2
1 Elasticity
1.2 The General Theory of Elasticity
Linear elasticity as a general three-dimensional theory has been developed in the
early 1820s based on Cauchy’s work. Simultaneously, Navier had developed an elasticity theory based on a simple particle model, in which particles interacted with
their neighbours by a central force of attraction between neighbouring particles.
Later, it was gradually realized, following work by Navier, Cauchy and Poisson in
the 1820s and 30s, the particle model is too simple. Most of the subsequent development of this subject was in terms of the continuum theory. George Green highlighted the maximum possible number of independent elastic moduli in the most
general anisotropic solid in 1837. Green pointed out that the existence of elastic
strain energy required that of the 36 elastic constants relating the 6 stress components to the 6 strains, at most 21 could be independent. In 1855, Lord Kelvin showed
that a strain energy function must exist for reversible isothermal or adiabatic response
and showed that temperature changes are associated with adiabatic elastic deformation. The middle and late 1800s were a period in which many basic elastic solutions
were derived and applied to technology and to the explanation of natural phenomena.
Adhémar-Jean-Claude Barré de Saint–Venant derived in the 1850s solutions for the
torsion of noncircular cylinders, which explained the necessity of warping displacement of the cross-section in the direction parallel to the axis of twisting, and for
the flexure of beams due to transverse loading; the latter allowed understanding of
approximations inherent in the simple beam theory of Jakob Bernoulli, Euler and
Coulomb. Heinrich Rudolf Hertz developed solutions for the deformation of elastic
solids as they are brought into contact and applied these to model details of impact
collisions. Solutions for stress and displacement due to concentrated forces acting
at an interior point of a full space were derived by Kelvin and those on the surface
of a half space by Boussinesq and Cerruti. In 1863, Kelvin had derived the basic
form of the solution of the static elasticity equations for a spherical solid, and these
were applied in following years to such problems as calculating the deformation
of the Earth due to rotation and tidal forcing and measuring the effects of elastic
deformability on the motions of the Earth’s rotation axis. The classical development
of elasticity never fully confronted the problem of finite elastic straining, in which
material fibres change their lengths by other than very small amounts.
1.3 Assumptions of Linear Elasticity
In order to evaluate the stresses, strains and displacements in an elasticity problem,
one needs to derive a series of basic equations and boundary conditions. During the
process of deriving such equations, one can consider all the influential factors, the
results obtained will be so complicated, and hence practically no solutions can be
found. Therefore, some basic assumptions have to be made about the properties of
the body considered to arrive at possible solutions. Under such assumptions, we can
1 Elasticity
1.2 The General Theory of Elasticity
Linear elasticity as a general three-dimensional theory has been developed in the
early 1820s based on Cauchy’s work. Simultaneously, Navier had developed an elasticity theory based on a simple particle model, in which particles interacted with
their neighbours by a central force of attraction between neighbouring particles.
Later, it was gradually realized, following work by Navier, Cauchy and Poisson in
the 1820s and 30s, the particle model is too simple. Most of the subsequent development of this subject was in terms of the continuum theory. George Green highlighted the maximum possible number of independent elastic moduli in the most
general anisotropic solid in 1837. Green pointed out that the existence of elastic
strain energy required that of the 36 elastic constants relating the 6 stress components to the 6 strains, at most 21 could be independent. In 1855, Lord Kelvin showed
that a strain energy function must exist for reversible isothermal or adiabatic response
and showed that temperature changes are associated with adiabatic elastic deformation. The middle and late 1800s were a period in which many basic elastic solutions
were derived and applied to technology and to the explanation of natural phenomena.
Adhémar-Jean-Claude Barré de Saint–Venant derived in the 1850s solutions for the
torsion of noncircular cylinders, which explained the necessity of warping displacement of the cross-section in the direction parallel to the axis of twisting, and for
the flexure of beams due to transverse loading; the latter allowed understanding of
approximations inherent in the simple beam theory of Jakob Bernoulli, Euler and
Coulomb. Heinrich Rudolf Hertz developed solutions for the deformation of elastic
solids as they are brought into contact and applied these to model details of impact
collisions. Solutions for stress and displacement due to concentrated forces acting
at an interior point of a full space were derived by Kelvin and those on the surface
of a half space by Boussinesq and Cerruti. In 1863, Kelvin had derived the basic
form of the solution of the static elasticity equations for a spherical solid, and these
were applied in following years to such problems as calculating the deformation
of the Earth due to rotation and tidal forcing and measuring the effects of elastic
deformability on the motions of the Earth’s rotation axis. The classical development
of elasticity never fully confronted the problem of finite elastic straining, in which
material fibres change their lengths by other than very small amounts.
1.3 Assumptions of Linear Elasticity
In order to evaluate the stresses, strains and displacements in an elasticity problem,
one needs to derive a series of basic equations and boundary conditions. During the
process of deriving such equations, one can consider all the influential factors, the
results obtained will be so complicated, and hence practically no solutions can be
found. Therefore, some basic assumptions have to be made about the properties of
the body considered to arrive at possible solutions. Under such assumptions, we can