148
12 Origin and Development of Plasticity Theory
very well traced using the original articles [65] by its authors where one can find the
bibliographic descriptions of the originals.
Using Tresca’s results, B. Saint-Venant (1870) created the first mathematic
plasticity theory. Apart from Coulomb’s idea and the yield condition of Tresca,
Saint-Venant built his theory based on the idea of the coinciding directions of the
maximum shear speed and maximum tangential stress, as well as an idea of the
non-compressibility of a body in plastic yielding. Saint-Venant’s theory was created
for the case of plane plastic deformation and basically was a generalization of the
viscous liquid movement equation of Navier–Stokes.
M. Levy (1871) generalized Saint-Venant’s results by writing the equation of
the 3D-problem of the ideal plasticity theory. In the suggested ratios, the first three
equations represented an equation of solid body equilibrium. The fourth equation
expressed Tresca’s yield condition and the fifth ratio is the mathematical expression
of the body non-compressibility condition in plastic yield. These ratios were then
supplemented by the condition of scalar linear dependency between the strain tensor
and the stress speed tensor. In a particular case of plastic yield, Levy’s ratios go to
Saint-Venant’s equation.
The detailed analysis of the Saint-Venant–Levy ratios can be found in an article
by S. G. Mihklin [49].
12.4 Development of Plasticity Theory in the Twentieth
Century
The end of the nineteenth century brought almost no new development to plasticity
theory. In the early twentieth century, plasticity problems started to attract the
attention of large scientists. Here we will describe only the most significant studies
of that period.
In 1909, a paper by A. Haar and T. Karman appeared, which made an effort
to obtain theory equations using a variation principle. It was stated that stresses
minimized some functionality in an elastic–plastic condition of the body.
Later, Haar and Karman formulated the so-called full plasticity theory that D. D.
Ivlev called fundamental for the entire theory of ideal plasticity. D. D. Ivlev believed
that the condition of full plasticity allowed formulating the general theory of ideal
plasticity with a unified mathematical tool of statically definable equations of a
hyperbolic type corresponding to the shift nature of ideally plastic deformation.
However, it should be noted this point of view is not accepted by everyone. For
example, A. A. Vakulenko and L. M. Kachanov believe that arguments of physical
nature for the full plasticity scheme “. . . are dictated by the tempting simplicity of
mathematical analysis rather than the essence of the issue” [26, p. 100]. A similar
evaluation of the full plasticity condition is also given by R. Hill [30, p. 320–321]
who calls the Haar–Karman condition as “artificial and unreal.”
12 Origin and Development of Plasticity Theory
very well traced using the original articles [65] by its authors where one can find the
bibliographic descriptions of the originals.
Using Tresca’s results, B. Saint-Venant (1870) created the first mathematic
plasticity theory. Apart from Coulomb’s idea and the yield condition of Tresca,
Saint-Venant built his theory based on the idea of the coinciding directions of the
maximum shear speed and maximum tangential stress, as well as an idea of the
non-compressibility of a body in plastic yielding. Saint-Venant’s theory was created
for the case of plane plastic deformation and basically was a generalization of the
viscous liquid movement equation of Navier–Stokes.
M. Levy (1871) generalized Saint-Venant’s results by writing the equation of
the 3D-problem of the ideal plasticity theory. In the suggested ratios, the first three
equations represented an equation of solid body equilibrium. The fourth equation
expressed Tresca’s yield condition and the fifth ratio is the mathematical expression
of the body non-compressibility condition in plastic yield. These ratios were then
supplemented by the condition of scalar linear dependency between the strain tensor
and the stress speed tensor. In a particular case of plastic yield, Levy’s ratios go to
Saint-Venant’s equation.
The detailed analysis of the Saint-Venant–Levy ratios can be found in an article
by S. G. Mihklin [49].
12.4 Development of Plasticity Theory in the Twentieth
Century
The end of the nineteenth century brought almost no new development to plasticity
theory. In the early twentieth century, plasticity problems started to attract the
attention of large scientists. Here we will describe only the most significant studies
of that period.
In 1909, a paper by A. Haar and T. Karman appeared, which made an effort
to obtain theory equations using a variation principle. It was stated that stresses
minimized some functionality in an elastic–plastic condition of the body.
Later, Haar and Karman formulated the so-called full plasticity theory that D. D.
Ivlev called fundamental for the entire theory of ideal plasticity. D. D. Ivlev believed
that the condition of full plasticity allowed formulating the general theory of ideal
plasticity with a unified mathematical tool of statically definable equations of a
hyperbolic type corresponding to the shift nature of ideally plastic deformation.
However, it should be noted this point of view is not accepted by everyone. For
example, A. A. Vakulenko and L. M. Kachanov believe that arguments of physical
nature for the full plasticity scheme “. . . are dictated by the tempting simplicity of
mathematical analysis rather than the essence of the issue” [26, p. 100]. A similar
evaluation of the full plasticity condition is also given by R. Hill [30, p. 320–321]
who calls the Haar–Karman condition as “artificial and unreal.”