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.

Atomic Structure of matter

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 matter1. 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.

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 atomism2; 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.

2The 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 1l of chlorine combine into hydrogen chloride, indeed 2 l of hydrogen chloride are formed. However, that means: n hydrogen molecules + chlorine n molecules yield 2n hydrogen chloride molecules (2n, because every litre of hydrogen chloride contains n molecules, whence they contain, according to Avogadro's rule 2n 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:

nCl2+ nH2 = 2nHCl.

An atom Cl is denoted by Cl, the molecule, consisting of 2 atoms, by Cl2. Similarly, you have the atom of hydrogen H and its molecule H2, 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 A1 and A2, then those quantities of the substances contain equally many (N) atoms or equally many molecules which are in the ratio NA1:NA2, that is, A1:A2; 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 = 6.06·1023 (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.

Density and specific weight

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, dw the density of water, then s = d/dw.

We set the density dw of water (its mass contained at 4ºC in 1 cm3) 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

Symmetry

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 structured1.

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.

1We will not discuss here the chemical structure of molecules, for example, of the water molecule H2O, 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,
finally,
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 symmetry1. 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 elements2, 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.

1The 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 centre.

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