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- 1. POLYADIC FORMULATION OF LINEAR PHYSICAL LAWS.byElysio R. F. RuggeriOuro Preto, MG, BrazilKlaus HelbigHannover, GeABSTRACTPhysical fields are represented by tensors or polyadics of different valence (rank, order): R(of valence R), H(of valence H), etc. We say that a (dependent) quantity Ris proportional to a second (independent) quantity H wheneach component (with reference to an arbitrary vector base) of Ris proportional, with different (perhaps constant)weights, to all coordinates of H . The proportionality between two physical quantities expresses a linear physical law. Inpolyadic calculus this proportionality is formulated as a "multiple dot multiplication" written in the form HHHRR.G ; the valence of the polyadic R+HG is the sum of the valences of the other two polyadics. Thecoordinates of R+HG define the weights with which the coordinates of the independent polyadic enter in the constitution ofeach coordinate of the dependent polyadic.For R=H the proportionality exists between two fields of the same valence. We concentrate on this type ofproportionality, expressed by H 2H H H G . , and postulate a second relation between the dependent and independentpolyadics, the scalar 2W: HH2HHHHHHW2 ... G . The existence of this scalar implies the symmetry of 2HG ,i.e., the equality of the proportionality polyadic with its transpose: H2H2HGG . Many physical laws are expressed assymmetric relationships of this type with H=1 or H=2. Symmetric polyadic relationships can be expressed with referenceto an arbitrary external vector base, but often the expression with respect to the orthonormal polyadic base is moreconvenient. In particular, the orthonormal polyadic base defined by the eigenH-adics of 2HG is to be preferred.Some of the most interesting cases of proportionality between polyadics occurs int the theory of elasticity. For H=2,2W is the energy density stored at each point of a stressed body; for H=1, 2W is the normal stress. For any H, theparticular concepts of normal and tangential stress are extended to "radial" and "tangential" values of 2HG . Stationaryradial and tangential values at a point of a field allow the generalization of classical theorems known in the theory ofstresses, as Cauchys and Lamès quadrics and the representation in Mohrs plane. From Mohrs circle one can derive ageneral criterion of proportionality, closely related to the failure criterion in the theory of materials.When one uses dyadic bases to study the natural laws with H=2, it is necessary to introduce a new 9-dimensionalspace that is closely linked to the core of the problem. This space allows us to use intuitively some concepts of nine-dimensional Euclidean geometry. The main concepts of this geometry were established within the Polyadic Calculus(Ruggeri 1999), but are outside the scope of this contribution. It is not difficult to generalize the properties to arbitrary Hand to establish the N-dimensional analytic geometry associated with the physical laws. They follow immediately if oneregards the linear law as linear transformations (a mapping) of the "space defined by one polyadic" into the space definedby another polyadic through a "polyadic operator" (the proportionality polyadic). Some aspects of the geometry hidden inthese laws suggest interesting experiments to define the polyadic operator and a statistical polyadic to define "probable"values.The main objective of this paper is to show that all linear physical laws in continuum physics (particularly for H=1,i.e., vector quantities linked by dyadics, and for H=2, i.e., for dyadics linked by tetradics). can be treated mathematicallyby a unified method. This method is algebraic as well as geometrical. It is based on a synthesis of Polyadic Calculus andmultidimensional Euclidean (analytic) geometry.
- 2. 2SECTION I: POLYADICS AND GEOMETRY.I.1 - Physical Magnitudes, Polyadics and Euclidean Space.All physical magnitudes can be represented by tensors of different orders, or by polyadics of different valences; scalars arepolyadic of zero valence, vectors are of valence one, dyadics have valence two etc..Scalars (work, energy, temperature, entropy etc.) and vectors (force, velocity, acceleration, electric field etc.) arewell known from elementary mechanics and electromagnetics. Some dyadics are also known as stress and strain in thetheory of elasticity and in fluid mechanics; others, like dielectric permittivity, dielectric impermeability, thermal diffusivityetc. are known but in crystal physics. Triadics are common one of the better known examples is the piezoelectric triadic.Tetradics are even more common: the stiffness (and compliance) tetradic, the elasto-resistivity tetradic, the piezo-opticaland the electro-optical tetradic; and in theoretical geometry, the Riemann-Cristophell curvature tetradic.Dyadics.With two given ordered sets of vectors, say },,{ 321 eee and },,{ 321aaa , between which we can establish a bi-uniquecorrespondence (the ei is the correspondent of ai), we can generate dyadics and represent then by the symbolic sumiiae , for i=1,2,3, the repeated indexes in different levels denoting sum in the range (Gibbs, 1901; Drew, 1961). The eiare said to be the antecedent and the aithe consequent of the dyadic. If we insert between the antecedent and theconsequent a dot we obtain from the dyadic a number, called the scalar of the dyadic and denoted by s; if ewe insert a (a inverted v) we obtain a vector, called the vector of that dyadic and denoted by V. A simple example of thiscorrespondence from the theory of elasticity is Cauchys tetrahedron of tension: to each unit vector iˆe normal to a face "i"corresponds one and only one stress-vector is . This correspondence generates the stress dyadic iiˆ se (for i=1,2,3). Thestress dyadic is symmetric, that is, it is equal to its transpose (obtained by interchanging antecedent and correspondentconsequent) denoted by T. We have: Tiiˆ es , in which case V=o (o is the null vector). The converse is true, thatis, the necessary and sufficient condition that a dyadic be symmetric is that its vector vanishes. A dyadic, say A, can alsobe anti-symmetric, when it is equal to the negative of its symmetric: A=-AT.Triadics.Using ordered and correspondent sets of vectors and dyadics we generate triadics. For piezoelectric crystals (that generatean electric field when deformed) there is a bi-unique correspondence between each electric vector field ie in a point andthe strain dyadic i in this point (or vice-versa). This generates the piezoelectric triadic ii3e . In this form the dyadicsare the triadic antecedent and the vectors the consequent. If each one of the dyadics i could be related to other sets ofvectors, say },,{ 321 rrr and },,{ 3i2i1iaaa we could write by substitution: ijji3)( era for i,j=1,2,3 where theparentheses are necessary. Triadics can also present symmetries depending on the characteristics of the original dyadics.The Euclidean space of a polyadic.With polyadics of valence H (the H-adics) and certain basic operations defined between then we create a Euclidean space(with up to 3Hdimensions). To generate a linear space these operations are the addition of polyadics (of the same valence)and the multiplication of a polyadic by a number. These two operations are similar to their counterparts defined forvectors. Another important operation must be defined: the multiple dot multiplication with polyadics (we will abbreviatemdm) which is based on the dot multiplication of vectors. For example: the dot product of the dyadic kkgb by thevector v is the vector defined by the law )( kk.vgb.v . The double dot product of the triadic ii3e by the dyadic is the vector defined by the law ))(( kiki3.ge.b: . In view of the first definition this later expression can be writtenas ))](([ kikjji3.ge.bra: . Proceeding in this manner we can define the multiple dot product (abbreviated mdp) as faras the number of dot does not exceed the valence of the polyadic factor of smaller valence. We say that two polyadics areequal if their mdp by a same and any polyadic are equal. After these definitions and the demonstration of some theorems itis possible to write ijji3era , ))(( kikjji3.ge.bra: and similar expressions.
- 3. 3The polyadic can be represented by an arrow in its space.We can also calculate the double dot product of a dyadic by itself and, in general, of a H-adic Hby itself; this product,indicated in the form HHH. , is a scalar, always positive and called the norm of that H-adic. The positive square rootof the norm of the polyadic His its modulus and denoted by | H|. Hence, H-adics - as vectors - can be written in theform ˆ|| HHH where ˆHis a unit H-adic parallel to H. It may be shown that the square of the H-dot product oftwo H-adics Hand His less than the product of its norms; hence, there is an angle defined by two H-adics such that itscosine equals the H dot product between then divided by the product of their moduli; and we write:),cos(|||| HHHHHHH .. Again we have found a similarity with vector operations. After the choice of a scale,we can represent a polyadic in its space by an arrow whose length and direction be the magnitude and direction of thepolyadic. The angle of two polyadics, for example, is the angle apanned by their arrows.Dimension and base in a polyadic space.We can say that two non-vanisning H-adics are perpendicular if its H-dot product vanishes. One vector in its space isorthogonal to at most two other vectors. What about dyadics, triadics and polyadics?. To answer this question we must firstlook for the maximal number of linear independent H-adics of the generated space, that is, its dimension. We can concludethat this number is up to 3H(3 for vectors, 9 for dyadics etc.) and say that any set of 3Hindependent H-adics of a H-adicspace is a base of this space. Given the G3HH-adics of a G-space (a subspace of the H-adic space), GH2H1H...,,, ,they will be linearly independent if the determinant (or order G) does not vanish:GHHGH2HHGH1HHGHGHH2H2HH2H1HH2HGHH1H2HH1H1HH1HH..................||......... , (I.1.01).Now, given a H-adic base any of a G-space, GH2H1H...,,, , we can determine its reciprocal, that is, the baseGH2H1H...,,, such that jijHHiH . where the deltas are the Kronecker deltas.Other types of products.We can define also a multiple skew product of G-1 H-adics of a G-subspace. For example: as for vectors, the (simple)skew product of the two H-adics Hand Hof a 3-subspace is a third H-adic, say H, whose direction is normal to thedirections of the factors (hence this H-adic belongs to the 3-subspace), its unit ˆHpointing to the side on which a rotationless than 180 from the first to the second appears positive, and whose magnitude equal to the product of their lengthsmultiplied by the sine of the angle between then. We write: ˆ),sin(|||| HHHHHHHH . If},,{ 3H2H1H is a base of the 3-space we can write also the pseudo-determinant:3HHH2HHH1HHH3HHH2HHH1HHH3H2H1HHHHH||...... , (I.1.02),a formula well known for vectors (H=1, 11He etc.). To extend the definition we can use this determinant as referenceand amplify it for at most H-1 H-adics since the skew product must belong to the H-space.From the two multiple operations defined (the dot and the skew), we can define the multiple mixed product of G H-adics H, H, ..., Hof a G-space by the expression:GHHH2HHH1HHHGHHH2HHH1HHHGHHH2HHH1HHHHHHHHHHHH..................)...(.......... , (I.1.03).If we substitute in this expression Hfor 1H , Hfor 2H etc., we can say also that the set 1H , 2H , ..., GH form abase if their multiple mixed product does not vanish (as for vectors).
- 4. 4The polyadic associated matrix.The bases in a H-adic space can be formed with P-adics as long as PH, since we could not express a P-adic in a H-adicbase. If, say, e1, e2, e3 and e1, e2, e3are reciprocal vector bases then we can generate the two following two groups of ninedyads: e1e1, e1e2, ... and e1e1 etc. to compose dyadic reciprocal bases; or the two group of 27 triads: e1e1e1, e1e1e2, ... ande1e1e1, e1e1e2 etc.) to compose triadic reciprocal bases etc.. . Taking these polyades as bases and a coupled reference system- in which case we shall say that the H-adic is referred to a vector base - we can associate to a H-adic a (rectangular)3H3H-1matrix if H is odd (a matrix whose elements are the H-adic coordinates with respect to that base); and a square3H3Hmatrix if H is even. In the latter case the trace is the H-adic scalar. This may lead to huge computational calculationssince the number of rows and column of these matrices can be large. The abstract image of a base in a space is a "star ofarrows", and a "pencil of arrows" in subspaces. If we imagine the arrow of a polyadic with its initial point coincident withthe vertex of the star of the base arrows, the coordinates of its end point are the coordinates of the polyadic, i.e., theelements of its associated matrix (this justifies the name coordinate instead of component).I.2 - Physical laws, Linear Transformations and Polyadic Geometry.The operations between polyadics studied in Polyadic Calculus (the mdm in special) are appropriate to express linear andnon linear physical laws. A compact general form to express that a H-adic (the value of a function) is a function of a P-adic(the argument of the function), that is, )(PHH , is the generalized Taylor-series:...21 PP2PP2HPPPH0HH , (I.2.01),where H0, H+P, H+2P etc. are polyadics independent of the current H and P (perhaps functions of time, temperature etc).We say also that H and P are, respectively, the dependent and independent variables. If this function is linear there areonly the first two terms of the series. This means - for polyadics referred to a common base - that each one coordinate H isa linear function of (or proportional to) all coordinates P. For most physical laws the linear description is sufficient. Theforegoing considerations means that if the end point of the arrow representing P describes a line, a plane or a sphere in thespace (of dimension 3P), then the H ending point arrow describes a line, a plane or a sphere in its space (of dimension upto 3H), respectively.A new matrix operation to represent the polyadic linear laws.With respect to specified reciprocal vector bases we can associate matrices with the four polyadics present in the linear law(H, H0, H+P and P) which can also be written in matrix form since we define a new operation between matrices (whichdiffers from the classic operation), called double scalar product, to translate the mdp of two polyadics. Let us consider twoarbitrary matrices, NM]A[ and NM]B[ of the same order (with M rows and N columns), being ijA and ijB theircorresponding elements. We define as the double dot product A:B of these matrices (in arbitrary order) as the numberMNMN12121111ijij BA...BABABA , (I.2.02).Notice that the polyadic associate matrices in the linear laws are multi-ordinal, that is, the numbers of rows and columns ofone are multiples of the correspondent ones in the other: for instance, QP]B[ and [ ]A LPMQ(with L and M integers). Thesecond matrix may be resolved in LM blocks with P rows and Q columns, that is, this matrix has L rows and M columnswhose elements are matrices Aij with P rows and Q columns. We define the double dot product [ ]A LPMQ: [B]PQof the multi-ordinally linked matrices [ ]A LPMQand QP]B[ , in this order, as the matrix with L rows and M columns whose elements arethe double dot product of each [ ]A LPMQsub-matrix [ ]Aij PQ(with i = 1, 2, ..., L e j = 1, 2, ..., M) with the matrix [B]PQ. Thus, QPQPLMQPL2QPL1QP2MQP22QP21QP1MQP12QP11BA...AA............A...AAA...AA: MLQPQPLMQPQPL2QPQPL1QPQP2MQPQP22QPQP21QPQP1MQPQP12QPQP11BA...BABA............BA...BABABA...BABA:::::::::, (I.2.03).The correspondent operation - the double dot multiplication of multi-ordinally linked matrices - always exists. It iscommutative, distributive with respect to addition, and generally non associative.
- 5. 5Example 1: 23116360x1+1x2(-1)x1+2x25x1+3x20x1+3x2(-1)x1+2x24x11x212015123014321 123x1=32x2=4: .Example 2:Let us consider the product )R()A( mmkjiijk3e.eee.r ; we have: jikijkRA ee and the following matrixrepresentations :AAAAAAAAAAAAAAAAAAAAAAAAAAA3x3=93x1=333333233132332232131331231123323223122322222121321221113313213112312212111311211133 R R R1 2 3 13,and 133213x1=33x3=933332331333232231233132131123322321323222221223122121113312311313212211213112111133RRRAAAAAAAAAAAAAAAAAAAAAAAAAAA: ,Notice that each dyadic product coordinate is a function of all vector coordinates factor.Example 3:The product 3 : , with 3 ijki j kA e e e and Brsr se e , is the vector v e e A B Vijkjk iii. This expression can bewritten in matrix form asVVVA A AA A AA A AA A AA A AA A AA A AA A AA A AB B BB B BB B B123111 112 113121 122 123131 132 133211 212 213221 222 223231 232 233311 312 313321 322 323331 332 3333x31x311 12 1321 22 2331 32 333133: ,or 31321VVVA A A A A A A A AA A A A A A A A AA A A A A A A A A111 112 113 211 212 213 311 312 313121 122 123 221 222 223 321 322 323131 132 133 231 232 233 331 332 3333=1x39=3x3:B B BB B BB B B11 12 1321 22 2331 32 3333,but these forms are not unique. Again notice that each vector coordinate is a function of all coordinates of the dyadicfactor.Example 4:For the same polyadics of example 3 we have, on the other hand: 3 3 ijkks i jsA B . e e e . The corresponding matrixnotation is 393111 112 113121 122 123131 132 133211 212 213221 222 223231 232 233311 312 313321 322 323331 332 3339311 12 1321 22 2331 32 33 33A A AA A AA A AA A AA A AA A AA A AA A AA A AB B BB B BB B B . .
- 6. 6We notice that in this case the individual coordinates of the triadic product is not a function of all coordinates of thedyadic factor, but only of some ones.Polyadic Geometry.The mdm of polyadics can be interpreted geometrically; particularly a linear mdm can be regarded as a lineartransformation (LT) between polyadics (of different spaces) achieved by a polyadic operator whose valence is the sum ofthe valences of the input and output polyadics. Linear Transformations from one vector space (valence 1 and dim3) toanother vector space, operated by a dyadic (valence 2), are well known and sometimes mentioned as a "VectorialGeometry". In this LT the operator may perform translation, rotation - i.e., rigid transformations - and deformation(implying changes in distances and angles in the neighborhood of a point). Such linear transformations occur frequently inthe theory of classical mechanics and electromagnetics. The more complex cases - the LTs between a vector space and adyadic space (performed by a triadic), or between two dyadic spaces (dim9) performed by a tetradic - are rarelymentioned. Such transformations occur in the theory of elasticity and electromagnetism, and in Crystallography.The polyadic operator may perform also translation, rotation and deformation. For example: a rotation dyadic canrotate vectors by simple dot multiplication, a rotation tetradic can rotate dyadics by double dot multiplication. In the sameway a tetradic may stretch or shrink the dyadic defined by two points (in a dyadic space) and diminish or enlarge the anglebetween two dyadics. There exist also a polyadic that performs the identity transformation: this polyadic is called the unitpolyadic of the space (or subspace) and has always even valence; it is denoted by 2H and its associated matrix is the GGunit matrix (for G3H).It follows that there is a multi-dimensional purely Euclidean geometry hidden in the physical laws, which can be asuseful as the common two- and three dimensional ones. It could be called a "Polyadic Geometry" and has still to beexplored.From this point of view, the triangle defined by three points, e.g., is a universal entity whether its sides are vectors,dyadics or arbitrary H-adics, each defined in the corresponding space with the corresponding dimension. The so called"cosine law for triangles" holds universally, whether the squares of the sides of the triangle - the norm of the H-adic - aredetermined in a space of 3, 9 or any other dimension.This approach to the physical laws is now unified; it extends or complements the isolated cases mentioned above.But what are the consequences of theses geometrical concepts for the physical laws they represent?I.3 - Linear Transformations, Experimental Measurements and Statistical Polyadics.This geometrical interpretation of physical laws suggests us a single way to determine the LT polyadic operator. Afundamental theorem: If in a G-space G independent P-adics Pi. (i=1,2, ..., G) is associated bi-uniquely with G H-adicsHi, then the linear transformation polyadic operator H+P (or the proportionality polyadic) is determined as H+P= HiPifor i=1,2, ..., G. From the physical point of view, the physical law connecting two physical magnitudes can be determinedby measuring under specified physical conditions (of time, temperature etc.) G pairs of the correspondent magnitudesunder the geometrical condition that one set of one member of the pairs, say P1, P2, ..., PG, is composed of independentpolyadics, i.e., (P1P2...PG)0 does not vanish. For example: to determine the tetradic which connects the stress dyadicwith the strain dyadic in linear elasticity, we must chose and measure six independent strains dyadics (instead of nine inview of the symmetry of the strain dyadic) and the corresponding stress dyadics (which are not necessarily independent).This approach may require appropriate laboratory devices and accurate measurements. Moreover, themeasurements must be collected within the "media" in a "state of proportionality". These measurements will be performedwith respect to a convenient chosen vector base, say e1, e2, e3 (and its reciprocal e1, e2, e3if the base is not orthonormal).With this base, the associate matrix and the ensuing calculations many other physical problems can be solved.Though one such determination of G independent pairs of polyadics is theoretically sufficient to define the trueassociate polyadic coordinate matrix, there may be practical difficulties. The matrix obtained by two such experimentsmay be not equal, mainly due to observational errors. To deal with these, the classical theory of "probability and errorstatistics" has to be extended to polyadics. Such a theory has to be based on the representation of polyadics by theirinvariants.I.4 - The Physical Phenomenon is Equivalent to a System of Linear Polyadic Equations.Physical phenomena occur in definite regions of the physical space. These regions are seen as a field of the various"quantities" participating in the phenomenon. This quantities (scalar, vector, dyadic etc.) are continuous functions ofposition (even in the limit) and time (often non zero only after a definite initial time) with continuous first derivatives.In a given point P and time t of a field, one quantity of a set (which we select as dependent) can be proportional toone or more of the magnitudes of different orders of a second set (selected as independent), each one of these latter varyingwith P and t.Hence we can conclude, from the mathematical point of view, that:
- 7. 71) the physical phenomenon is equivalent to a system of linear polyadic equations involving (arbitrarily selected)dependent and independent magnitudes;2) this system must be compatible, that is, to a given set of values of the independent variables there alwayscorrespond one and only one set of values for the dependent variables;3) as this system must be true in space and time, there must be defined its values in the beginning of the timemeasurements (with correspondent position) and at the boundaries of the region (with the correspondent time).It should be noticed that when one of the variables undergoes a differential operation (for example, when it derivesfrom a potential) the system pass to be a system of linear or non linear differential equations (according as the derivativesappears as simple derivatives, or as second derivatives, as third etc. or, also, as products of different derivatives) butalways with degree one (the power of the derivatives is always one).I.5 - Eigenvalues and Eigenpolyadics.In vector geometry we look for "special bases" with respect to which we can simplify the geometrical studies; in physics,besides this geometrical simplification, we may be interested in the facilitation of experimental measurements. This isalways possible when the LTs are to be performed between polyadics of the same valence, say H and H, in which casethe polyadic operator has even valence, 2H. We ask: when HHH2HX. for some scalar X, i.e., is there any H-adicH which is transformed into a H-adic parallel to itself?. Or, what is the same, when HHH2H2H)X( . ?. Theexistence of this equality for some H implies that the 2H-adic between the parentheses must be incomplete, that is, itsassociated mixed matrix must be degenerate (its determinant must vanish).If we put 2H HiH i for i=1,2, ..., G with respect to some H-adic reciprocal bases {H*} and {H*} of the G-space, we define the 2H adjoint, and denote it by G~2H , by the expression fatores1GmHjHiHmHjHiHG~2H......1)!(G1 for (i, j, ... ,m = 1, 2, ..., G), (I.5.01).where we are using the ready defined multiple skew multiplication of H-adics. The G~2H associated matrix is the adjointof the 2H associated matrix. This adjoint and its leading (diagonal) minors express the condition for 2H2HX to beincomplete:0)1(X)1(X)1(...XXXXG2HGG~E2HG2)~1(GE2HG3G3~E2H2G2~E2H1GE2HG, (I.5.02),where 2HG is the 2H determinant; the coefficient of the linear term is the sum of the leading minor of degree one of thisdeterminant, that is, the scalar of G~2H ; the coefficient of the quadratic term is the sum of the leading minors of degreetwo of this determinant, that is, the scalar of )~1-(G2H ; etc.. This equation is the "2H-adic characteristic equation". Thesolution of this problem brings us to the determination of the 3H(invariant) eigenvalues and eigenH-adics of the (statisticalmeasured) 2H-adic operator. Then it is possible to demonstrate the Cayley-Hamilton theorem (for future usefulness) forpolyadics, that is: every 2H-adic satisfy its own characteristic equation.SECTION II: THE ESSENTIAL CONDITION FOR A GEOMETRICAL APPROACH TOPHYSICAL LAWS.II.1 - A particular situation, largely useful in Physics.Let it H and H the H-adics (polyadics of valence H) representing two H-order proportional and continuousvariable quantities defined in the current point O of a G-space (G<3H), one of then, say H , taken as independentvariable. We have: HHHHHHH||||,ˆ|| . , and H H H .1 (II.1.01)1,1 - When a polyadic is expressed in an arbitrary vector base by its "coordinates" (covariant, contravariant, etc.), its norm (always a positive number) isequal to the sum of the product of the coordinates of one type with the corresponding coordinates of opposite type; the square root of its norm is its
- 8. 8where ˆand||||,|| HHHdenotes the norm, the modulus and the unitary (a H-adic of norm 1 parallel to H ) of the H-adic H .The proportionality of the magnitudes – the linear physical law - can be expressed as the linear polyadic equation HHH2H.G , (II.1.021),where the dependent variable H , besides to have variable direction, has also variable norm (hence, a variable modulus);and 2H G - the proportionality polyadic, independent of the point O and the current H and H (a constant or, perhaps,a function of time, temperature or other parameters) - must be a complete 2H-adic into the G-space if it is necessary toexpress H as a function of H . Hence, admitting that the 2H G determinant in this G-space does not vanish, we caninvert (II.1.021) and write0)det(2HG and H H H H 2G . H 2H H H G .1 , (II.1.022).The pair of inverse laws (II.1.021) and (II.1.022) exist in the G-space if 2H G has non vanishing eigenvalues (in thisspace). Else the law exist in a space of dimension one unit less (a subspace if G3H) for each vanishing 2H G eigenvalue;and in this subspace the 2H G will be seen complete.The polyadic G2Hcharacterizes the medium completely for the phenomenon governed by the law (II.1.01). Thuswe can say that these polyadics are the parameters of the medium with respect to the phenomenon under consideration;with our laboratory devices we can determine its cartesian coordinates by specifying a convenient vector base. Before handwe must observe that the polyadic coordinates for the specified phenomenon are different for different observers (becauseeach one chooses his own vector base).A Postulate, Specific Magnitudes and 2HG Symmetry.In view of physical usefulness we might admit the followingPostulate:There exists a continuous variable function of scalar value 2W with several continuous derivatives,defining a new physical magnitude by the lawW2 HH2HHHHHHHHH .... G , (II.1.031).To simplify the mathematical handling we will introduce the new variablesHHHH00H H2W=| | 2W and 2w=2W| |2W|| || | |, , , (II.1.04),so that - besides the unchanged law (II.1.01) - we have,H H H H H H H H 2 2G G. . , (II.1.02),2w H H H H H H H H 2H H H . . . .G , (II.1.03). ˆˆW2 HHHHHH0 .. , (II.1.031).We could name magnitudes H and 2W0 "specific magnitudes", or "magnitudes by unit of H intensity(modulus)" since | H | represents a quantity of the magnitude H .From (II.1.02) and (II.1.03) we deduce 0=ˆ)(ˆ:ˆ HHH2H2HHHH .. GG , which is possible if and only if2H 2H H 2HG G , that is,2H 2H HG G, (II.1.05).Hence, the acceptance of the postulate (II.1.03) carries the 2HG single symmetry2 for any H.The unit H-adic H (in a G-space) has G coordinates when resolved in a H-adic base of this space but only G-1 areindependent because its norm is equal to one. Taking the point O of this euclidean G-space as origin of polyadics, H canmodulus. If the base is orthonormal the norm is equal to the sum of the square of its coordinates.2 This concept is valid only for polyadics of even valence (as dyadics, tetradics etc.).
- 9. 9be seen as the H-adic position of a point on a hiperspherical surface (or, simply, a spherical surface) of unit radiuscentered at O; it defines a hiperdirection in this space (or, simply, a direction). In elasticity, for H=1 and G=3 forexample, H is a vector p representing the stress vector on a plane with normal unit vector H n ; and 2w is the normalstress, , on this plane. Still in elasticity3, for H=2 and G=6, H is the stress dyadic on a subspace of dimension six4 atthe point O (of the 6-space spanned by six independent H ) with normal unit dyadic H ; and 2W0 represents twice thespecific density energy (the stored strain energy by unit of volume at this point), although the value 2W is more commonlyused.In view of the isomorphism with vector spaces we call H ( H ) the H-adic (specific) projection of 2HG in thedirection H . Similarly, 2W (2w) is the scalar (specific) projection of H ( H ) in the direction H ; it is also called theradial (specific) value of 2HG relative to the direction H .For two different directions H and H we write (in accordance with polyadic algebra) ˆˆˆˆw2 HH2HHHHH2HHH.... GG , (II.1.061),or HHHHHHˆˆw2 .. (II.1.06).Hence the proposition:The 2HG projection H relative to a direction H projected on a second direction H , isequal to the 2HG projection H relative to this direction projected on the first direction H .We call the scalar 2w the tangential (specific) value of 2HG relative to the directions H and H . It is interesting tonote that in the theory of elasticity equality (II.1.06) translates into Bettis law. For H=1 - in which case H is a forcevector, ˆHis a unit displacement vector and the tangential value a work - we state:"the work done by a certain force f1 (or a system of forces f1, f1, ...) in virtue of the application of the force f2 (or asystem of forces f2, f2, ...) is equal to the work that should be produced by this latter, in virtue of the application ofthe first".For H=2 - in which case H is a stress dyadic and ˆHa unit strain dyadic - we state:"for a linear elastic body the work done by a first state of stress in the strain of a second state of stress is equal tothe work done by the second state of stress in the strain of the first state of stress".The problem consist in the study of the simultaneous laws (II.1.021) and (II.1.031) when the independent andcontinuous variable H assumes all possible finite values of a certain defined domain that will be not discussed here fromphysical point of view.II.2 – The 2HG Characteristic Elements or Eigensystems.Orthogonal and unit H-adic bases.There is a well-known theorem:In a G-dimensional H-adic space there exists H-adic orthogonal bases.If }...,,,{ GH2H1H constitute an orthogonal base, then }ˆ...,,ˆ,ˆ{ GH2H1H - the set of the unit dyadics of theformer - also constitute a base whether the metric matrix of this set is the GxG unit matrix, or the principal of this matrix(which is equal to 1) is of degree G. Hence, 1 is the norm of this base. The unit and orthogonal H-adic bases are calledorthonormal bases; for these bases we can writeijjH1Hˆˆ : (i,j=1,2, ..., G), (II.2.01),where the ij are the Kronecker deltas. One notable particular case is that in which the base dyadics (H=2) of a 9-space are3 We shall show further down that the space of stress surrounding the point O is six dimensional.4 In a 6-space, a (non null) dyadic can be orthogonal to at most five other dyadics.
- 10. 10dyads formed with vectors of a orthonormal vector base { }i j k , that is, ijkjkkjjii ˆˆˆ....,,ˆˆˆ,ˆˆˆ,ˆˆˆ,ˆˆˆ 94321 ,where evidently the norm of this base is equal to one.The following theorem is also known:To each pair of different eigenvalues corresponds orthogonal unit eigenH-adics.If all G eigenvalues of 2HG are different we have G distinct mutually orthogonal eigenH-adics that can be assumedto have unit norms: G21 ˆ...,,ˆ,ˆ ; this means,ijjiG21 ˆˆH...HH : (i,j=1,2, ..., G), (II.2.021).The metric matrix associated to this set is ]ˆˆ[ ji : , that is, the GxG unit matrix whose determinant (the norm of the base)is 1. Hence, the set constitute an orthonormal base in the entire space. So we can represent 2HG in the form:iiiH2ˆˆG G (sum for i=1,2, ..., G), (II.2.022).This diagonal representation is preferable because of the properties of the eigenH-adics (Kelvin, 1856; Mehrabadi andCowin, 1994; Helbig, 1994).Let us suppose now that 2HG has a double eigenvalue, say G1G GG . Then (II.2.022) is valid for 1=1,2, ..., G-1,i.e., there exist G-1 mutually orthogonal eigenvectors in a (G-1)-space of the G-space. It can be proved that the crossproduct of this G-1 eigendyadics, 1G21 ˆ...ˆˆ , is still an eigendyadic of the tetradic.In general, if a 2HG has S simple eigenvalues, hence S different eigenH-adics, the cross product of this S eigenH-adics is still a 2HG eigenH-adic; the cross product of these S+1 eigenH-adics is also a 2HG eigenH-adic; and so on until wecan complete the set of G eigenH-adics.II.3 – The stationary proportionality polyadic specific radial value (2w).The extreme of w at the point O is a linked extreme because H might satisfy (II.1.01). If a direction exist at O that makesw an extreme then dw=0 in this direction. By differentiating (II.1.03) we get: 0ˆdˆ2dw2 HH2HHH .. G . From(II.1.01), we deduce H H Hd .0; hence, we conclude that H and H 2H H H G . are orthogonal to d H , that is,orthogonal to the same plane (hyperplane) tangent to the spherical surface H H H .1. This means that the H-adics H and H 2H H H G . must be parallels.By (II.1.031) we write5: 2w=| | cos( , )H H H , whence we deduce that the 2w extreme value is |H| if the H-adicsH and H are parallels (a maximum corresponding to the null angle and a minimum to 180). The parallel condition maybe expressed in the form XH H H H H 2G . , where X and H are a scalar and a H-adic to be determined, which, aswe know, are the 2HG eigenvalues and correspondent eigenH-adics. Hence:The 2HG radial value, 2w, given by (II.1.03), is stationary at the point O of the G- space for directionsH drawn by O and parallels to the 2HG eigenH-adics.The G 2HG eigenvalues Gu are all real (because it is symmetric) and we will suppose they are single and non null;representing its corresponding (real) unit eigenH-adics by H u, we write:2HuHuHuGG (sum for u=1,2, ..., G), (II.3.01),since2H H H1HGG . 1 1 , 2H H H2HGG . 2 2 , ..., (II.3.011),andH H 2H H H 1 20. .G = H H 2H H H 1 3. .G .....= H H 2H H H= ... 2 3. .G , (II.3.012).5 - For multiple dot multiplication of polyadics essentially same concepts hold as for scalar multiplication of vectors.
- 11. 11The principal polyadic and principal directions.The polyadic 2HG, written in the form (II.3.01), is said to be sad represented in its principal form in the point O; itseigenH-adics are its principal directions and constitute the principal (orthonormal) H-adic base in the point O. Referredto this principal H-adic base, the 2HG associated matrix is a (GxG) diagonal matrix, its diagonal elements being the 2HGeigenvalues; hence, (II.1.061) and (II.3.012) permits us to conclude:The 2HG tangential values relative to any two different principal directions at a point are always nil.Substitution of (II.3.01) into (II.1.03) gives:2w=( ) G( ) G ( ) G ...,H H Hu2uH H H121H H H222 .. . (u=1,2, ..., G) (II.3.04),whence we conclude:Each eigenvalue of 2HG is a stationary value of 2w in the point O of the G-space, which occur for thecorresponding 2HG eigenH-adic direction.If we denote by Eu and Su the projections (coordinates) of H and H on the eigenH-adic of base H u, that is, ifwe putH H Hu uE . and H H Hu uS . , (II.3.05),the law (II.1.02) is then equivalent to the systemS G ES G ES G E1 1 12 2 2G G G..., (II.3.06).We conclude:When, in the vicinity of a point, the G-space is referred to the eigenH-adic orthogonal base of thesymmetric polyadic 2HG, the ratio of the H-order magnitudes with the same subscript is equal to thecorresponding 2H-adic eigenvalue.II.4 - The Projection Norm and Octahedral Directions.For an arbitrary direction H in the vicinity of the point O of the G-space we can write, with respect to the principal base}ˆ...,,ˆ,ˆ{ GH2H1H :H H H HuHu ( ) . (u=1,2, ..., G), (II.4.01),being1)ˆˆ(G12uHHH . , (II.4.011),because H H H .1. The numbers H H Hu . are the G principal director cosines of the direction. In general they areall different, but for a particular direction they can be all equal. For a given and ordered set of G squares, whose sum isequal to one, there are 2Gdirections (that is, all the arrangements with repetition of the signs + and – taken G by G with themodulus of the director cosines, GG2 2(AR) ) whose director cosines have the same modulus.We shall call octahedral directions, or octahedral H-adics of 2HG, the unit H-adics equally inclined to itsprincipal directions. Denoting a octahedral direction by Hoct we can write from (II.4.01),HoctHoctH Hu u ( ) . (u=1,2, ..., G), (II.4.012),and from (II.4.011), since the cosines (cos oct) are all equal:
- 12. 12for (u=1,2, ..., G)GGG1cosˆˆ octuHHoctH . , (II.4.013).So, for G=2, 45oct , for G=3, 4454oct , for G=4, 60oct , for G=9, "443170oct etc..For any octahedral direction to which w=woct corresponds we haveG1uu2uHHoctHoctHH2HHoctHoct GG1G)ˆˆ(ˆˆ=2w ... G , (II.4.014),that is:In any octahedral direction, the 2HG radial value 2woct is equal to the invariant average of theeigenvalues.For an arbitrary direction any in the space we can write the norm of the correspondent 2HG projection as ˆˆ=ˆˆ|||| HH22HHHHH2HH2HHHH..... GGG , with 22H2HH2H=GGG . , (II.4.02).The polyadic 2H2 G is, by definition, the double dot power of the polyadic 2HG. Considering the second of the expressions(II.4.02) and (II.3.01) we can also write2HuHuHu=(GG) 2 2 , (u=1,2, ..., G) (II.4.021);whence we deduceG12u2E2H)(GG , (II.4.022).Hence:|| || ( ) ( )H H H Hu u2G .2 , (II.4.03),or, writing in full:2G2GHHH2222HHH2121HHH)G()ˆˆ(...)G()ˆˆ()G()ˆˆ(|||| ... , (II.4.031).The Gu are invariant, that is, they dont depend on the H . Hence, ||H|| varies with the square of the 3Hdirectioncosines ( )H H Hu .2whose sum, in conformity with (II.4.011), is equal to one. Thus, when these are all equal - foroctahedral directions - representing by Hoct the 2HG projection correspondent to any one of the octahedral directions, wehave:2EH2G12uoctHG1)G(G1||||G , (II.4.032),in which case ||H||max is a invariant. We conclude:The norms of the 2HG octahedral projections at a point are equal to the average of the squared 2HGeigenvalue (or equal to the G-th part of the 2HG dot square scalar).Let us calculate now the difference between the 2HG square eigenvalues average and the square of the average 2HGeigenvalues, that is,1 1 12 2 2 2 2G G GGGGHE2 HE uuG G( ) ( ) ( ) .Developing the squares in the second member, grouping pieces conveniently and noting the presence of new perfectsquares we have
- 13. 131 2 2GGGGuu( ( )) 1 2 2 2GG G G G ... G G2 1 2 1 3 1 G[( ) ( ) ( ) ( ) ( ) ( ) ( ) ]G G G G ... G G ... G G2 3 2 4 2 G G G2 2 212.Since we have CG2squares of differences inside the brackets, we write:1 122 2GGGGGGG)Cuu2G2( ( )) , (II.4.04),where ( )G 2 indicates the sum of squares of differences of all eigenvalues pairs. From (II.4.04) we state:The 2HG eigenvalues square average is equal to the square of its average summed to the ( )/G G1 2of the average of its squares difference two by two.The 3Hprincipal 2HG invariants are the coefficients of its characteristic equationX X X ...G 2HE1 G 2HE2 G G G~ ~1 20)det(X+X 2H)~1G(E2H2)~2G(E2H GGG , (II.4.05),where: for i=1,2, ..., G-1, i~E2HG is the sum of the diagonal minors of degree i of det(2HG).Considering the first and the second invariant of 2HG we write the expression (II.4.04) in the form])()[(21E2H22EH22~E2H GGG , (II.4.06).Hence:The sum of the pairwise products of eigenvalues of 2HG is equal to one half the difference between thesquare of their sum and the sum of their squares.II.5 - The Transversal Value of the Proportionality Polyadic.From 2w=| | cos( , )H H H we see that : || || (2w)H H 2 . Hence, there always exists a positive number, say t2,which complement (2w)2to ||H||. We can write: : | | =(2w) tH H 2 2 2 , (II.5.01).This Pythagorean relationship – used to calculate the square of a vector resolved in two perpendicular directions –suggests to name |t| the transversal value of 2HG relative to the direction H .In the theory of elasticity, for H=1 t2is the square of the tangential stress vector on a plane defined by a normalunit vector n on which the stress vector p acts. If is the normal stress vector, then (II.4.07) can be written p2=2+2. ForH=2, t2is an always existing positive number which has to be summed to the square of the specific energy density 2w toobtain the norm of the specific stress dyadic. We write (II.4.07) as t2+(2w)2=||2=||||.Applying (II.4.04) to the case of octahedral direction, considering (II.4.032) and (II.4.014) and simplifying we gettGG)oct2 1 , (II.5.011),and state:The 2HG transversal value toct relative to octahedral directions is equal to the G-th part of thesquare root of the sum of the squared pairwise difference of the eigenvalue.From (II.5.011) we see that the limiting case toct=0 occurs for (G)2=0, and vice-versa. This implies that the 2HGeigenvalues are all equal, to be, 2HG is a scalar polyadic. We write:
- 14. 14t G with G=G G ... and || Goct2H 2H1 2Hoct2 0 G || , (II.5.012).If at least two of 2HG eigenvalues are different, then toct0.Let us search the directions with respect to which t2, the square of the transversal value of 2HG, t2, is a maximum(since its minimum is zero). In accordance with the method of Lagrangian multipliers to find stationary values of amultivariable function we must extreme the functionF t L2 H H H . , (II.5.02),(with the conditional equation H H H .1) as F would be a free extreme and L is a constant.From (II.5.02), considering (II.5.01), we write (recalling polyadic analysis):HHHHHHH2HˆL2ˆwˆ||||=ˆL2ˆtˆF , (II.5.021),where H0 is the null H-adic. Calculating the derivatives and simplifying we write (II.5.021) in compact polyadic notation:Fw) -L =HH 2 2H 2H H H H[ ( ] 2 2 22 G G . , (II.5.022);or, with respect to the 2HG eigenH-adic base:2 2 22[( ) ( ) ]( ) G w G Lu uH H HuHuH . , (II.5.023).The linear combination (II.5.023) implies the nullity of all eigenH-adic (H u ) factors, since these H-adics form abase). Hence[( ) ( ) ]( )G w G Lu uH H Hu22 2 0 . (u=1,2, ..., G), (II.5.03).The expression (II.5.03) represents the following system up to G equations independent for distinct eigenvalues:,0)ˆˆ](LG)w2(2)G[(...0)ˆˆ](LG)w2(2)G[(0)ˆˆ](LG)w2(2)G[(GHHHG2G2HHH2221HHH121...(II.5.04).Theorem 1:Each pair of single eigenvalues of a symmetric 2H-adic is in correspondence with a direction H that maximizes its transversal value. This direction is perpendicular to at least one of the eigenH-adics different from those that correspondent to the single eigenvalues.Let us consider 2HG with two single eigenvalues, say G1 and G2, to which correspond the orthogonal eigenH-adicsH H2and 1. Let us suppose that H - the unit H-adic that makes the 2HG squared transversal value t2stationary – is notorthogonal to any 2HG eigenH-adic. Then we get from system (II.5.04)( ) ( ) ( ) ( ) ( ) ( )G w G G w G ... G w G L1 1 2 2 G G2 2 22 2 2 2 2 2 , (II.5.041).As G1 and G2 are single eigenvalues, G G G G ..., G1 2 3 4 G , , . The first two members of (II.5.041) give G +G w1 2 4 ;the first and the third give G +G w1 3 4 , and so on. But this is a contradiction because we might accept that G2=G3=G4=...= GG . Hence H must be orthogonal to at least one 2HG eigenH-adic. If H was perpendicular to H1 , the system(II.5.041) would be reduced to at most G-1 equations and we could write ( ) ( ) ( ) ( )G w G ... G w G L2 2 G G2 22 2 2 2 ,whence we could deduce G +G w2 3 4 , G +G w2 4 4 , ..., to be G2=G4=...= GG. But this is also a contradiction (G2 is a
- 15. 15single eigenvalue). By the same reason H cannot be perpendicular to H2.Theorem 2:In the 2-space defined by a pair of orthogonal eigenH-adics of a symmetric 2H-adic there exist twoother H-adics, unitary and orthogonal to each other, which bisect the supplementary angles of thetwo first, and make its transversal value |t| a maximum.If in the foregoing theorem, H would be perpendicular to two, three ... up to G-3 of the eigenH-adics (betweenwhich couldnt exist H1nor H2) we still should find contradictions because the corresponding conditions would implythe equality of G1 and G2. But if H should be perpendicular to H H HG.., , , . 3 4- in which case H would belong tothe 2-principal space defined by H 1and H 2and still would make the square of the 2HG transversal value stationary -the system (II.5.041) would be reduced to the following two equations( ) ( ) ( ) ( )G w G G w G L1 1 2 22 22 2 2 2 ;we would deduce G +G w1 2 4 , or, taking into account the correspondent expression (II.3.05) of the 2HG radial value 2w:G +G G G1 2H H H1H H H2 2 1222[( ) ( ) ] . .. Noting that, in the 2-space, H H H H1H1H H H2H2 ( ) ( ) . .with ( ) ( )H H H1H H H2 . .2 21 , we have: G +G G ]G1 2H H H1H H H2 2 11212{( ) [ ( ) } . .; we can simplifythis, remembering that G1–G20: ( ) /H H H1 .21 2 . Hence: 2/2ˆˆˆˆ 2HHH1HHH .. . Related toH H H1 / . 2 2 we get the solution H H(+) under which H (+) makes the angle of 45 with H1; relatedto H H H1 / . 2 2 , the solution H (-) makes the angle of 135 with H1. We arrive to an analogous conclusion withrespect to H2. Hence H (+) and H (-) bisect the supplementary principal directions defined by H1and H2; evidentlythey are perpendicular to each other.Corollary 1:If the symmetric proportionality 2HG-adic has N single eigenvalues (1N G), there exists CN2(combinationsof N taken two by two) H-adics that make the square of its transversal value |t| stationary, each H-adicbelonging to a 2-principal space defined by a pair of the eigenH-adics.For, to each pair of single eigenvalues there exists a H-adic bisector of the supplementary angles defined by thecorresponding eigenH-adics; and if N is the number of single eigenvalues, these exist in number of CN2.Theorem 3:The radial value of the symmetric proportionality 2H-adic, 2w, relative to any bisector direction isequal to one half the sums of the eigenvalues related to the corresponding bisected principaldirections.Indeed, for the bisector H H H ( ) / 1 22 2 in the 2-space (H1, H 2), equation (II.1.03) gives thecorresponding 2HG radial value, 2w12: we have )ˆˆ()ˆˆ(4w 2H1HH2HH2H1H12 .. G . Substituting 2HG for(II.3.01), expanding and considering (II.3.011) and (II.3.012), we get 2/)GG(2w 2112 . In general, then, we write forthe principal directions Hu and Hv :2w G Guv u v 12( ) , (u,v=1,2, ...,G) (II.5.05).Theorem 4:The transversal value |t| of a symmetric 2H-adic relative to each bisector direction at a point is equalto one half the modulus of the difference of the eigenvalues related to the correspondent bisectedprincipal directions.We can calculate t2for the particular case of the (orthogonal) bisector directions considered in the demonstration ofthe Theorem 3, that is, H H H ( ) / 1 22 2 . Noting that, from (II.4.03),
- 16. 16|| || ( ) ( ) [ ( ) ] ( )H H H Hu uH H Hu uG G . .2 21 22 212,we have, developing the sum in u: 2/])G()G[(|||| 2221H . Hence, using (II.4.07) and considering (II.4.11) we write:4/)GG(4/)GG(2/])G()G[(t 22122122212 , that is, 2/|GG||t| 21 . We should obtain the same resultfor the bisector direction H H H ( ) / 1 22 2 .In general, then, we write for the principal directions Hu and Hv :|t ||G Guvu v |2, (u,v=1,2, ..., G) (II.5.06).Note:By (II.4.12) we can calculate the highest |t| in the 6-space, which is (G6-G1)/2. But if the eigenvalues are allpositive or all negative, this value is not the |t|max because this maximum is |G6|/2 and occurs in the 9-space (thezero eigenvalue must now be considered).II.6 - Cauchys and Lamés quadric.For det2HG0 and (II.1.03), considering (II.1.01), we can deduce, in the G-space:H H H H H . .G2 21 , (II.6.01),and1|2w|2w/Q HH2HHH .. G , (II.6.02),where H is a H-adic parallel to the unitary H-adic H and a modulus that is the inverse of |2w| square root, i.e.,|2w|/ˆHH , (II.6.03)If H varies with fixed origin O, assuming all positions about the point, that is, when its end point describes the (hyper)surface of the unit (hyper) sphere centered at O, the end points P and Y of the H-adics H and H describe the (hyper)surfaces (II.6.01) and (II.6.02), respectively. The distances PO and YO are the modules of H and H , respectively. Theseare quadric surfaces centered at the point O. The first – representing the variation of – is the Lamés ellipsoid. Thesecond – representing the variations of 2w and called Cauchys quadric or indicator quadric – is either an ellipsoid or ahyperboloid (of one or two sheets) depending on the coordinates of the 2HG-adic at the point. Correspondingly, 2HG iscalled elliptic and hyperbolic.From the geometrical shape of the quadrics (II.6.01) and (II.6.02) relative to the point O, |H| and 2w related to Hcan be easy determined. To calculate 2w it is enough to determine the point Y where H intercepts the indicator Q, sinceaccording to (II.6.03), |2w|=1/(OY)2. To calculate |H| it is enough to fix its direction in space and to write |H|=|OP.|.Denoting by Q+ and Q- the indicator (II.6.02) corresponding to algebraic signals (+) and (–), respectively, we canconclude:1) If Q+ is a real ellipsoid, Q- wills no real graphical representation, because it is a imaginary ellipsoid. In this case2w>0 for any H and all eigenvalues of 2HG are positive. The angle between H and H is always acute;2) If Q- is a real ellipsoid, Q+ is imaginary and 2w<0; the angle between H and H is always obtuse;3 If Q+ is a hyperboloid, Q- is its conjugate hyperboloid; both are separated in the space by the common asymptoticcone whose equation is C =0H H 2H H H . .G .The point Y, intersection of H with Q, could be on Q+, on Q-, or even might not exist (if H should be parallel toany generator of the cone). In the first case, the angle between H and H is always acute and 2w>0; in the second case theangle is obtuse with 2w<0; and in the third case the angle is 90 with 2w=0.The hyper curves of intersection of the hyper cone C and the hyper sphere define over the hyper sphere the regionswhere 2w>0 and 2w<0. The orthogonal projections of these hyper curves on the coordinate planes are ellipses orhyperbolic arches.
- 17. 17Reduced equations of quadrics.The quadrics associated to the 2H-adic 2HG at the point O could be represented in a simpler – reduced – form if the spaceis referred to the (principal) base (at the point) defined by the 3Horthonormal eigenH-adics, Hu of 2HG. In this case,(II.6.01) and (II.6.02) are written in the respective forms 1)G/ˆ( 2uuHHH . and ( )H H Hu uG .21 , where theleft-hand-sides are sums on u. If Su and Yu are the coordinates of H and H in the H-adic principal base we can write:1)G/S( 2uu and 1)Y|G|( 2uu , (II.6.04),where the sign of each term in the left-hand-side of the second equation is the sign of Gu. In reduced form, the indicatorquadric is easy to classify.II.7 - Graphical Representations.The simultaneous laws under study, (II.1.01), (II.1.02) and (II.1.03), are transformed into a system of scalar equationsˆˆ1ˆˆT)W2(ˆˆ=2WHHHHH22HHH22HH2HHH.....GG, (II.7.01),enclosing the 2HG radial value 2w and the 2HG transversal value |t| corresponding to the variable H-adic H . With respectto the orthonormal base of the 2HG eigenH-adics, we can - in view of (II.3.04) and (II.4.021) - transform the right-hand-sideof these equations and put H H Hu uE . for all u=1,2, ..., G to obtain2G22212G2G2222212122G2G222121)(E...)(E)(E1)(H)(E...)(G)(E)(G)(Et)w2(G)(E...G)(EG)(E=2w, (II.7.02).We discuss this system of three equations in the G+2 variables: 2w, |t| and the square of the G coordinates of H(the Eu2), when H varies, i.e., when the Eu vary. Without any loss of generality, we can suppose G G G ... G1 2 3 G .II.7.1 - H varies in a 3-space.Let us imagine initially that the H-adic H varies in the 3-space defined by H H2H3and , 1, in which case E4=E5= ...= EG=0 and the system (II.7.02) is reduced to2w=(E G (E G (E Gw t (E (G (E (G (E (G(E (E (E1 1 2 2 3 321 1 2 2 3 31 2 3) ) )( ) ) ) ) ) ) )) ) )2 2 22 2 2 2 2 2 22 2 221 (II.7.03).The results deduced ahead are well known in the theory of elasticity for H=1 (Jaeger, 1969; Ruggeri, 1984).In a coordinate plane 2wx|t| the current point (2w,|t|) traces a certain portion of surface as H varies (because thispoint varies with two independent parameters). The analytical calculation of this area can be done, of course, by the system(II.7.02) which is linear in (E1)2, (E2)2and (E3)2.Solving (II.7.03) we get, remembering that G1G2 G3:)G-(G)G-(Gt)G-w2)(G-w2()E()G-(G)G-(Gt)G-w2)(G-w2()E()G-(G)G-(Gt)G-w2)(G-w2()E(231322123123221322121323221(II.7.04),whence
- 18. 18( )( )( )( )( )( )2 2 02 2 02 2 0222w-G w-G tw-G w-G tw-G w-G t2 33 11 2 (II.7.05),since the left-hand-sides in (II.7.04) must be all positive. The first equation in (II.7.05) can be written also in the form( )( ) ( ) ( )2 22 22 2 2w-G w-G tG -G G -G2 33 2 3 2 ,or, transforming the left-hand-side:t wG +G G -G3 2 3 22 2 222 2 ( ) ( ) , (II.7.06).The inequality (II.7.06) represents points not interior to the semicircle centered in C G G23 2 32 0 (( ) / , ) withradius R G G223 32 ( ) / . Similar interpretation hold for the other two inequalities in (II.7.05), the second representingpoints not exterior to the semicircle centered at C G G13 1 32 0 (( ) / , ) with radius R G G13 3 12 ( ) / and the third,points not interior to the semicircle centered at C G G12 2 12 0 (( ) / , ) with radius R G G12 2 12 ( ) / . As the pairs(2w,|t|) must satisfy system (II.7.05) its images in the plane 2wxt are points of the dashed area shown in Figure 1 (draw forthe particular case of all G>0).This graphical representation on the variations of the radial and transversal values of 2HG is called Mohrsrepresentation because of its analog in the theory of elasticity; the bounding circles, Mohrs circles and the plane 2wx|t|,the Mohr plane.We show now how to determine in the Mohr plane the point N corresponding to a given direction in a 3-space with-out calculating 2w and |t|. Any direction H can be defined by the angles 1 and 3 it defines with H 3. Let us look for thelocus of points equally inclined against H 3; in other words, we ask: if N describes the parallel of co-latitude 3 of thespherical surface H H H .1, what curve does N describe in the Mohr plane?The equation of this curve is obtained by elimination of E1 and E2 in the system (II.7.03); we have:t wG G G Gcos G G G G2 2 ( ) ( ) ( )( )22 21 2 2 2 1 23 3 1 3 2 , (II.7.07).This is the equation of a circle centered at C12 with radius equal to the square root of the right-hand-side. Let usdraw in the Mohr plane, as shown in Figure .2, the line r3 from (G3 ,0) that makes an angle 3 with the |t| axis. This lineintersects the circles (C13,R13) and (C23,R23) at A2 e A1 , respectively. Since G2A1 e G1A2 are both perpendicular to r3, theyare both perpendicular to the bisector of A1A2 which passes through C12. Hence we deduce (with the help of Figure 2,which has been drawn for G1<0):C A C G GG G12 12212 32 23 3 cos ( ) cos , 1 2 2 232( ) ( ) sen ( ) ( cos ),A A G G G G1 2 2 1 2 2 232 1 2 232 2 21 C A C AA A G GG G G G12 1212 122 1 2 2 2 1 21 3 2 3232 2 ( ) ( ) ( )( ) cos , (II.7.08).
- 19. 19Hence the radius of the circle (II.7.06) is C A12 1 as we can conclude by comparing the left-hand-side of (II.7.07) and(II.7.08).Let us see now between what limits the radius C A12 1 can vary:for 3 0 , C AG GG G G G GG GC G12 12 1 21 3 2 3 31 212 32 2 ( ) ( )( ) ;for 3 2 / , C AG GGG GC G12 12 121 212 22 2 .These results indicate the possibility to grade the circle (C23,R23) in 3. This allows to locate easily the arc of circle(II.7.08), as we show in Figure 3.We proceed in a similar manner with respect to the inclination 1 of H against H1, but now we must have 1 32 / in order to satisfy the third equation in (II.7.03). The locus of the points in the Mohr plane that correspond to2w for directions with the same inclination against H1is the circle2223121322232322BC)GG)(GG(cos]2/)GG[()2/)GG(w2[t , (II.7.09),We grade the semicircle (C12,R12) in 1 in the same way as before. Now it is possible to locate immediately thepoint N whose coordinates are 2w and |t| (Figure 3).It is evident that the points of the semicircle centered at C13 with radius R13 are related to directions with 2 2 / ,i.e., with direction perpendicular to the plane defined by directions H1and H 3. These are precisely the directions withrespect to which 2w assumes extreme values. It is evident, also, that the extreme value of |t| is R13, i.e.,2/)GG(R|t| 1313max , (II.7.10).There is also a third circle passing through the point N (not shown in the figures) with center at C13 with theequation)GG)(GG(cos]2/)GG[(]2/)GG(w2[t 2123222132132 , (II.7.101).The square of its radius is the right-hand-side of (II.7.101). As in the previous cases this radius can be detected graphically.
- 20. 20Note:In the theory of elasticity with H=1, the radial value 2w of the stress dyadic represents normal stress and itstangential value |t| tangential stress; with H=2, i.e., for Greens tetradic, the radial value 2W for a certainunit strain dyadic , represents stored energy. The tangential value |t| is a new variable which we mightcall complementary energy, since 2t|2w|max . Hence, in connection with Greens tetradic we can generalizethe concept of Mohrs circle to the diagram energy x complementary energy. The minimum energy is zero.Since the eigenvalues of a stable proportionality polyadic are all positive, the origin belongs to the validarea, i.e., the Mohr circle corresponding to the null eigenvalues must be included.II.7.2 - H varies in a 4-space, 5-space ....Let us imagine now that the H-adic H varies in the 4-space defined by H H2H3H4, and , 1, in which caseE5= ... = EG=0 and the system (II.7.02) is reduced to2w=(E G (E G (E G (E Gw t (E (G (E (G (E (G (E (G(E (E (E (E1 1 2 2 3 3 4 421 1 2 2 3 3 4 41 2 3 4) ) ) )( ) ) ) ) ) ) ) ) )) ) ) )2 2 2 22 2 2 2 2 2 2 2 22 2 2 221 (II.7.11).Substituting (E4)2from the last equation into the first two and rearrange we obtain an equivalent system in the unknownE12, E22and E32:,)(E)(E)(E)(E1])(G)[(G)(E])(G)(G[)(E])(G)[(G)(Et)G()w2()G-(G)(E)G-(G)(E)G-(G)(E=G-2w2322212424232324222224212122424323422241214The determinant of this system is ( )( )( )G G G G G G2 1 3 1 3 2 0; the solution as a function of E42is:])(E-1)[G-G)(G-G(t)]G-G+(G-w2)[G-w2()E)(G-(G)G-(G])(E-1)[G-G)(G-G(t)]G-G+(G-w2)[G-w2()E)(G-(G)G-(G])(E-1)[G-G)(G-G(t)]G-G+(G-w2)[G-w2()E)(G-(G)G-(G242414242142323132434142431422122324342424324211213,i.e.,( )[ )] ( )( )[ ) ]( )[ )] ( )( )[ ) ]( )[ )] ( )( )[ ) ]2 2 02 2 02 2 02 22 22 2w-G w-(G +G -G t G -G G -G 1-(Ew-G w-(G +G -G t G -G G -G 1-(Ew-G w-(G +G -G t G -G G -G 1-(E4 2 3 4 4 2 4 3 44 1 3 4 4 1 4 3 44 1 2 4 4 1 4 2 4 (II.7.12),If we add [( ) / ]G G G2 3 422 to both sides of the first inequality in (II.7.12) we find after some manipulation:t wG G G GG G G G E2 2 3 2 3 2 24 2 4 3 4222 2 ( ) ( ) ( )( )( ) .As 0(E4)21 we can write( ) ( ) ( )G Gt wG GGG G3 2 2 2 2 3 242 3 2222 2 , (II.7.121).From the second and the third inequalities we obtain also two other inequalities; a second, viz.:( ) ( ) ( )G Gt wG GGG G2 1 2 2 1 2 241 2 2222 2 , (II.7.122),and
- 21. 21t wG GGG G2 1 3 241 3 222 2 ( ) ( ) , (II.7.123).The meaning of these inequalities is obvious: (II.7.121), for example, represent the set of points not interior to thecircle centered at ( (( ) / , )G G2 32 0 ) with radius ( ) /G G3 22 and not exterior to the circle with the same center andradius G G G4 2 32 ( ) / .Hence, in this 4-space the set of points in the 2wx|t| plane that satisfies the system (II.7.12) contains all points notexterior to the circle centered at ( (( ) / , )G G2 32 0 ) with radius G G G4 2 32 ( ) / and not interior to the following twocircles: a first centered at (( ) / , )G G1 22 0 with radius ( ) /G G2 12 and a second centered at (( ) / , )G G2 32 0 withradius ( ) /G G3 22 . As the 3-spaces defined by ( H H H , , 1 2 3), ( H H H , , 2 3 4) etc. are subspaces of theconsidered 4-space, the corresponding areas could be valid areas; hence we must exclude the points not interior to thesemicircle centered at (( ) / , )G G3 42 0 with radius ( ) /G G4 32 .If we had taken (E1)2from the last equation (II.7.11) in place of (E4)2we would obtain inequalities similar to(II.7.121), (II.7.122) and (II.7.123) with the only difference that G1 and G4 changed places.After checking all possibilities we can conclude that the "valid" points are not interior to the semicircles (C12,R12),(C13,R13), (C34,R24) and not exterior to the semicircle (C14,R14).It is easy to extend these conclusions to 5-space, 6-space, ..., G-space.We show now how to determine in the Mohr plane of a 4-space the point N corresponding to a given direction withdirection cosines cos1, cos2 etc. It is convenient to observe that the sum of two arbitrary direction angles that satisfy thethird equation (II.7.11) can not be less than 90 . Indeed, for any two the sum of the squares of the respective cosines mustbe less than one, say cos cos2 2 1 21 , i.e.,cos sen cos2 2 2 1 2 290 ( ), or 1 290 .Let us consider initially the directions inclined of the given angle iagainst the reference H-adic Hi and magainst Hm . When N describes the spherical surface H H H .1 (with fixed iand m) what curve does thecorresponding N describe in the Mohr plane ?.As in the previous case, the equation of this curve can be obtained by eliminating the cosines Ej and Ek in thesystem (II.7.11). In this way we findt (2wG G2) R2 j k 2jk2 (II.7.13),withR (G G2) (G G )(G G )cos (G G )(G G )cosjk2 k j 2k i j i2i k m j m2m , (II.7.131).Suppose now that four angles satisfy the third equation of the system (II.7.11). For elimination we can select sixpairs of angles. This is the same as saying that from (II.7.13), with j and k running from 1 to 4 (with jk), we can obtainthe equations of six circles, each centered at the "Mohrs centers" with radius R*jk. But this six circles will have necessarilya common point, precisely the point of the Mohr plane to which the direction specified by the given angles corresponds.II.7.3 - Some Properties of Mohrs Circles.Some particular properties of these circles can be listed. For example: the direction perpendicular to the base H-adicH4 ( 490 ) belongs to the 3-space of base H H H 1 2 3, , . With the choice j=1, k=2 and i=3 in (II.7.13) and (II.7.131)we obtain the equation (II.7.07); with the choice j=2. K=3 and i=1 we obtain equation (II.7.09). In this case thegeometrical meaning are the same as in section II.7.1.For i m 90the considered direction belongs to the 2-space of base HjHk , . Hence the correspondentMohr circle is (Cjk,Rjk). This is evident from (II.7.13) and (II.7.131).The normal to the 2w-axes drawn by G2 and G3 cut the semicircles (C13,R13) and (C24,R24) at fixed points F4 and F1,respectively, equidistant from C14. Indeed, we can write:
- 22. 22C F C G14 4 14 22 22 42 G F , and C F C G14 1 14 32 22 12 G F (II.7.14).ButC GG G14 22 1 4 22 ( )G2, and C GG G14 32 1 4 22 ( )G3(II.7.15).Then, since the perpendicular from a point of a circle over a diameter is the geometric average between the segmentsdetermined by its foot,G F G G G G G G G G2 421 2 2 3 2 1 3 2 ( )( ) , (II.7.16),G F G G G G G G G G3 123 4 2 3 4 3 3 2 ( )( ) , (II.7.161).Substituting (II.7.15), (II.7.16) and (II.7.161) into the equalities (II.7.14) we conclude:C F C FG GG G G G G G14 4214 12 1 4 23 2 1 3 2 42 ( ) , (II.7.17).For octahedral directions we have: 4/1coscoscoscos 42322212 . Denoting the radii of the correspondentcircles by Rjkoct we write from (II.7.131):4 422 2RG GG G G G G G G Gjkoctk jk i j i m k m j ( ) ( )( ) ( )( ) .Developing the left-hand-side, adding pieces conveniently and representing by R2 the sum of the squares of the radii ofthe Mohr circles, i.e.,,)2GG()2GG()2GG()2GG()2GG()2GG(R 2122132232342242142 (II.7.18),we obtain,42 22 2 2 2R RG G G Gjkoctk j m i ( ) ( ) , (II.7.19).Summing up the six expressions obtained from (II.7.19) giving to the subscripts 1, 2, 3 and 4 we deduce thatR R R R R R Roct oct oct oct oct oct12 34213 24 14 23122 2 2 2 2 2 , (II.7.20).From the Mohr plane representation it is evident that if all the 4G eigenvalues are positive, then all of its radialvalues are positive; to each point of the valid area correspond one and only one point of the corresponding indicatorellipsoid. If there are some negative eigenvalues – in which case the indicator is a hyperboloid - then to the segment of the|t| axis inside the valid area correspond the (non-planar) hyper curves defined by the intersection of the cone with thesphere. We could say also that to the points belonging to that segment correspond directions parallels to the conesgenerators. Yet in this case, to each point of the valid area correspond a point in the indicator hyperboloid.II.8 - A Criterion of Proportionality.As we have emphasized (see II.1) the 2H-adic characterizes the medium with respect to the phenomenon governed by thelaw of proportionality. In the previous paragraphs we have shown that to each pair (H, H) corresponds a point P in thevalid area of the Mohrs plane with coordinates (2w,t), see Figure 4.
- 23. 23A necessary and sufficient condition for the point P to be not exterior to the valid area – that is, a condition for theexistence of the proportionality established by the law (II.1.021) - is that its distance d to center of the greatest radius circlebe less than this radius, i.e., less than the greatest t, the invariant tmax which is a characteristic of the medium.Hence, if the eigenvalues are all positive, tmax=|Gmax|/2, if they are all negative, tmax=|Gmin|/2 and if Gmin<0 andGmax>0, tmax=|Gmax-Gmin|/2. In any case, we see that the double of tmax is the modulus of the difference between the greatestand the smalest of the 2HG eigenvalues (zero included). On the other hand, we can write2max22max2)t(t)t|w2(|d .Hence,|w2|t2||GG||||max2HminmaxH, (II.8.01).The point P must be exterior to any circle interior to the greatest circle. Since 2w certainly lies in the interval between theeigenvalues Gi and Gi+1 (diameter points of a certain circle) the inequality (2w-Gi)(Gi+1-2w) t2must also be satisfied.Thus,|w2|GGGG||||i1ii1iH, (II.8.02).Hence, for any i,i1ii1iHmaxHGGGG||||w2t2||||, (II.8.03).From these inequalities we deduce a "criterion of proportionality" between those magnitudes.In the theory of elasticity for example, for H=2, this criterion is stated as:The necessary and sufficient condition that in the current point of a loaded elastic solid the stress dyadic beproportional to the correspondent strain dyadic through the Greens tetradic of this solid is that, for any straindyadic:1) - the quotient of the stress dyadic norm to the double of the modulus of the difference between the extremetetradic eigenvalues do not surpass the double the respective specific density energy stored in the vicinity of thatpoint;2) - the quotient of the sum of the stress dyadic norm with the product of two any consecutive eigenvalues to themodulus of the sum of these eigenvalues is not less than twice the respective specific energy density stored in thevicinity of that point.We know (Mehrabadi and Cowin, 1990) that for isotropic materials, 0, 2 and 3+2 are, respectively, 3-tuple,single and quintuple eigenvalues of the proportionality tetradic (in this particular case called Hookes tetradic or Greensisotropic tetradic). Hence, from (II.8.01), it follows that )22w(3|||| ; and (II.8.02) generates 22w|||| and)42w(322(3|||| . These inequalities must be physically interpreted.Analogous inequalities may be derived for materials with other symmetries.
- 24. 24REFERENCES1 Drew, T. B., Handbook of vector and Polyadic Analysis, Reinhold Pub. Corp., 1961.2 Gibbs, J. W. and Wilson, E. B., Vector Analysis, Yale University Press, New Haven, 1901.3 Helbig, K., Foundations of Anisotropy for Exploration Seismic, Handbook of Geophysical Exploration, vol. 22,Pergamon, 1994.4 Jaeger, J. C., Elasticity, Fracture and Flow, Methuen, 1969.5 Kelvin, Lord (William Thomson), Elements of Mathematical Theory of Elasticity, part 1, On stress and strains,Philosophical Transactions of the Royal Society, 166, p. 481-498, 1856.6 Mehrabadi M. M. and Cowin, S. C., Eigentensors of Linear Anisotropic Materials, Quarterly Jounal of Mechanicsand Applied Mathematics.,vol. 43, Pt. 1, 1990.7 Ruggeri, E. R. F., Introdução à Teoria do Campo, Imprensa da UFOP (Universidade Federal de Ouro Preto), 1984.8 Ruggeri, E. R. F., Fundamentals of Polyadic Calculus, under way, registered in the Biblioteca Nacional do Rio deJaneiro under number 173298, book 291, sheet 445 on may/1999; to be printed.I.1 Physical Magnitudes, Polyadics and Euclidean Space. 2I.2 - Physical laws, Linear Transformations and Polyadic Geometry. 4I.3 - Linear Transformations, Experimental Measurements and Statistical Polyadics. 6I.4 - The Physical Phenomenon is Equivalent to a System of Linear Polyadic Equations. 6I.5 - Eigenvalues and Eigenpolyadics. 7II.1 - A particular situation, largely useful in Physics. 7II.2 – The 2HG Characteristic Elements or Eigensystems. 9II.3 – The stationary proportionality polyadic specific radial value (2w). 10II.4 - The Projection Norm and Octahedral Directions. 11II.5 - The Transversal Value of the Proportionality Polyadic. 13II.6 - Cauchys and Lamés quadric. 16II.7 - Graphical Representations.17II.7.2 - H varies in a 4-space, 5-space .... 20II.8 - A Criterion of Proportionality. 22