1.3 Homogeneous Strain
5
of action of this “molecule.” Poisson does not find it possible to substitute all sums
with integrals and thinks that such substitution is possible only when summing upon
the solid angle around this “molecule,” but not when summing upon the distance
counted from the “molecule.” Equations of equilibrium and motion of an isotropic
elastic body derived in this manner coincide with the Navier equations.
Apart from the above-mentioned founders of elasticity theory (Cauchy, Navier,
and Poisson), we may also name such prominent scientists as M. V. Ostrogradsky,
G. Lame, B. Clapeyron, A. Saint-Venant, B. Green, D. Maxwell, V. Thompson
(Lord Kelvin), D. Rayleigh, D. Michell, Mathieu, F. S. Yasinsky, S. P. Timoshenko,
G. V. Kolosov, N. I. Muskhelishvili, and many others. Those readers who want to
get familiar with the history of occurrence and development of elasticity theory
may refer to a comprehensive feature placed in the introduction to the book by A.
Love “Mathematical Theory of Elasticity” [6], as well as books by S. P. Timoshenko
“History of Science on Strength of Materials,” [8] A. T. Grigoryan [1–3], etc.
1.2 Elasticity of Solid Bodies
In the case of interaction of forces, distances between particles of a solid body
change. This change defines strain of a body. 1 The body’s property to take initial
dimensions and shape after forces are removed is referred to as elasticity. The
physical nature of elasticity lies in the following.
It is known [4] that the location of atoms in a solid body is characterized by
a specific order (short-range order for amorphous bodies and long-range order for
crystalline bodies). The property of a solid body to keep the atom ordering in a
loaded state and the distance between atoms in a non-loaded state defines elasticity
in physical terms. Disturbances in the initial ordering of atoms are referred to
as [4] structural imperfections. They are found in all real solid bodies, except
maybe artificial single crystals. When exposed to forces, structural imperfections
are slowly re-distributed. If strains caused by the above re-distribution of defects in
the material structure disappear some time after removing the load, they are referred
to as reversible. The disappearance of such strains is called elastic after-effect.
1.3 Homogeneous Strain
Assume that a multitude of pairs of material points of a solid body satisfies the
following conditions.
1 Temperature strains and fluctuations are not considered.
5
of action of this “molecule.” Poisson does not find it possible to substitute all sums
with integrals and thinks that such substitution is possible only when summing upon
the solid angle around this “molecule,” but not when summing upon the distance
counted from the “molecule.” Equations of equilibrium and motion of an isotropic
elastic body derived in this manner coincide with the Navier equations.
Apart from the above-mentioned founders of elasticity theory (Cauchy, Navier,
and Poisson), we may also name such prominent scientists as M. V. Ostrogradsky,
G. Lame, B. Clapeyron, A. Saint-Venant, B. Green, D. Maxwell, V. Thompson
(Lord Kelvin), D. Rayleigh, D. Michell, Mathieu, F. S. Yasinsky, S. P. Timoshenko,
G. V. Kolosov, N. I. Muskhelishvili, and many others. Those readers who want to
get familiar with the history of occurrence and development of elasticity theory
may refer to a comprehensive feature placed in the introduction to the book by A.
Love “Mathematical Theory of Elasticity” [6], as well as books by S. P. Timoshenko
“History of Science on Strength of Materials,” [8] A. T. Grigoryan [1–3], etc.
1.2 Elasticity of Solid Bodies
In the case of interaction of forces, distances between particles of a solid body
change. This change defines strain of a body. 1 The body’s property to take initial
dimensions and shape after forces are removed is referred to as elasticity. The
physical nature of elasticity lies in the following.
It is known [4] that the location of atoms in a solid body is characterized by
a specific order (short-range order for amorphous bodies and long-range order for
crystalline bodies). The property of a solid body to keep the atom ordering in a
loaded state and the distance between atoms in a non-loaded state defines elasticity
in physical terms. Disturbances in the initial ordering of atoms are referred to
as [4] structural imperfections. They are found in all real solid bodies, except
maybe artificial single crystals. When exposed to forces, structural imperfections
are slowly re-distributed. If strains caused by the above re-distribution of defects in
the material structure disappear some time after removing the load, they are referred
to as reversible. The disappearance of such strains is called elastic after-effect.
1.3 Homogeneous Strain
Assume that a multitude of pairs of material points of a solid body satisfies the
following conditions.
1 Temperature strains and fluctuations are not considered.