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1 Summary of Elasticity Theory: Basic Concepts
The first researcher who acquired general equations of equilibrium and oscillations of elastic bodies was Navier. He used the Newton concept of the
discrete structure of substances and believed that the interaction force between
two molecules, the distance between which changed due to the body strain, was
proportional to the product of distance increment and some function of the initial
distance. However, the Navier line of reasoning was not commonly accepted. The
legality of using the integration operation in the area (discrete) occupied by the body
was argued.
As we know, mathematical analysis of that time was built on the concept of
continuous geometric space (continuum) where infinitely short sections could be
considered and where differentiation and integration processes could be conducted
based on that. Newton’s molecular theory of body structure represented them as
discrete media consisting of individual particles interrelated by mutual attraction
and repulsion forces. Therefore, the applicability of the mathematical analysis
apparatus to such media, which was substantially associated with the concept of
continuous functions, seemed illegal and unjustified.
However, a large number of particles is contained even in an extremely small
volume mentally separated from a body. This circumstance made researchers think
of using the law of large numbers and the method that would be called statistical
thereafter. Using the statistical method helped to build a bridge between the
continuous space of mathematical analysis and the solid body as a discrete medium
and resulted in the rehabilitation of using the powerful apparatus of mathematics for
creating a new field of physics. This was also supported by light wave propagation
theory reported by Fresnel to the French Academy of Science in 1821.
The importance of applying elasticity theory in physics and engineering and the
high complexity of assigned tasks in terms of mathematical analysis drew attention
to this new field of science on part of the largest mathematicians and mechanics of
that time, O. Cauchy and S. Poisson.
Cauchy substantiated the legality of Navier’s approach as follows [6, p. 23]. For
the volume containing a multitude of molecules and having small dimensions as
compared to the radius of the sphere where noticeable molecular action is expressed,
the number of molecules can be deemed proportional to the volume. If we leave
aside the molecules adjacent to the considered molecule, the action of all molecules
contained in one of the small volumes described above is equivalent to the force,
whose line of action crosses the center of gravity of the volume, and the value is
proportional to this volume and to some function from the distance of the center of
gravity of the volume from the considered molecule.
By the autumn of 1822, Cauchy introduced the most basic concepts in modern
elasticity theory. In particular, the concept of stress in a body point as a ratio between
forces and the area of various flat elements built in this point is of fundamental value.
Poisson’s first memoir dedicated to the matter under consideration was read
by the French Academy in April 1828. When considering the issue in general
equations of elasticity theory, Poisson (just as Cauchy) started with the derivation of
equilibrium equations expressed in stress components. The formulas expressing the
stress components through strains contain sums taken for all “molecules” in the area
1 Summary of Elasticity Theory: Basic Concepts
The first researcher who acquired general equations of equilibrium and oscillations of elastic bodies was Navier. He used the Newton concept of the
discrete structure of substances and believed that the interaction force between
two molecules, the distance between which changed due to the body strain, was
proportional to the product of distance increment and some function of the initial
distance. However, the Navier line of reasoning was not commonly accepted. The
legality of using the integration operation in the area (discrete) occupied by the body
was argued.
As we know, mathematical analysis of that time was built on the concept of
continuous geometric space (continuum) where infinitely short sections could be
considered and where differentiation and integration processes could be conducted
based on that. Newton’s molecular theory of body structure represented them as
discrete media consisting of individual particles interrelated by mutual attraction
and repulsion forces. Therefore, the applicability of the mathematical analysis
apparatus to such media, which was substantially associated with the concept of
continuous functions, seemed illegal and unjustified.
However, a large number of particles is contained even in an extremely small
volume mentally separated from a body. This circumstance made researchers think
of using the law of large numbers and the method that would be called statistical
thereafter. Using the statistical method helped to build a bridge between the
continuous space of mathematical analysis and the solid body as a discrete medium
and resulted in the rehabilitation of using the powerful apparatus of mathematics for
creating a new field of physics. This was also supported by light wave propagation
theory reported by Fresnel to the French Academy of Science in 1821.
The importance of applying elasticity theory in physics and engineering and the
high complexity of assigned tasks in terms of mathematical analysis drew attention
to this new field of science on part of the largest mathematicians and mechanics of
that time, O. Cauchy and S. Poisson.
Cauchy substantiated the legality of Navier’s approach as follows [6, p. 23]. For
the volume containing a multitude of molecules and having small dimensions as
compared to the radius of the sphere where noticeable molecular action is expressed,
the number of molecules can be deemed proportional to the volume. If we leave
aside the molecules adjacent to the considered molecule, the action of all molecules
contained in one of the small volumes described above is equivalent to the force,
whose line of action crosses the center of gravity of the volume, and the value is
proportional to this volume and to some function from the distance of the center of
gravity of the volume from the considered molecule.
By the autumn of 1822, Cauchy introduced the most basic concepts in modern
elasticity theory. In particular, the concept of stress in a body point as a ratio between
forces and the area of various flat elements built in this point is of fundamental value.
Poisson’s first memoir dedicated to the matter under consideration was read
by the French Academy in April 1828. When considering the issue in general
equations of elasticity theory, Poisson (just as Cauchy) started with the derivation of
equilibrium equations expressed in stress components. The formulas expressing the
stress components through strains contain sums taken for all “molecules” in the area