**D1****. Mechanical Properties of
solids**

**What is understood by properties
of solids?**

We have assumed so far that a
body consists of matter and
is rigid, nothing more.
This is admissible only in *abstract
Dynamics*. In real
life, there do not exist bodies which are rigid in the sense of
that definition and we know matter only from what we call their properties, expressed by the adjectives we add to them such as *rigid, fluid, gaseous, hard, soft,
tough, brittle, elastic*,
etc.

You can describe *properties of matter*: Their *capacity**
*to execute under given conditions certain actions on other
natural objects. You call these actions *properties*, if we can assume that those conditions
are self-evident - or better: known - and therefore need not be
stated. For example, we call a body *elastic *and
understand by its *elasticity** *its capacity to change its
form when it is compressed or otherwise deformed, but resumes its
original shape as soon as the pressure or the deformation stops.
You call a body *heavy* and understand by its *weight** *its capacity, due to Earth's
attraction, to *press on
its base* or, if hung
up, to *pull its
suspension*.

The actions of form changing
forces, in the one case, of the attraction of the body by Earth
in the other case, are the conditions under which the properties
of elasticity and weight become active without special mention
once the body has been called *elastic**
*or *heavy**.* You can also define *properties of matter *as *their
capacity* *of reacting* to given forces in a characteristic manner; for example - as above - to forces changing their
shape or to the attraction of Earth. Analyzed in this manner,
properties of matter turn out to depend on each other so much
that one recognizes the impossibility, to describe a single one
exhaustively without reference to other properties. A description
of a *single* property can therefore only be
superficial and one can do so only in *especially characteristic *cases.

Moreover, according to our
definition, the *number
of properties of matter must be infinite* and the discovery of a previously
unknown force always brings with it new properties of matter
which were previously unknown to us. It was like this in the discoveries of X-rays and radio-active
materials, In other words:
It is impossible, without objection, to subdivide such different
kinds of properties of matter into incontestable groups.

**What
is****
matter?**

A discussion of this question
demands knowledge of all of physics, whence it can only be
mentioned here, in order to tell you that it is a capital problem
of physics. Originally, it was believed that what we call
properties of matter are attached to a substantial carrier. However, such a carrier which is always identical to
itself does not exist. Already the *idea of substance* cannot be upheld, because (according to the *Theory of Relativity*)
the *mass* of a body* **is not equal** *to* *the amount of a
substance.

We know today for certain that *mass* is not at all unchangeable, although, to
a great approximation, the *Law of Conservation of Mass *holds in most cases. Modern Physics
views the essence of a substance in forces (*Dynamic Theory of Matter*). However, in order to act, the force must be *transmitted** *to where it is to act. This
transmission of force takes place in the continuous field by spreading of *energy and impulse*. The field theory and the dynamic
theory combined yield today
the relatively most satisfying reply to the question: What is matter?

You certainly cannot resort to
sensual experience, in order to legitimate the concept of
substance. Our senses do not at all reach out into the distance,
seizing the substantial objects, but the *Principle of Continuity *is valid for the psycho-physical
interaction as well as for the purely physical of immediate
closeby action: *My
visual observations are determined by the rays of light meeting
the retina, that is, by the state of the optical or
electro-magnetic field in my immediate neighbourhood with the
embodied sense of that mysterious reality, the Ego, to which the
world of objects** **appears as images*; and indeed, here is decisive above all
the flow of energy - its *direction*
for the *direction in
which I see objects*,
its periodic changeability for their *colour*.

If you
grasp a *piece of ice*, you note the *flow of energy* at the point of contact between it and
your senses as
heat, the flow of impulse* as pressure (resistance). Hence one
can say that the *energy-impulse
quantums of the field are how you receive information directly
through your senses . . . Instead of holding the qualities
together by means of asubstantial carrier, it is solely required
to conceive there functional relations" (Hermann Weyl' book: "What
is matter?"). However, in spite of this philosophical
insight into the essence of matter, we *speak** *nevertheless of its properties
always as *if* they are attached to a substantial
carrier. We speak of their extent,
their divisibility, their atomic structure, their aggregate
states, etc.

Since matter fills *space**, *it has* *like space
itself* extent*; every body extends into three
dimensions, has a finite, by measure and number specifiable
length, width and height. It is *divisible*,
that is, every body can be decomposed into smaller ones.
According to the *atomic
theory** *of
matter, you arrive during continued subdivision of matter
eventually at *atoms*, the smallest particles of matter, to
which we may ascribe individual existence and which can be viewed
as the building elements of matter^{1}. Considerations, which lead to the Kinetic Gas Theory and certain optical
phenomena, show that about 10 million atoms, densely arranged next to each other, would have
a length of 1 mm.

^{1}Cf:
The atom of matter as aggregate of atoms of
electricity.

Analyzed chemically, bodies
turn out to be composed of certain *basic substances*, for example, iron, zinc, oxygen (Chemistry knows of
about 90 of them) which we consider to be simple, since we cannot decompose them chemically any further.
Not every element combines with every other element into a new
material. And *if *two elements combine with one another,
they do so only in *definite
ratios *of quantities.
For example, hydrogen combines with chlorine into chlor-hydrogen,
with bromine into bromhydrogen, with iodine into iodinehydrogen,
with fluorine into fluor-hydrogen.

If 1 g of hydrogen compounds
with chlorine, it *always* compounds with 35.5 g of chlorine,
while, if it compounds with bromine or iodine or fluorine, it
always compounds with 80 g of bromine or with 127 g of iodine or
19 g of fluorine. Compounds of any two elements are similar.
Thus, compounding with an element always takes place in *another*, but always in a constant ratio which
is *characteristic* for this element*. *This is
Chemistry's *Fundamental
Law of Constant Proportions*, discovered by Dalton 1808 simultaneously with another
similar law. It leads also to the chemically defined concepts of atom and molecule and forms in that
respect the start of modern atomism^{2}; the fact that any two elements combine
only in a definite *limited *quantity ratio points towards
a limit of the smallest particles which can combine and leads
from chemistry to the concepts of atom and molecule.

We will explain this phenomenon
by means of the Law of Avogadro 1811,
justified by the *Kinetic
Gas Theory*, according
to which *all** *gases contain in *equally large** *volumes *equally many* gas molecules, provided that they are
at the *same* pressure (barometer level) and have the
*same* temperature. Accordingly, under the same
conditions, for example, 1 *l* of hydrogen contains the
same number of molecules as 1 *l* of oxygen or 1* l*
of hydrogen chloride. However, hydrogen chloride arises through*
**chemical** *compounding of hydrogen and
chlorine.

If we make the assumption - the simplest - that *one** *molecule of chlorine at a time
combines with *one *molecule of hydrogen at a time into one
molecule of hydrogen chloride and if we call *n *the
number of molecules in 1 *l* gas at given pressure and
temperature, we can express this *suggested**
*compound ratio by the equation

*n *Mol.
hydrogen + *n *Mol. chlorine - *n *Mol. hydrogen
chloride,

where the + sign
is to represent the combination of molecules into a *chemical compound*. If the process was like that, *n *molecules
of chlorine would *combine* with *n* molecules of hydrogen
into *n* molecules of hydrogen chloride.

^{2}The concept of atoms has been developed further only through *physical problems** *(Avogadro, Faraday, Maxwell, Helmholtz, v.
Laue). The theoretical
treatment of purely physical phenomena has led to *Quantum Theory *(1900) and its development into *Quantum Mechanics* (1926). The atomistic concept obtained
its the strongest support from *Quantum Theory*; it was to start with a theory of the structure of
single atoms, but *Quantum
Mechanics* dealt with
the *mutual forces
between several atoms**,
*that is, chemical
binding. - The concept of
atom which originated with Democritus
is solely of the nature of
theoretical insight and in no way a forerunner of the fundamental
concepts of modern atomistic.

However, experience teaches
something else: If 1 *l* of hydrogen and 1*l* of
chlorine combine into hydrogen chloride, *indeed* 2 *l* of hydrogen chloride are
formed. However, that means: *n *hydrogen* *molecules
*+* chlorine* n *molecules yield 2*n*
hydrogen chloride molecules (2*n*,* *because every
litre of hydrogen chloride contains *n* molecules, whence
they contain, according to Avogadro's rule 2*n *molecules.
Hence the assumption that 1 chlorine molecule combines with 1
hydrogen molecule into 1 hydrogen chloride molecule is *wrong*.

Instead, we must assume : 1
chlorine molecule and 1 hydrogen molecule form 2 hydrogen
chloride molecules and each chlorine molecule and each hydrogen
molecule consists of *two
parts of the same kind**.
*Whence you are forced to accept the further assumption: There exists a difference between the
smallest quantity of a substance, *which can exist freely**, *and the smallest quantity of
a substance which can *combine
chemically*. The first
quantity is called a
molecule, the second an atom.

We must assume that a molecule
consists of several atoms. Imagine the hydrogen *molecule** *to consist of two hydrogen
atoms and the *chlorine
molecule** *of
two chlorine atoms, further more, that during combination of
hydrogen and chlorine into hydrogen chloride the molecules split
into atoms and one chlorine atom at a time combines with one
hydrogen atom. This is expressed in the equation:

*n*Cl_{2}+
*n*H_{2} = 2*n*HCl.

An atom Cl is
denoted by Cl, the molecule, consisting of 2 atoms, by Cl_{2}.
Similarly, you have the atom of hydrogen* *H and its
molecule H_{2}, and the molecule HCL consisting of one Cl
and one H. Accordingly, you must ascribe to the two atom molecule
twice the mass of the atom, that is, set the mass of the atom to
half that of the molecule.

The *atom- and molecule-masses of all
elements *are referred
to the atom- and molecule-mass of oxygen. The *atom-mass of oxygen* is set equal to 16,
its *molecule-mass* to 32. As a rule, one
speaks instead of atom- and molecule-mass, not quite correctly,
of atom- and molecule-weight.
If two substances have the
atomic weights A_{1} and A_{2}, then *those** *quantities of the substances
contain equally many (N) atoms or equally many molecules which
are in the ratio NA_{1}:NA_{2}, that is, A_{1}:A_{2};
for example, 108 g silver and 16 g oxygen (you say: 1g-atom Ag
and 1 g-atom O). Correspondingly, 32 g O is 1 g-molecule O or 1
mol O.

Every
gram-atom (every mol)
contains accordingly the same number N of atoms (molecules). A generally acknowledged principle of
research concerned the atomic structure of matter when Loschmidt 1865 succeeded in determining the number of actually
present molecules in one mol by several *totally different methods* and all his results agreed with each
other: The most probable value was N = 6.06·10^{23} (Loschmidt *Number**)*.

The results of chemical analysis of substances force us to assume that there exist as many different kinds of atoms as there are different kinds of chemically indivisible simple substances. However, that does not mean that atoms themselves are indivisible. For example, one could imagine that a gold atom can be subdivided, however, its parts cannot then be referred to as gold.

Scientists have
always been occupied with the question whether the atoms of
different substances are built out of *one single *material. The discovery of radio
activity and X-rays (at the end of the Nineties of the 19-th
Century ) has illuminated this question and opened up a new
research field - *Atomic
Physics*. It tells us
that the atoms of all elements consist of *two** *primary components of an electrical kind - the
negatively charged *electron** *and the positively charged *hydrogen nuclei** *(*proton*).
In the atoms, these primary components are composed in different
numbers and arrangements into immensely strong structures which resist also the strongest physical
and chemical forces. We will not deal here with *shattering of atoms*.

Thus, the *numerous properties* of the different elements are only
differences which are conditioned by the *number and special arrangement of
the nuclei and electrons*. In spite of the strength of their structure, we must not imagine that in the atoms
the primary components are packed together tightly. What we call
an atom is only to a very small extent filled with nuclei and
electrons; they move
permanently at large velocities with respect to each other and
thus border a relatively large space, the volume of the atom,
just about in the same manner as a fly-wheel at relatively high
speed occupies a space which is considerably larger than the
volume of its metallic mass. As the fly-wheel stops other bodies
from entering the space between its spokes so do the electrons *circulating** *within a definite volume
border - the periphery of an atom. The diameter of an atom is therefore only a measure for
the border of the outermost range of action of all the electrical forces emanating
from the atom.

In spite of the
void of the atoms, the impenetrability of the substance, which we
experience daily, persists. We
do not know any forces which would be able to compress two atoms
in spite of their spacious structure so that they will penetrate
each other. Should it become possible - the latest atomic
research has opened for us such a possibility - it would become a
catastrophe for the atoms, just
as a similar undertaking would do to a fly-wheel. However, while
iron components would be the result of this penetration of a fly
wheel, it would be electrons and nuclei, that is, the *atom would be destroyed or converted
into another one*.

In this sense,
matter is called *impenetrable*. The statement: A body penetrates into another one is
wrong; the penetrating body enters the space, from which it
displaces the former, for example, an axe enters into the space
which before its penetration was filled with wood.
Impenetrability is a force
by which a body protects the space which it occupies against
penetration by *another** *body, that is, it is a repulsive force which acts at a
body's border.

**Cohesion. States of aggregates.**

The fact that
atoms hang together and form bodies can only be explained by the
existence of a force which attracts the atoms to one another and
which resists a forced separation of atoms from one another (that
is, partitioning of the
body). This force is
conditioned by the attraction between the *electrical** *components of the individual
atoms. In general, it is called *cohesion*;
it explains by the different degrees of its strength the *aggregate states*: *Rigid,
fluid, gaseous*. The smallness of the electrons and
nuclei inside the atoms allows us to conceive that the attraction
between atoms can act *only* in the immediate neighbourhood of the
atom's periphery and that it has disappeared once the distance
becomes observationally noticeable.

Even two polished pieces of glass cannot be brought into so perfect contact (so near to each other) that they adhere to each other like one body. Firm and fluid bodies can only be compressed a little; inside them, the atoms lie so close to each other that the electrical forces holding the atoms together resist most strongly every greater proximity.

If all cohesion
were of the kind as we have assumed in the definition of the
rigid body, there would* **only** *exist
rigid bodies: However, there are various grades
which cause those differences which make us speak of *rigid, fluid** *or *gaseous* bodies. A sharp subdivision of these
concepts is impossible, because the *differences** *in cohesion vary gradually from rigid to
gaseous states.

Obviously, the three aggregate states differ by the
manner in which the mass particles of rigid bodies, of fluids and
gases cohere. This
diversity manifests itself also in their *density*. Depending on the number of grams which
a substance contains in a cubic centimetre, they are said to be *more or less dense*. However, the density of substances is
not only obvious between those in *different *states
of aggregate, but also between substances in the same aggregate
state. You should think here of gold, water and mercury, of
hydrogen and carbonic acid.

The concept of *specific weight** *differs totally from that of *density*. The specific weight of a substance is
the ratio of *its
density* to that of *water*. The statement: The specific weight of Hg is about 13.6
means: One volume of Hg contains 13.6 times as much mass as the *same* volume
of water. And this fact does not depend on the units in which we
measure the mass and the
volume (cubic inch, cubic centimetre) of the fluid. If *s*
and *d *denote the specific weight and density of a
substance, *d*_{w}* *the density
of water, then *s = d*/*d*_{w}*.
*

We set the density
*d*_{w }of water (its mass contained at
4ºC in 1 cm^{3}) equal to 1, whence *s = d. *The *specifi weight *of a substance, referred to *water*, is therefore numerically equal to its
density. In the following table, the numbers mean density as well
as specific weight (related
to water). However, note that only the numbers are the same, not
the two concepts!

hydrogen |
0.000089 | quartz | 2.65 | ||||||

steam |
0.0006 | aluminium | 2.67 | ||||||

nitrogen |
0.00125 | granite,marble, slate | 2,7 | ||||||

air |
0.00129 | glas | 2.7 - 4.5 | ||||||

oxygen |
0.00143 | basalt | 2.9 | ||||||

carbonic acid |
0.00198 | bromine | 3.0 | ||||||

cork |
0.24 | zinc | 7.2 | ||||||

lithium |
0.59 | tin | 7.3 | ||||||

ethyl alcohol |
0.791 (at 18º) | iron | 7.8 | ||||||

petroleum |
0.8 | nickel | 8.7 | ||||||

calcium |
0.86 | copper | 8.9 | ||||||

benzol |
0.881 | silver | 10.6 | ||||||

castor oil |
0.961 | lead | 11.3 | ||||||

gutta-percha |
0.98 | mercury | 13.6 | ||||||

water |
1.00 | gold | 19.4 | ||||||

acetic acid |
1.053 | platinum | 21.5 | ||||||

carbon disulphide |
1.265 | iridium | 22.4 | ||||||

magnesium |
1.75 |

*The numbers in this table* are only approximate, even if, for
example, copper, silver, gold are chemically pure, their
densities can differ, depending on their mechanical treatment (casting, hammering, rolling, etc.).
When great accuracy is important, the density of a given piece of
copper or silver or gold must be determined for each individual
sample. - For *fluids*, you must state the *temperature*, for gases the *temperature and pressure*, at which the value of density was
measured.

The table shows that there are
substances which, although they are fluid at normal temperature,
they are *denser* than most firm materials (mercury,
bromine). Without making them fluid, a pressure of *several hundred atmospheres* would make *air and oxygen denser than many solids*. The ratio of the densities of iridium
and hydrogen is 250,000:1. With the aid of an air pump, you can
reduce the density of hydrogen arbitrarily, that is, you can have
side by side kinds of substance the densities of which are in the
ratio of milliards to one.

**Geometrical structure of firm
bodies**

Does there exist a
relationship between the inner structure and outer form of a firm
body? A material body
consists of many material *points*, whence the space which it occupies is
not filled by something which is continuously connected; it is
formed by point-like masses which are separated by spaces.
According to our experience (*Physics, Chemistry, Crystallography*), this is even true for a mass which
appears to be *continuous
under a microscope*. In
certain bodies, the particles lie in *ideal
disorder* - they are said to be *amorphous** *or *without structure** - *in other bodies, they lie
according to a strict law - they are *structured*^{1}.

The Difference
between bodies with or without structure - it is better to say
without *immediately
perceptible** *structure
- already suggests itself through their different appearance. *Symmetry* meant originally a definite regularity
in the outer appearance of a body (morphological symmetry). Bodies which when seen from
two or more sides have the same appearance were called symmetric,
others, which from each side had a different appearance,
asymmetric. For this
reason, the sphere is considered to be *most symmetric*, a tetrahedron with all different sides
has the *lowest symmetry** *and in between lie all others, for example,
regular prisms, pyramids and polyhedra, etc.

In order to
determine strictly the symmetry of a body, for example, of a *cube*, you must look at it from* **all *sides, record every observation by a
picture and then find out which of these are equal. For this
purpose, imagine that there is at each point of space an observer
who photographs the body (say, a cube). Actually, a simple
photograph would not be sufficient, because it is two-dimensional
while a body is three-dimensional. Rather, each observer must be
equipped with a *three-dimensional
coordinate system *into
the three planes of which he can project the body; only the
combination of the three projections (*ground-plan, side view, elevation*) yield a three-dimensional picture and thereby the most
necessary, observational material.

Moreover, if you
conceive the coordinate axes to be rulers, say, subdivided into
millimetres, you can study the pictures dimensionally and compare
them numerically, in order
to discover which of them are equal. However, such a quantitative
comparison makes sense only if all the coordinate systems, used
by the observers, have the same scale, that is, when they have
the same angular and length dimensions. Metrically equal
coordinate systems are said to be *symmetric*;
the conversion of one system into another, which is metrically
equal to it, is called a *symmetry
operation*. Prior to
discussion of the *symmetry
of a body*, we must
first of all discuss that *of
coordinate systems*.

^{1}We will not discuss here the chemical
structure of *molecules**, *for example, of the water
molecule H_{2}O, of the molecules constituting
atmospheric air, gases, etc. (*Macro point
of view*).

You obtain a
survey of the symmetry operations, possible in an *empty Euclidean space*, by discovery of all operations which
one can perform with a coordinate system without distorting its
axes (that is, without changing its *metric*).
Without bending and distortions, you can transfer it

**1.**
by *parallel
displacement* to all
points of space,

**2.** orientate it by *rotation*
at each point in all directions,

**3.** *reflect* it in a plane mirror, and,

*finally*, you can *combine arbitrarily all these operations and repeat
them*.

*There do not
exist other symmetry operations** *without bending or distortion of the system.
Hence there are *three
kinds* of possible symmetry
operations:

**1.**
*Translation* (parallel displacement of the
coordinate systems from one point in space to any other,

**2.** *rotation* (of direction of coordinate axes in
space),

**3.** *reflection* in a plane

and,

f**inally**, *their
combinations*: *Twisting *(*translation-rotation*)*,
slide-reflection *(*translation-reflection*)* and
rotation-reflection.*

Also every repetition (*power**) *of a symmetry operation is a *symmetry operation*, since the metric does not change; you
combine a symmetry operation and all its powers and represent all
of them by a *symmetry-element**. *This is how there arises out
of translation the symmetry element: *Translation-chain** *(illustrated by an infinite
line on which equidistant points have been marked (Fig. 154), out
of rotation by an angle 360º/*n* the
symmetry element: *n-rotation-axis** *(illustrated by a line - axis
of rotation - with a regular *n-*polygon as index,
pointing out that only rotations by 360º/*n *and their
integral multiples are admissible - *n* an integer). By reflection, one returns by its repetition to the
initial state and you can represent the symmetry-element by a *reflection-plane**.*

An observer who stands at the
origin of a coordinate system in the *XY*-plane (the +*Z*-axis
enters through his feet) and looks along the +*X*-axis,
has either the *+Y*-axis on his right- or on his left-hand
side. Correspondingly, you distinguish between *right-handed* and *left-handed* coordinate systems. Every reflection as well as every
symmetry operation involving reflection (rotation- or
displacement-reflection) converts a coordinate system A always
into one which is metrically equal, but opposite as
regards right- and left-handedness (Fig.151). Two such opposing
equal coordinate systems are said to be *enantimorphous* (Greek: enantios = opposite). The remaining symmetry operations, that is,
translations, rotations and screwings, transform each coordinate
system into a metrically and also with respect to its
right-left-character equal coordinate system.

After this survey of the
symmetry operations on coordinate systems, we return now to the
determination of the *symmetry* of a given body and pursue the line of
thought of K. Weissenberg. As a consequence of the deliberations
above, *we know all
symmetry operations*
which are possible in space. We can now fix the observer and his
coordinate system in space and, in contrast, put the body through
the now known to us symmetry operations in space (*displace, rotate, reflect it, etc.*), thus survey it from all sides and
take measurements of the three-dimensional images. This is how we
arrive at the following *definition
of symmetry*.

The *symmetry *or
also *symmetry group* of a body is the totality of those
symmetry operations (translations,
rotations, reflections and their combinations), which place a body from its *initial* position to all those positions which,
measured from a fixed coordinate system, are *indistinguishable*. Through its symmetry elements, the
symmetry group of a body becomes intuitive (chains of
translations, axes of rotations, planes of reflection, which form
a scaffold, Fig, 156,
third row). It you now perform, for example, a rotation of the
body and take from the resting coordinate system continuously the
corresponding pictures, you can observe how often and in what
positions the rotated body offers the *same* view, while it rotates about a definite
axis by 360º. Depending on whether the same image occurs 1, 2,
3, ,. . . *n* times, you say that the axis of rotation is
an 1-, 2-, 3- . . . fold axis of symmetry. If you examine in this
manner every axis through the body, you can, as far as rotations
are concerned, determine the body's symmetry^{1}. For example, if you rotate a 3-, 4-,
5- . . . faced regular pyramid about its axis ( a line through
its tip and the centre of its base) and observe from an arbitrary
coordinate system, you discover correspondingly 3, 4, 5 . . .
times the same images and say therefore that the axis of the
pyramid is a 3-, 4-, 5- . . . fold axis of symmetry.

In an analogous manner, you
discover the existence of all other kinds of symmetry and finds
thus as the symmetry group of a body a *scaffold* of *rotation- and rotation-reflection axes, screw-axes
and sliding reflection planes*. For a finitely
bounded body shape, only rotation- and rotation-reflection axes
are possible as symmetry elements^{2}, while translation chains as well as
all symmetry elements, combined with translations, occur only in
the case of *unbounded
large* structures.

For example, an infinitely long
row of equal trees would have the symmetry of a *translation chain*, provided the trees are at equal
distance from each other.

^{1}The single screw axis (rotation by 360º/1 and translation)
corresponds to a *translation
*chain, the single
rotation-reflection axis (rotation by 360º/1 and reflection) to
a *reflection plane*, and the two-fold rotation-reflection
axis (rotation by 360º/2 = 180º and reflection) to a *symmetry centr*e.

^{2}Thus, symmetric and *asymmetric*
are *not opposites*, but *asymmetry*
is only the l*owest
degree* of symmetry,
namely characterized by the one-fold axis of rotation. Indeed, *every body *covers itself on rotation by 360º/1.