6 Lattice and the Continuum
55
The quantity ε is a field in one-dimensional space as defined by the continuum
(the rod) and satisfies d’Alembert’s wave equation. In the view of this, ε is in fact
comparable to other fields in physics, for example electromagnetic fields. However,
the space in which electromagnetic waves propagate is three dimensional, whereas
the one-dimensional lattice, which we idealised to a one- dimensional continuum
(an infinitely long rod), is the space in which the field ε propagates. The ε-waves are
nothing more than sound waves. We could also say that the rod is the ether for sound,
just as we could talk about the propagation of light through the ether, as long as we
only mean the propagation through our three-dimensional space. Ether and space
will be used synonymously. The lattice (or the equivalent continuum) as a carrier,
the space for sound waves is the aspect that we will delve deeper into in the course
of our consideration. Here, we will not however just deal with and investigate sound
waves, but we will concentrate on other physical states of the lattice, just as we can
consider other physical states in three-dimensional space, not only electromagnetic
waves, but also elementary particles for example.
The relative elastic strain ε is the physical quantity that represents elastic deformations in far more complicated lattices (or rather continua) that can be measured
directly or indirectly for every state of deformation. The relative strain of a rod can
be described using one function alone, ε = ε(x, t), whereas the elastic strain of twoand three- dimensional atomic lattices have to be represented by a more complicated
mathematical quantity. Strain is a tensor ε with three, or rather six independent functions for the two dimensional- or the three-dimensional case. The same applies for
the elastic stress caused by strain. These relations are not necessary for the representation of our problem. In Chaps. 25–27, we will show the mathematically interested
reader more about the three-dimensional lattice and the continua. We will, however,
only make use of the shown results in supplementary explanations. With regard to
the physical meaning of strain ε it must be said that when dealing with complicated,
multidimensional atomic lattices, we have no choice but to work with this relative
strain ε. An elastic displacement s out of a position of equilibrium cannot be defined
for complicated atomic lattices that deviate from the ideal structure (and this applies
to most of them). All crystalline solids found in nature show characteristic deviations from the ideal structure, which are only found in small local areas of a lattice.
These deviations are produced by the so-called dislocations, which are the key to
understanding plastic deformations of crystals. They are of utmost importance for
practically all mechanical properties of crystalline solids. We will have to examine
the properties of dislocations more closely.
For the strict one-dimensional lattice or continuum, there is only one longitudinal
sound velocity. A two- or three- dimensional lattice, however, additionally allows
for transversal motions which produce transversal sound velocities. The longitudinal
sound velocity always differs from the transversal sound velocity. This is a problem
of the two- and three-dimensional lattices. As we know from the theory of elasticity
and what we will explicitly show in Chap. 25, there are always two sound velocities
which are different from each other even in the mathematically simplest, multidimensional lattice. In the view of this, our lattice, the ‘space for the sound waves’
as we wanted to say differs substantially from our space for the light with its single
55
The quantity ε is a field in one-dimensional space as defined by the continuum
(the rod) and satisfies d’Alembert’s wave equation. In the view of this, ε is in fact
comparable to other fields in physics, for example electromagnetic fields. However,
the space in which electromagnetic waves propagate is three dimensional, whereas
the one-dimensional lattice, which we idealised to a one- dimensional continuum
(an infinitely long rod), is the space in which the field ε propagates. The ε-waves are
nothing more than sound waves. We could also say that the rod is the ether for sound,
just as we could talk about the propagation of light through the ether, as long as we
only mean the propagation through our three-dimensional space. Ether and space
will be used synonymously. The lattice (or the equivalent continuum) as a carrier,
the space for sound waves is the aspect that we will delve deeper into in the course
of our consideration. Here, we will not however just deal with and investigate sound
waves, but we will concentrate on other physical states of the lattice, just as we can
consider other physical states in three-dimensional space, not only electromagnetic
waves, but also elementary particles for example.
The relative elastic strain ε is the physical quantity that represents elastic deformations in far more complicated lattices (or rather continua) that can be measured
directly or indirectly for every state of deformation. The relative strain of a rod can
be described using one function alone, ε = ε(x, t), whereas the elastic strain of twoand three- dimensional atomic lattices have to be represented by a more complicated
mathematical quantity. Strain is a tensor ε with three, or rather six independent functions for the two dimensional- or the three-dimensional case. The same applies for
the elastic stress caused by strain. These relations are not necessary for the representation of our problem. In Chaps. 25–27, we will show the mathematically interested
reader more about the three-dimensional lattice and the continua. We will, however,
only make use of the shown results in supplementary explanations. With regard to
the physical meaning of strain ε it must be said that when dealing with complicated,
multidimensional atomic lattices, we have no choice but to work with this relative
strain ε. An elastic displacement s out of a position of equilibrium cannot be defined
for complicated atomic lattices that deviate from the ideal structure (and this applies
to most of them). All crystalline solids found in nature show characteristic deviations from the ideal structure, which are only found in small local areas of a lattice.
These deviations are produced by the so-called dislocations, which are the key to
understanding plastic deformations of crystals. They are of utmost importance for
practically all mechanical properties of crystalline solids. We will have to examine
the properties of dislocations more closely.
For the strict one-dimensional lattice or continuum, there is only one longitudinal
sound velocity. A two- or three- dimensional lattice, however, additionally allows
for transversal motions which produce transversal sound velocities. The longitudinal
sound velocity always differs from the transversal sound velocity. This is a problem
of the two- and three-dimensional lattices. As we know from the theory of elasticity
and what we will explicitly show in Chap. 25, there are always two sound velocities
which are different from each other even in the mathematically simplest, multidimensional lattice. In the view of this, our lattice, the ‘space for the sound waves’
as we wanted to say differs substantially from our space for the light with its single