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12 Origin and Development of Plasticity Theory
12.2 The Subject and Tasks of the Theory of Plasticity
The above shows that the study of solid body complex deformation should include
the mechanics of plastic deformation, in other words, the mathematical theory of
plasticity. The mathematical theory of plasticity is a section of mechanics of a
continuous medium, a science of deforming plastic bodies.
The task of plasticity theory is to describe the load behavior of materials getting
irreversible deformations provided that the dependency between stresses and strains
contains no time. This is true in the first approximation for metals and their alloys
usually used in engineering at normal temperatures. Therefore, plasticity theory,
in fact, describes the behavior of steels, brass, aluminum alloys, and some other
materials and alloys at normal and low temperatures. At high temperatures under
load, creep occurs in materials and the link between stresses and strains greatly
depends upon time. Creep theory describes these phenomena and we will not
highlight them, except cases when the creep is found even at normal temperatures.
The mathematical theory of plasticity takes experimental data of observations
over the macroscopic behavior of a solid body as initial positions. A systematic
description of the mechanical properties of metal will be found by the reader in the
third part of the Oding’s book [59]. Among domestic studies of metal behavior in
the case of elastic and non-elastic deformations, the classical monograph by Y. B.
Fridman must be mentioned [13]. However, the mathematical theory of plasticity
is not directly intended to physically describe the plasticity properties based on
ideas of the real structure of materials. As in elasticity theory, a real solid body
is replaced by its continuous ideal model, to which various material properties,
obtained from an experiment with macro-samples, are attributed in an idealized
state. It is suggested that the plastic resistance of a continuous homogeneous
medium reproduces the behavior of a real solid body in an integral form.
The aim of plasticity theory is twofold: first of all, building sufficiently precise
relations between stresses and strains aligned with the experiment as close as possible; secondly, developing mathematical methods of solving boundary problems,
important for all applications.
In this manner, actually, two inter-related problems must be solved: (1) finding
the determinant law and formulation of the closed boundary problem; (2) solution
of the boundary problem in cases representing applied interest. The first problem
is solved by a method traditional for the mechanics of deformable media. Experimental data is generalized and inputted into the theory as principles based on which
a possible type of the determinant law is set (determinant equation). Determinant
equations of plasticity theory have a long and complicated history. The development
history of plasticity theory starting with the primary works by Saint-Venant and
Levy (M. Levy, 1871) can be tracked using Russian translations of original articles
published in the translation collection [65] “Plasticity Theory” (edited by Yu.N.
Rabotnov). The collection includes 28 articles by Saint-Venant, Levy, R. von Mises,
L. Prandtl, H. Hencky, A. Reuss, and W. Prager. These works reflect the origin and
development of mathematical plasticity theory and allow getting familiar with its
12 Origin and Development of Plasticity Theory
12.2 The Subject and Tasks of the Theory of Plasticity
The above shows that the study of solid body complex deformation should include
the mechanics of plastic deformation, in other words, the mathematical theory of
plasticity. The mathematical theory of plasticity is a section of mechanics of a
continuous medium, a science of deforming plastic bodies.
The task of plasticity theory is to describe the load behavior of materials getting
irreversible deformations provided that the dependency between stresses and strains
contains no time. This is true in the first approximation for metals and their alloys
usually used in engineering at normal temperatures. Therefore, plasticity theory,
in fact, describes the behavior of steels, brass, aluminum alloys, and some other
materials and alloys at normal and low temperatures. At high temperatures under
load, creep occurs in materials and the link between stresses and strains greatly
depends upon time. Creep theory describes these phenomena and we will not
highlight them, except cases when the creep is found even at normal temperatures.
The mathematical theory of plasticity takes experimental data of observations
over the macroscopic behavior of a solid body as initial positions. A systematic
description of the mechanical properties of metal will be found by the reader in the
third part of the Oding’s book [59]. Among domestic studies of metal behavior in
the case of elastic and non-elastic deformations, the classical monograph by Y. B.
Fridman must be mentioned [13]. However, the mathematical theory of plasticity
is not directly intended to physically describe the plasticity properties based on
ideas of the real structure of materials. As in elasticity theory, a real solid body
is replaced by its continuous ideal model, to which various material properties,
obtained from an experiment with macro-samples, are attributed in an idealized
state. It is suggested that the plastic resistance of a continuous homogeneous
medium reproduces the behavior of a real solid body in an integral form.
The aim of plasticity theory is twofold: first of all, building sufficiently precise
relations between stresses and strains aligned with the experiment as close as possible; secondly, developing mathematical methods of solving boundary problems,
important for all applications.
In this manner, actually, two inter-related problems must be solved: (1) finding
the determinant law and formulation of the closed boundary problem; (2) solution
of the boundary problem in cases representing applied interest. The first problem
is solved by a method traditional for the mechanics of deformable media. Experimental data is generalized and inputted into the theory as principles based on which
a possible type of the determinant law is set (determinant equation). Determinant
equations of plasticity theory have a long and complicated history. The development
history of plasticity theory starting with the primary works by Saint-Venant and
Levy (M. Levy, 1871) can be tracked using Russian translations of original articles
published in the translation collection [65] “Plasticity Theory” (edited by Yu.N.
Rabotnov). The collection includes 28 articles by Saint-Venant, Levy, R. von Mises,
L. Prandtl, H. Hencky, A. Reuss, and W. Prager. These works reflect the origin and
development of mathematical plasticity theory and allow getting familiar with its