasticity
σ
ij = 2G s ε
ij , σ i = (ε i ), σ 0 = 3Kε 0
with dJ
2 > 0,
σ
ij = 2Gε
ij , σ 0 = 3Kε 0
with dJ
2 < 0
(17.1)
and their formal affinity with the respective ratios of elasticity theory attracted wide
attention of engineers and researchers. Another important advantage of the theory
is an opportunity to solve multiple applied elastic–plastic problems. Some of these
solutions were given in the previous chapter. It was also shown that a simple process
of approximate solution was built based on the ratios (17.1) to solve marginal
problems—a method of elastic solutions.
After 1928, when the fundamental study of Lode [15] was published, multiple
experimental works were done to find the applicability limits of the strain theory of
plasticity. It was found and generally admitted that we must know to what extent the
conditions of proportional loading are implemented in each element of the body
volume to judge about the possibility of using the strain theory. In the case of
proportional loading, the laws (17.1) of the strain theory have been experimentally
proved.
However, for a non-hardening material, the term of proportional loading has no
sense, since when the plasticity condition is met, which binds components of the
stress tensor, the stressed state may change only by changing the ratios between
stresses. Nevertheless, strain theory is used in the case of ideal plastic material
taking the obtained results as approximation.
Experimental studies show that the laws of the strain theory of plasticity remain
rather accurate even when loading differs from proportional. Slight divergence from
© 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_17
245