D3. Mechanical Properties of solids

Crystal Physics

The symmetry of the lattices of the fine structure of crystals (arrangement of atoms and the inner-atomic fields of force in the crystal) leads to the general, physical properties of symmetry of crystals, which we will discuss here corresponding to the three kinds of symmetry :

1. Homogeneity of crystals. In order not to be disturbed by the inhomogeneity which appears at each face, we imagine the crystal to be infinitely large. Corresponding to the translation lattice of the fine structure of the crystal, we must then expect that each crystal assumes a for all methods of investigation indistinguishable position as it is displaced by an arbitrary multiple of its translation (which has the approximate size of 10^-10 cm). Since physical examinations cannot be executed at an infinitely small point or on an infinitely thin line or face of the crystal, respectively, but always only in a (compared with the translation distances) large area, every parallel displacement of the crystal within the limits of accuracy of the method of examination is a multiple of a translation and therefore a symmetry operation. The crystal appears thus to be continuously homogeneous and its properties, for example, strength, velocity of growth and dissolution, heat conductivity, etc., are at all its points and in parallel directions identical. The periodic homogeneity only becomes detectable and measurable, if you use X-rays, since their wave length is about as small as one translation period in the crystal (Laue 1912).

In empty space, the waves of X-rays propagate at the velocity of light along straight lines and uniformly. However¹, if a bundle of parallel X-rays encounters a crystal face, one part of the radiation passes through it and leaves marks on on a photographic film, mounted behind it, as penetration points; another part of the radiation is dispersed (secondary radiation). The secondary rays leave the crystal only in definite directions, which are independent of the outer bounds of the crystal and solely given by the position of the crystal lattice with respect to the entering ray. As the rays meet the photographic film, they mark themselves as interference points.

These points can be interpreted as a certain projection of the crystal lattice on to the photographic film, so that they are subject to the same rationality law as the lattice; you compute from the positions of the interference points and the wave length l of the X-rays the translations a, b, c of the crystal lattice, in fact, Laue's rationality law applies². The investigation of crystals with X-rays has completely confirmed Schoenflies' definition of crystals. The rationality law of translations and planes in a lattice, which we encountered in crystal morphology as that of the crystal-edges and -planes, manifests itself in crystal physics as rationality law of X-ray-interference, and hence also homogeneity of crystals manifests itself in all physical properties. Depending on the differentiation ability of the method of investigation, the crystal appears to be continuously homogeneous or - recognizable and measurable - periodically homogeneous.

¹ Fig. 814
² a) (cos a_n- cos a₀) = hl,
b) (cos b_n - cos b₀) = kl,
c) (cos g _n-cos g₀) = ll,
where a₀b₀g₀ and a_nb_ng _n are the angles between the translations and the incident and refracted rays, respectively.

2. Anisotropy of crystals. The lattice structure leads to differences in direction, and therefore excludes isotropy of the fine structure of crystals. Hence, in their fine structure, crystals are always anisotropic, whence their physical properties cannot be represented by spherical surfaces. In general, strength, growth velocity, conductivity, etc. differ with direction. However, the result of an examination depends again on the sensitivity of the experimental method. Ordinary optical, electrical, thermal, etc. methods cannot detect in cubical crystals anisotropy (deviation of the cube symmetry from the sphere), whence cubical crystals seem be optically, etc, sphere-symmetrical; however, if you measure the mechanical properties, for example, strength, or if you illuminate instead of with light with X-rays, the anisotropy manifests itself also here.

Cleavability The volume of the fundamental parallelepiped of the space lattice is a constant W, characteristic for the body. If the particles (molecules, atoms, ions) lie in a lattice-plane very close to each other, that is, if the fundamental parallelogram s is very small, then the distance d from the next plane is comparatively larger, in order that s d = W continues to exist. However, the molecular forces decrease with increasing distance very quickly, whence the two planes are easily separated. The presence in crystals of planes of preferred cleavability, which intersect one another in a given substance at a constant angle, is explained in this manner; for example, in calcite at 105º 5', the fundamental parallelogram of the space lattice forms a rhombo-hedron with this angle.

Elasticity of crystals According to their physical properties, the number of crystal classes (32) and the systems (7) reduces considerably. We are especially interested in the elastic properties of crystals, because the crystalline state is the ordinary state of solid substances. The modulus of elasticity of a crystal differs in different directions at any point of the crystal. If you represent it by vectors, starting at the point of observation, you do not obtain a sphere, but another surface; for example, Fig. 158 shows ithat of rock salt.

In order to characterize the elastic properties of a crystal, the two elastic constants (tensile and shear modules) of the isotropic material do not suffice. The less symmetric is the crystal system, the more constants are required. Already the cubic system demands 3, the triclinic 21.

The elasticity of crystals also determines their optical behaviour. Their behaviour with penetrating light is characteristic for crystals and especially for their different physical behaviour. The essential cause for the strange optical phenomena in crystals - we can only refer to them here - arises from the fact that light propagates in different directions at different speeds.

There exist three main light velocities along the axes. In the cubic system, all three of the main light velocities are equal, in the crystals of the tetragonal and hexagonal systems only two, the third corresponding to that axis of symmetry, after which the crystal is called optical mono-axial; in the other crystals (optically bi-axial), all three differ. If you plot again the velocities of the rays emanating from a point in different directions as vectors, their ends form the ray surface. In isotropic bodies (because the light velocities are the same), no direction differs from another, whence the ray surface is a sphere. It is also a sphere in crystals of the cubic system, whence these crystals are called optically isotropic.

The situation differs in the optically anisotropic crystals, the optically mono-axial and bi-axial crystals. In those crystals, two rays propagate in each direction (except only along the optical axes themselves), which differ physically and are called ordinary and extra-ordinary rays. All ordinary rays propagate equally fast, the extra-ordinary ones at different velocities. Therefore consists the ray surface of an optically anisotropic crystal always of two shells, one allotted to the ordinary, the other to the extraordinary rays.

In optically mono-axial crystals, a sphere is allotted to the ordinary rays, a concentric rotational ellipsoid to the extraordinary ones , the axis of rotation - the main crystallographic axis (Fig. 159). In bi-axial crystals, the ray surface is very complicated. Fig. 160 shows the main cuts: In two of them, a circle and ellipse do not touch one another, in the third they intersect. at the points of the circle, the tangents to which simultaneously are tangents to the ellipse, enter and leave the optical axes. The angle of the optical axes is characteristic for optically bi-axial crystals. For example, optically mono-axial are beryl, ice, calcite, corundum, sodium nitrate, quartz, tourmaline, cinnabar, zircon and optically bi-axial are borate, chloracic acid, potash, iron vitriol, acetic caustic soda, copper sulfate,sucrose, nitrate of silver.

3. Orthomorphy and Enantiomorphy of crystals. By Schoenflies' definition, ortho-morphic and enantiomorphic cystals are compatible. Since the empty space is orthomorphic, the conditions for the generation of both kinds are equally favourable. Enantiomorphic crystals are important for Physics (rotation of the plane of polarization) as well as for Chemistry (Stereo-chemistry). If the molecules of a substance are already enantiomorphic (Constitutive Enantiomorphy), say, like a tetrahedron with an unequal number of sides or corners, we must distinguish between left- or right-molecules, respectively. During crystallization, there may basically turn up 3 forms; the mutually enantiomorphic ones (left-crystal or right-crystal), consisting only of left- and right-molecules, respectively, and one ortho-morphic form (racemic crystal ¹) with equally many left- and right-molecules in the fundamental body and hence in the macroscopic volume unit. If you transform a right-crystal of a constitutive enantiomorphic substance into an amorphous phase, that is, steam, solution or melt, you always obtain from these phases during crystallization right-crystals; it is analogous for left-crystals, so that you can go to and fro between the crystalized and amorphous phases without changing the enantiomorphy (Pasteur).

¹racemus is Latin for grape; racemic-acid led to this discovery.

Crystal chemistry

X-ray Physics of crystals has also made possible measurement of the elementary body of lattices. Pure lattice theory does not give information about the material content of the elementary body, but it tells that the entire crystal consists of mutually identical elementary bodies; in other words: The proportion of the number of different kinds of atoms in the elementary body are the same as in the crystal. The size of elementary bodies is only a few atomic distances and can therefore only contain a few atoms. Hence follows Dalton's law in the form: If different kinds of atoms come together in crystallized chemical compounds, then the ratio numbers must be small integers.

It is now possible to develop beyond Dalton's Law a strict molecular theory of crystals. The first attempts in this direction were made by Nernst. He represented the heat content of a crystal (in accordance with the kinetic theory of heat) by the energy of oscillations of the atoms about equilibrium positions in the crystal. At high temperatures, all atoms oscillate with respect to each other; at falling temperature, the stronger links between the atoms rigidify faster than the weaker ones, whence at sufficiently low temperatures only the relatively loosely interlinked molecules oscillate with respect to each other, while the internal oscillations of the atoms are frozen in individual molecules. The decrease of the energy of oscillation at falling temperature can be measured by the drop of specific heat of the crystal and out of this Nernst has computed the largest groups of atoms, which at sufficiently low temperature oscillate together - that is, in the kinetic sense, they must be viewed as the molecules of the crystal.

You must distinguish two cases:

1. All the different kinds of atoms, contributing to a crystal, form at first equal molecules (that is, firmly connected groups of atoms) and these combine at first with relatively weak forces into a crystal; you have then the molecule lattice. This case is especially frequent in the case of organic compounds, where the molecules have been preformed already in the amorphous phase ( steam, solution, melt), during crystallization they come together to the crystal compound and separate again from each other during conversion into steam, solution and melting without the molecular binding being lost.

You might think now - and Bravais' views about the building of crystals correspond approximately to this - the crystal arises here in that the molecules are composed in one to another parallel positions periodically in the three space directions into the crystal lattice, so that the centres of gravity of the molecules form a Bravais translation lattice and the smallest translation cell contains always one molecule, corresponding to the formula Vs = 1·(M) with V the volume of the translation cell, s the density of the crystal, (M) the weight of one molecule.

A crystal is only exceptionally built in so simple a manner. In fact, the molecules can (rather than through the action of three translation chains, according to Bravais) be composed by the action of 3 symmetry elements (screw axis, slide reflection), involving translation into a lattice structure and thus into a crystal. If you find all these lattices, you arrive at an exhaustive systematic of the possible arrangements of the centres of gravity of the molecules in the crystal - simple lattice types (Weissenberg).

The number of molecules in the translation cell is obtained as follows: Since a 1-, 2-, 3-, 4-, 6-fold screw axis reproduces a molecule 1-, 2-, 3-, 4-, 6-fold until a full turn and displacement about one translation is completed, there exist within each translation, and therefore also within in the translation cell 1, 2, 3, 4, 5, 6 molecules, if the symmetry group of the crystal contains a 1-, 2-, 3-, 4-, 6-fold screw axis: The 1- fold screw axis corresponds to one translation chain , so that Bravais translation lattices represent special cases of the simply lattice types. Therefore the number of molecules in the translation cell adjusts itself here only according to the toughness of the screw axes, that is, to the symmetry of the lattice, and not to the chemical structure of the molecules.

2. The different kinds of atoms combine directly, without formation of molecules into a crystal; there arises a radical lattice. Most inorganic salts belong to this type. In the simplest case, for example NaCl, which only involves two different kinds of atoms, each atom of the one kind surrounds itself symmetrically by 6 atoms of the other kind without development of a stronger binding between one Na- and one Cl-atom. (Fig. 161). As above, the lattice types can be derived; however, you expect that composite lattice types correspond to the different radicals.

The lattice types yield the following law (Weissenberg): The number of crystal molecules and equal radicals, respectively, in the elementary body does not depend on the chemical constitution, but is determined by the symmetry group (space group) of the crystal. In fact, if you divide the weight of the elementary body (volume*density) by this number, which is universally given for every symmetry group for all times, you obtain the molecular weight or the weight, respectively, of the stoichiometric total formula of the crystallized substance.

Lattice constant

The grouping of atoms (atomic structure) and their smallest mutual distance (lattice constant) are known for many crystals from their structure analysis with the aid of X-ray Physics; for example, that of rock salt, NaCl (Fig.161 Sir William Bragg 1862 1915). The smallest atomic distance (edge of the elementary cube) is found as follows: Denote it by d cm and the density of rock salt by D (= 2.164), then the mass of the elementary cube M = d³D g. Imagine that the crystal has been subdivided by planes parallel to the lattice planes into small cubes, at the centres of which is located one atom, then the mass¹ of a Na-cube = 23·1.65·10^-24 = 3.80·10^-23g, of a Cl-cube =35.5·1.65·10^-24=5.86·10^-23g, thus, on an average, each cube has the mass [(3.80 + 5.86)/2)·10^-23 = 4.83·10^-23g. If we set the mass of the cube, found from the weights of the Na-, Cl- and H-atoms, equal to the density found from the NaCl-crystal, then: 4.83·10^-23 = 2.164·d³, whence d = 2.814·10^-8 cm. This is the order of magnitude of all hitherto measured lattice constants of other crystals, for example:

diamond 1.540·10^-8 cm,
calcite 3.029·10^-8 cm,
gypsum 7.578·10^-8 cm.

By means of the lattice constants, one measures the wave lengths of the X-ray radiation.

¹ The atomic weight of Na is 23, that of chlorine 35.5 and the mass of an H-atom 1.6510^-24g.

Structural analysis of crystals of the inorganic salts yields in most cases facts, which contradict the Classical Valence Theory of Chemistry: The structure of rock salt (Fig. 161) shows that every Na-atom is surrounded by 6 Cl-atoms and every Cl-atom by 6 Na-atoms. However, since Na and Cl are equi-valued, it is impossible to interpret the structure of the NaCl-crystal solely according to valence in the classical sense. In order to interpret crystal structures chemically, one has to involve the coordination theory of Alfred Werner 1866-1919 (P.Pfeiffer 1904).

Deformation of solids

Elasticity. Strength

The capacity of displacement of mass particles with respect to each other is smallest among solids, but it does exist. In reality, a solid body does not in any way correspond to the definition of the rigid body of Theoretical Mechanics. Forces which represent the kind of its cohesion can be overcome and, if they have been overcome, the body decomposes: it tears, breaks, etc. The resistance with which it opposes is called strength. The forces of cohesion attempt to resist decomposition - hence they reconstitute the body's form as far as this is possible as soon as the deforming causes cease. However, it succeeds more or less completely: Almost completely - almost, we will return to this subject - but only when the deformation had not yet exceeded a certain bound, otherwise incompletely, that is, a part of the deformation remains. The property due to which a body recovers from a deformation is called Elasticity¹. (In fluids and gases, only changes in volume recover; they only possess volume elasticity. Solid bodies have in addition form elasticity). Everyday practice makes wide use of the elasticity of the most varied materials (metal, wood, leather, rubber, etc.).The limit of deformation which in solid bodies must not be exceeded is called the elasticity limit. Up to this limit, bodies are perfectly elastic, for deformation beyond this bound, they are are said to be incompletely elastic.

The limit of elasticity is not a conclusive mathematical quantity, because its numerical value depends on the accuracy of the measurement of the remaining deformation. If one were to determine every small change whatsoever, one would perhaps consider it already after the smallest deformation to be persistent. In practice, the elasticity limit is replaced by the better defined yield limit, that is the load, at which one has at first major deformation without noticeable increase of load.

¹Greek: elaun = drive

Elastic hysteresis, Elastic after-effect. Relaxation

While an elastic body is loaded and then gradually unloaded, it should, if it were perfectly elastic, pass always through the forms, which it assumes during loading and then unloading, in an identical manner: At a given load, it should assume the same form. However, it does not do that; the process is irreversible; to the same loads correspond always during unloading larger deformations than during loading, whence there remains a deformation residue. One part of it vanishes gradually, if you give it enough time. The retardation of the reduction of this part of the deformation is called elastic after-effect. However, the other - often very considerable - part of the residual deformation remains: The difference of the equilibrium deformation during loading and unloading is called hysteresis², following Warburg. The irreversible process comprises therefore two phenomena. Their effects are not readily separated, because the after-effect is very slow and may still be discernible after weeks.

² Greek: u^csterew = remain

Scientists have wanted to explain it by the inner friction of the individual particles. However, this interpretation is contradicted by the phenomena of relaxation whereby is understood: Stretch a wire, which has been clamped and fixed at one end, to a certain extent and measure the force required to maintain the wire's extension. Then you will observe that, in order to maintain this state of deformation, you need at first a certain force; as the extension is maintained, this force becomes less, because the wire relaxes without motion. Therefore relaxation does not match internal friction, because, in order to explain the delayed deformation, it must be assumed that the inner friction depends only on the rate of deformation. A satisfactory theory of after-effects did not yet exist in 1935. The remaining deformations do not have a big role in the use of rigid materials in buildings and machines, since it is always concerned with lasting loads.

Different forms of elasticity

The mode of deformation depends on the initial form of a body and also on whether the deforming action compresses it or pulls it apart, or bends it or twists it. Accordingly, one speaks of elasticity in compression, in bending, etc, and of strength under pressure, in tension, in bending, etc. - One and the same body can be subjected simultaneously to different deforming actions. But we assume here that it is always loaded only by one of these at a time and limit our consideration to a prismatic or cylindrical bar.

1. Forces act which pull outwards (Fig. 162a), lengthen and eventually tear the bar apart: Its resistance to this is called tensile strength.
2. Forces act inwards (Fig. 162b), try to shorten the bar and finally crush it: Its resistance to this is called compressive strength. If you place two cross-sections A and B perpendicular to the bar's axis, you recognize that the deformation of the bar in the case of pull increases the distance between A and B and reduces it in the case of pressure.
3. Forces act to shift one section of it above the other (Fig. 163/4): Its resistance to this is called shear strength.

You can envisage the deformation of the body in this case by imagining it composed of very thin, parallel layers - like a pile of sheets of paper, etc. (Fig. 163), which only hold together due to friction. You convert the pile into the form indicated by dashed lines, if you press your hand on it and then slide it parallel to the base, that is, apply a force in the direction of the arrow. Every single sheet slides with respect to the others - remaining parallel to them - in the direction of the applied force, indeed more, the further the force is located from the base. The pile of sheets of paper retains its deformed shape as you remove your hand. However, if you use instead a rubber prism, glued to the table top, the deformation is just as described, that is, each cross-section is displaced with respect to every other, although only by a very small amount, and returns when you release the top of the prism, because rubber is elastic.

For example, a roll has shear strength when you cut with scissors (Fig. 164). As you try to bring the scissors' blades s and s' closer to each other, they push one half of the body above the other. This is displayed most clearly in the metal cutter (Fig. 165) used in industry. The circular, slightly conical discs S and S' touch one another with knife-like, sharpened edges and take over the role of scissors. You turn them in opposite directions and feed the metal sheet between them. The upper disk presses downwards, the lower upwards against the sheet; like in Fig. 164 the scissors; both together cut where their pressure acts. In many activities in industrial work, the shear strength of materials is acted upon, for example, when you punch or drill holes into metal sheets.

4. Forces act which tend to bend and finally break the body (Fig. 166). Its resistance to this action is called bending strength. - The two cross-sections (Fig. 162) A and B (perpendicular to the longitudinal axis of the not yet deformed bar) are no longer parallel after the deformation; at certain points they have approached each other, at others moved away from each other. Let there be on the undeformed bar (Fig. 166 lower figure) parallel to A and B the lines ab, cd, · and cross-sections ab, cd, · · ·, etc.; after deformation, they have moved to the positions a'b', c'd', · · ·, etc. The distances ac, ce, · · ·, etc. have shortened, the distances bd, df, ab, cd, · · ·, etc. have lengthened, that is, the fibres of the bar, facing the centre of the curvature, have been shortened - compressed - and the outer ones lengthened - stretched. There must exist between CD and EF a layer which has neither been shortened nor extended, that is, which only changes its shape, the neutral layer. At each cross-section of the bar, which it intersects, it passes through the centre of gravity of its cross- section.

5. Forces P and Q act on the ends of the bar (Fig. 167) to turn it in opposite directions, in order to twist and eventually destroy it. Its resistance to this action is called torsional strength. The two cross-sections A and B (perpendicular to the bar's longitudinal axis) are still parallel to each other after deformation, but have been rotated with respect to each other; its sections have turned a little about the longitudinal axis clockwise or anti- clockwise.

This deformation is described yet more clearly by Fig. 168: Twist the upper end of a rubber cylinder with fixed base at its upper end parallel to its base. The twist is transferred through the bar to the lower, fixed end. During this process, the cylinder twists about its axis: Sections which were before deformation parallel to each other, are so afterwards. The base does not move, the upper end of the cylinder twists most, every cross-section in the ratio of its distance from the base, that is: If the rotation at the distance of 1 cm from the base is j, it is 2j at the distance of 2 cm, etc. - You recognize the presence of the twist on the surface of the cylinder in that points, which before the defamation were located on lines parallel to its axis, are after deformation located on lines screwed about the axis.

Hooke's law relating deformation to the applied force

The relationship between the magnitudes of the deforming force and deformation within the limit of elasticity is given by the law: The magnitude of the deformation is proportional to the magnitude of the deforming action (In Latin: ut tensio sic vis. Robert Hooke 1635-1703 1676).

In other words: If a force of given magnitude causes a certain elongation (or shortening or twist), then twice as large a force causes twice as large that elongation, etc. - always assuming that the deformation does not exceed the limit of elasticity. The law allows measurement of the relationship between force and deformation and to derive by the coefficient of elasticity a measure of the elasticity of a substance.

Hooke's law can be employed to measure the applied force by means of the deformation of an elastic body . For example, an instrument for this purpose is already the ordinary spring balance (letter balance), which determines a body's weight, that is, the force by which Earth attracts its mass (Fig.169). You use for this purpose a coil spring, one end a of which is fixed, while the other end b can move. Such a spring elongates (or shortens) as a pull (or a pressure) in the direction ab is applied; in fact, the more, the larger the pull, (or the pressure) and it resumes its original form as soon as the force ceases to act. Hence, if you suspend by it a mass W, b moves further downwards the heavier is the mass - due to the pull of the same mass a little less than at the Pole or at any location in between, because g, and therefore also M·g increases from the Equator to the Pole. If during this process a pointer moves along a scale S, while you suspend masses of known magnitude from a set of weights and you know the value of g at the location, where the calibration occurs, you can create a calibrated dyn scale.

Such a device (with necessary changes) can measure every pull and every pressure, for example, the pull of a horse on a cart, if you place it between the horse and the cart, so that it forms a part of the trace. Since such devices measure forces, they are called dynamometers.

Deviations from Hooke's law

Elastic after-effects clash with Hooke's law: This shows that there do not at all exist unique and reciprocal relations between actions of force and deformation. If tension stops, deformation does not vanish simultaneously, it passes through the range back to the initial zero value according to a hitherto unknown law. Also purely elastic deformations obey Hooke's law only approximately, as has been learned by continuously improving observational measurement techniques, - The deviations are considerable with cast iron, minerals and technically important substances for binding surfaces (cement, concrete).

Coefficients of elasticity. Modulus of elasticity

Within the limit of elasticity, the lengthening of a wire through stretching is proportional to the load. However, for many substances, the limit of elasticity is only known approximately . For example, a drawn, not annealed silver wire with a cross-section of 1 mm² can be loaded by 11.2 kg* before it reaches its limit of elasticity (Fig. 170). A wire of 1 m length under this load lengthens by 1.5 mm, with a load of 1 kg*, therefore by 1.5/12.2 = 0.134 mm, that is, by about 1/7400 of its initial length.

This fraction 1/7400 is called the elasticity coefficient of silver. We define: The coefficient of elasticity (a) is the fraction by which a wire with a cross-section of 1 mm² lengthens in tension under the load of 1 kg*.

This is for all substances a very small fraction, whence it is more comfortable to employ the modulus of elasticity or Young's Modulus (Young 1807): The silver wire is extended by 1/7400 m by 1 kg*, whence 7400 kg* would, if the limit of elasticity were faway ar enough (that is, Hooke's law is still applicable) stretch it by 1 m, that is, double its length. We define: The modulus of elasticity (E = 1/a) states the number of kilograms* the weight of which would double the length of a wire with a cross-section of 1 mm² (provided Hooke's law still applies and the material can endure such a deformation).

Constants of elasticity. Modulus of compression

Elastic extension is always linked to a perpendicularly to it contraction, which is proportional to the extension. If a bar has the length l and the diameter d, and it extends by l - the fraction l /l is called the relative extension - its diameter undergoes the relative reduction d /d. Every day experience shows that d /d = n··l / l, where n·, the ratio of Siméon Denis Poisson 1781-1840, is for each material a constant which lies between 0.2 and 0.5.

Within the limit of elasticity, the deformation is proportional to the load. In view of the multiplicity of the possibilities of deformation, one must expect very many factors of proportionality - called, in general, elastic constants. However, for isotropic materials, that is, materials for which all directions are equivalent, their number, according to theory, does not exceed 2: The elastic coefficient a and the shear coefficient b. However, there exists the relationship between a, b and n: a = b/2(1 + n·), whence an isotropic body has only two independent elastic constants. (A crystal of the triclinic system - the perfectly anisotropic body - has 21).

Besides Young's modulus E = 1/a and the shear modulus G = 1/b ( = m), one has still the modulus of compression. If you have to deal with a state of all round compression and denote the pressure by p and the specific volume by DV/V, then p = [ l + (2/3)m ]·DV/V . The factor K=l+(2/3)m is called the modulus of incompressibility and its inverse k = 1/K that of compressibility. You have K = (1/3)E/(1 - 2n·), whence you conclude that 0 < n· < 0.5. If you had n· > 0.5, then K would be negative, that is, a body would contract during pulling, expand during pressure, which contradicts experience.

Strength

Once the limit of elasticity is exceeded, the form of a body remains altered. Depending on the nature of the substance and the kind of loading, the residual change is different in form and size¹, but eventually it reaches a limit, the body tears, breaks, etc., in other words, it decomposes. In the regime between the limit of elasticity and the border, where bodies decompose, materials exhibit properties, due to which they are called extendable, malleable, rollable, brittle, friable, hard, soft, etc. - properties for which there do not exist unique definitions nor exact measures.

¹ The elasticity and strength of metals assumes strange features in the case of single crystal wires. One section of the crystal lattice moves as a whole along the wire, without imternal degradation and destructure of the coherence of the wire; in fact, this happens along definite sliding planes and there in definite sliding directions. The irregularities in the diagram (Fig. 171) correspond to the transition of the wire by sliding into a new equlilibrium state. The wire deforms then into a band; with extensions uo to several 100% before the wire tears.

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