Polyadic Formulation by Elysio Ruggeri

POLYADIC FORMULATION OF LINEAR PHYSICAL LAWS. by

Elysio R. F. Ruggeri Ouro Preto, MG, Brazil Klaus Helbig Hannover, Ge

ABSTRACT  (of valence R), H  (of valence H), etc. We say that a (dependent) quantity  is proportional to a second (independent) quantity H  when each component (with reference to an arbitrary vector base) of R  is proportional, with different (perhaps constant) weights, to all coordinates of H  . The proportionality between two physical quantities expresses a linear physical law. In polyadic calculus this proportionality is formulated as a "multiple dot multiplication" written in the form R   R  H G H. H  ; the valence of the polyadic R+HG is the sum of the valences of the other two polyadics. The coordinates of R+HG define the weights with which the coordinates of the independent polyadic enter in the constitution of each coordinate of the dependent polyadic. For R=H the proportionality exists between two fields of the same valence. We concentrate on this type of proportionality, expressed by H   2H G H. H  , and postulate a second relation between the dependent and independent

Physical fields are represented by tensors or polyadics of different valence (rank, order):

polyadics, the scalar 2W:



H H  .

 2W  H 

H 2H G H. H  .

. The existence of this scalar implies the symmetry of  H

i.e., the equality of the proportionality polyadic with its transpose: G  G . Many physical laws are expressed as symmetric relationships of this type with H=1 or H=2. Symmetric polyadic relationships can be expressed with reference to an arbitrary external vector base, but often the expression with respect to the orthonormal polyadic base is more convenient. In particular, the orthonormal polyadic base defined by the eigenH-adics of 2H G 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, the particular concepts of normal and tangential stress are extended to "radial" and "tangential" values of 2H G . Stationary radial and tangential values at a point of a field allow the generalization of classical theorems known in the theory of stresses, as Cauchy's and Lamè's quadrics and the representation in Mohr's plane. From Mohr's circle one can derive a general 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-dimensional space that is closely linked to the core of the problem. This space allows us to use intuitively some concepts of ninedimensional 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 H and to establish the N-dimensional analytic geometry associated with the physical laws. They follow immediately if one regards the linear law as linear transformations (a mapping) of the "space defined by one polyadic" into the space defined by another polyadic through a "polyadic operator" (the proportionality polyadic). Some aspects of the geometry hidden in these 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 mathematically by a unified method. This method is algebraic as well as geometrical. It is based on a synthesis of Polyadic Calculus and multidimensional Euclidean (analytic) geometry. 2H

SECTION 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 are polyadic 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.) are well known from elementary mechanics and electromagnetics. Some dyadics are also known as stress and strain in the theory of elasticity and in fluid mechanics; others, like dielectric permittivity, dielectric impermeability, thermal diffusivity etc. 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-optical and the electro-optical tetradic; and in theoretical geometry, the Riemann-Cristophell curvature tetradic. Dyadics. With two given ordered sets of vectors, say {e1 , e 2 , e 3 } and {a1 , a 2 , a 3 } , between which we can establish a bi-unique correspondence (the ei is the correspondent of ai), we can generate dyadics and represent then by the symbolic sum   e i a i , for i=1,2,3, the repeated indexes in different levels denoting sum in the range (Gibbs, 1901; Drew, 1961). The ei are said to be the antecedent and the ai the consequent of the dyadic. If we insert between the antecedent and the consequent 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 this correspondence from the theory of elasticity is Cauchy's tetrahedron of tension: to each unit vector eˆ i normal to a face "i" corresponds one and only one stress-vector s i . This correspondence generates the stress dyadic   eˆ i s i (for i=1,2,3). The stress dyadic is symmetric, that is, it is equal to its transpose (obtained by interchanging antecedent and correspondent consequent) denoted by T. We have:   s i eˆ i   T , in which case V=o (o is the null vector). The converse is true, that is, the necessary and sufficient condition that a dyadic be symmetric is that its vector vanishes. A dyadic, say A, can also be 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 generate an electric field when deformed) there is a bi-unique correspondence between each electric vector field e i in a point and the strain dyadic  i in this point (or vice-versa). This generates the piezoelectric triadic 3    i e i . In this form the dyadics are the triadic antecedent and the vectors the consequent. If each one of the dyadics  i could be related to other sets of vectors, say {r1 , r2 , r3 } and {a1i , a 2i , a 3i } we could write by substitution:

  (a ji rj )e i for i,j=1,2,3 where the

parentheses 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 3H dimensions). 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 for vectors. Another important operation must be defined: the multiple dot multiplication with polyadics (we will abbreviate mdm) which is based on the dot multiplication of vectors. For example: the dot product of the dyadic   b k g k by the vector v is the vector defined by the law  .v  b k (g k .v) . The double dot product of the triadic 3    i e i by the dyadic  is the vector defined by the law 3  :   ( i .bk )(e i .g k ) . In view of the first definition this later expression can be written as 3  :   [a ji (r j .b k )]( e i .g k ) . Proceeding in this manner we can define the multiple dot product (abbreviated mdp) as far as the number of dot does not exceed the valence of the polyadic factor of smaller valence. We say that two polyadics are equal if their mdp by a same and any polyadic are equal. After these definitions and the demonstration of some theorems it is possible to write 3   a ji r je i , 3  :   a ji (r j .b k )(e i .g k ) and similar expressions.

The 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 H  by itself; this product, indicated in the form H  H. H  , is a scalar, always positive and called the norm of that H-adic. The positive square root of the norm of the polyadic H  is its modulus and denoted by | H  |. Hence, H-adics - as vectors - can be written in the form H  | H | H ˆ where H ˆ is a unit H-adic parallel to H  . It may be shown that the square of the H-dot product of two H-adics H  and H  is less than the product of its norms; hence, there is an angle defined by two H-adics such that its cosine equals the H dot product between then divided by the product of their moduli; and we write: H  H. H  | H  | |H  | cos( H , H ) . 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 the polyadic. 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 is orthogonal to at most two other vectors. What about dyadics, triadics and polyadics?. To answer this question we must first look for the maximal number of linear independent H-adics of the generated space, that is, its dimension. We can conclude that this number is up to 3H (3 for vectors, 9 for dyadics etc.) and say that any set of 3 H independent H-adics of a H-adic space is a base of this space. Given the G3H H-adics of a G-space (a subspace of the H-adic space), H 1 , H  2 , ..., H  G , they will be linearly independent if the determinant (or order G) does not vanish: H H 1 . H H H  2 . 1 H

| H   |

1

G

Now, given a H-adic base any of a G-space,

1 , H  2 , ..., H  G such that

i

H H j  .

1

H H 1 .

G

H H G . H H H 2 . G H

... ...

... H

H H 2 . H H H 2 . 2 H

... ...

H H 2 .

1

... H

G

(I.1.01).

H H G .

1 , H  2 , ..., H  G , we can determine its reciprocal, that is, the base

  i j 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 H  and H  of a 3-subspace is a third H-adic, say H  , whose direction is normal to the directions of the factors (hence this H-adic belongs to the 3-subspace), its unit H ˆ pointing to the side on which a rotation less than 180 from the first to the second appears positive, and whose magnitude equal to the product of their lengths ˆ . If multiplied by the sine of the angle between then. We write: H   H  H  | H  ||H  | sin( H , H ) H 

{H 1 , H  2 , H  3 } is a base of the 3-space we can write also the pseudo-determinant: 



H 1 H

    |   | H







H 2

H H 1  . H H 1  .





H 3

H H 2  . H H 2  .





H H 3  . H H 3  .

(I.1.02),

a formula well known for vectors (H=1, H 1  e1 etc.). To extend the definition we can use this determinant as reference and 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 HH adics  , H  , ..., H  of a G-space by the expression:

( H  H  ... H  H  )  H  H  ...



H H  .







H H 1  . H H 1  .





... H



H H 1  .

H H 2  . H H 2  .

...





H H G  . H H G  .

... H



H H 2  .

... ...

If we substitute in this expression H  for H 1 , H  for H  2 etc., we can say also that the set base if their multiple mixed product does not vanish (as for vectors).



(I.1.03).

H H G  .

 ,

H 1

 , ...,

H 2



H G

form a

The polyadic associated matrix. The bases in a H-adic space can be formed with P-adics as long as PH, since we could not express a P-adic in a H-adic base. If, say, e1, e2, e3 and e1, e2, e3 are reciprocal vector bases then we can generate the two following two groups of nine dyads: e1e1, e1e2, ... and e1e1 etc. to compose dyadic reciprocal bases; or the two group of 27 triads: e1e1e1, e1e1e2, ... and e1e1e1, 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) 3H3H-1 matrix if H is odd (a matrix whose elements are the H-adic coordinates with respect to that base); and a square 3H3H matrix if H is even. In the latter case the trace is the H-adic scalar. This may lead to huge computational calculations since the number of rows and column of these matrices can be large. The abstract image of a base in a space is a "star of arrows", and a "pencil of arrows" in subspaces. If we imagine the arrow of a polyadic with its initial point coincident with the vertex of the star of the base arrows, the coordinates of its end point are the coordinates of the polyadic, i.e., the elements 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 and non 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, H   H ( P ) , is the generalized Taylor-series: H



0 

HP



P P  

1 2

H2P



2P P P    ..., 

(I.2.01),

where H0, 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 are only the first two terms of the series. This means - for polyadics referred to a common base - that each one coordinate H is a linear function of (or proportional to) all coordinates P. For most physical laws the linear description is sufficient. The foregoing considerations means that if the end point of the arrow representing P describes a line, a plane or a sphere in the space (of dimension 3P), then the H ending point arrow describes a line, a plane or a sphere in its space (of dimension up to 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, H0, H+P and P) which can also be written in matrix form since we define a new operation between matrices (which differs from the classic operation), called double scalar product, to translate the mdp of two polyadics. Let us consider two N N arbitrary matrices, [A] M and [B] M of the same order (with M rows and N columns), being A ij and B ij their corresponding elements. We define as the double dot product A:B of these matrices (in arbitrary order) as the number

A ij Bij  A11B11  A12 B12  ... A MN B MN ,

(I.2.02).

Notice that the polyadic associate matrices in the linear laws are multi-ordinal, that is, the numbers of rows and columns of MQ one are multiples of the correspondent ones in the other: for instance, [B]Q P and [ A ] LP (with L and M integers). The second matrix may be resolved in LM blocks with P rows and Q columns, that is, this matrix has L rows and M columns whose elements are matrices Aij with P rows and Q columns. We define the double dot product [ A ] MQ : [B] QP of the multiLP ordinally linked matrices [ A ] MQ and [B]Q P , in this order, as the matrix with L rows and M columns whose elements are LP the double dot product of each [ A ] MQ sub-matrix [ A ij ] QP (with i = 1, 2, ..., L e j = 1, 2, ..., M) with the matrix [B] QP . Thus, LP

 A 11 Q A 12 Q P P  Q A  A Q 22 P  21 P  ... ...   Q Q  A L1 P A L2 P

 A11Q : BQP A12 Q : BQP ... A 1M QP  P P   Q Q Q A  : B A Q : BQ ... A 2M P 22 P P P  : BQ   21 P P  ... ... ... ...     Q Q Q Q Q ... A LM P  A L1 P : BP A L2 P : BP

... A1M QP : BQP   Q Q ... A 2M P : BP  ,  ... ...  Q Q ... A LM P : BP  L M

(I.2.03).

The correspondent operation - the double dot multiplication of multi-ordinally linked matrices - always exists. It is commutative, distributive with respect to addition, and generally non associative.

Example 1:

1 4 3 5

2 1 2 1

3= 3x1

3 1 0 2  1x2  4x1 2x2 + (-1)x1 3x2 + 0x1   6 3 6 . : 1 1 2 3x2 + 5x1 2x2 + (-1)x1 1x2 + 0x1 11 3 2 0 4 = 2x2

Example 2: Let us consider the product   3  .r  (A ijk e i e je k ) . (R m e m ) ; we have:   A ijk R k e i e j and the following matrix representations

 

3   3

9 = 3x3

 A 111 A 112 A 113 A 121 A122 A 123 A 131 A132 A 133  3  A 211 A 212 A 213 A 221 A 222 A 223 A 231 A 232 A 233  : R1 R 2 R 3 ,  311 312  1 A A 313 A 321 A 322 A 323 A 331 A 332 A 333  A  3=3x1





and 3= 3x1

 A 111 A 121 A 131   A 112 A 122 A 132   113 123 133  1  A 211 A 221 A 231  R1   A 212 A 222 A 232   3  3  A : R 2  , A A  R   A 213 A 223 A 233   3 3  A 311 A 321 A 331   312 322 332   A 313 A 323 A 333  A A  9 =3x3  A Notice that each dyadic product coordinate is a function of all vector coordinates factor. Example 3: The product 3  :  , with 3   A ijk e i e j e k and   B rs e r e s , is the vector v  A ijk B jk e i  V i e i . This expression can be

 

written in matrix form as

V   1 V  2  V3  3 1

V

 A 111  A 121  A 131  A 211   A 221  A 231  A 311  A 321  A 331

A 112 A 122 A 132 A 212 A 222 A 232 A 312 A 322 A 332

A 113  A 123  A 133  B A 213   11 223 A  : B 21  A 233   B 31 A 313  A 323  A 333  3x3 1x3

B12 B 22 B 32

B13   B 23 ,  B 33  3 3



V3 1 

9=3x3 B B B   A 111 A 112 A 113 A 211 A 212 A 213 A 311 A 312 A 313   11 12 13  121 122 123 221 222 223 321 322 323  A A A A A A A A A  :  B 21 B 22 B 23  ,  A 131 A 132 A 133 A 231 A 232 A 233 A 331 A 332 A 333   B 31 B 32 B 33  3=1x3 3 but these forms are not unique. Again notice that each vector coordinate is a function of all coordinates of the dyadic factor. Example 4: For the same polyadics of example 3 we have, on the other hand: 3   3  .   A ijk B ks e i e j e s . The corresponding matrix

notation is

 3  39

 A 111  A 121  131 A  211 A    A 221  A 231  A 311  A 321  A 331 

A 112 A 122 A 132 A 212 A 222 A 232 A 312 A 322 A 332

A 113  A 123   A 133 3 B B B  213  A  11 12 13   A 223  .  B 21 B 22 B 23  . 233 A   B 31 B 32 B 33  3 A 313  323  A A 333  9 3

We notice that in this case the individual coordinates of the triadic product is not a function of all coordinates of the dyadic 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 linear transformation (LT) between polyadics (of different spaces) achieved by a polyadic operator whose valence is the sum of the valences of the input and output polyadics. Linear Transformations from one vector space (valence 1 and dim3) to another vector space, operated by a dyadic (valence 2), are well known and sometimes mentioned as a "Vectorial Geometry". 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 in the theory of classical mechanics and electromagnetics. The more complex cases - the LTs between a vector space and a dyadic space (performed by a triadic), or between two dyadic spaces (dim9) performed by a tetradic - are rarely mentioned. 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 can rotate vectors by simple dot multiplication, a rotation tetradic can rotate dyadics by double dot multiplication. In the same way a tetradic may stretch or shrink the dyadic defined by two points (in a dyadic space) and diminish or enlarge the angle between two dyadics. There exist also a polyadic that performs the identity transformation: this polyadic is called the unit polyadic of the space (or subspace) and has always even valence; it is denoted by 2H and its associated matrix is the GG unit matrix (for G3H). It follows that there is a multi-dimensional purely Euclidean geometry hidden in the physical laws, which can be as useful as the common two- and three dimensional ones. It could be called a "Polyadic Geometry" and has still to be explored. 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 - are determined 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. A fundamental theorem: If in a G-space G independent P-adics Pi. (i=1,2, ..., G) is associated bi-uniquely with G H-adics H i, then the linear transformation polyadic operator H+P (or the proportionality polyadic) is determined as H+P= Hi Pi for i=1,2, ..., G. From the physical point of view, the physical law connecting two physical magnitudes can be determined by measuring under specified physical conditions (of time, temperature etc.) G pairs of the correspondent magnitudes under the geometrical condition that one set of one member of the pairs, say P1, P2, ..., PG, is composed of independent polyadics, i.e., (P1P2...PG)0 does not vanish. For example: to determine the tetradic which connects the stress dyadic with the strain dyadic in linear elasticity, we must chose and measure six independent strains dyadics (instead of nine in view 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, the measurements must be collected within the "media" in a "state of proportionality". These measurements will be performed with respect to a convenient chosen vector base, say e1, e2, e3 (and its reciprocal e1, e2, e3 if 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 true associate polyadic coordinate matrix, there may be practical difficulties. The matrix obtained by two such experiments may be not equal, mainly due to observational errors. To deal with these, the classical theory of "probability and error statistics" has to be extended to polyadics. Such a theory has to be based on the representation of polyadics by their invariants.

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 of position (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 to one or more of the magnitudes of different orders of a second set (selected as independent), each one of these latter varying with P and t. Hence we can conclude, from the mathematical point of view, that:

1) 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 always correspond 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 time measurements (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 derives from a potential) the system pass to be a system of linear or non linear differential equations (according as the derivatives appears as simple derivatives, or as second derivatives, as third etc. or, also, as products of different derivatives) but always 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 is always possible when the LTs are to be performed between polyadics of the same valence, say H and H, in which case the polyadic operator has even valence, 2H. We ask: when 2H  H. H   X H  for some scalar X, i.e., is there any H-adic  which is transformed into a H-adic parallel to itself?. Or, what is the same, when ( 2H   X 2H) H. H   H  ?. The existence of this equality for some H implies that the 2H-adic between the parentheses must be incomplete, that is, its associated mixed matrix must be degenerate (its determinant must vanish). If we put 2H   H i H  i for i=1,2, ..., G with respect to some H-adic reciprocal bases {H*} and {H*} of the GH

space, we define the 2H adjoint, and denote it by 2H

G 

 G , by the expression

1  H  i H  j ... H  m  (G  1)! 

i

 j ... H  m 

for

(i, j, ... ,m = 1, 2, ..., G),

(I.5.01).

G 1 fatores

where we are using the ready defined multiple skew multiplication of H-adics. The

 G associated matrix is the adjoint

of the 2H associated matrix. This adjoint and its leading (diagonal) minors express the condition for incomplete:

XG 

 E X G 1 

~ 2H 2  E

X G 2 

~ 2H 3  E

 (1) G where

  X 2H to be

X G 3  ... ~ 2H (G 1 )



X 2  (1) G

~ 2H G



X  (1) G

G  0

(I.5.02),

 G is the 2H determinant; the coefficient of the linear term is the sum of the leading minor of degree one of this

determinant, that is, the scalar of

 G ; the coefficient of the quadratic term is the sum of the leading minors of degree ~

two of this determinant, that is, the scalar of 2H  (G -1 ) ; etc.. This equation is the "2H-adic characteristic equation". The solution of this problem brings us to the determination of the 3 H (invariant) eigenvalues and eigenH-adics of the (statistical measured) 2H-adic operator. Then it is possible to demonstrate the Cayley-Hamilton theorem (for future usefulness) for polyadics, that is: every 2H-adic satisfy its own characteristic equation.

SECTION II: THE ESSENTIAL CONDITION FOR A GEOMETRICAL APPROACH TO PHYSICAL 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 continuous variable quantities defined in the current point O of a G-space (G<3H), one of then, say H  , taken as independent variable. We have: H (II.1.01)1,  | H  | H ˆ , || H  ||  H  H. H  , and H  H. H  1 1 - When a polyadic is expressed in an arbitrary vector base by its "coordinates" (covariant, contravariant, etc.), its norm (always a positive number) is equal 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

where ||H  ||,| H  | and H ˆ denotes the norm, the modulus and the unitary (a H-adic of norm 1 parallel to H  ) of the Hadic H  . The proportionality of the magnitudes – the linear physical law - can be expressed as the linear polyadic equation H



H .

,

(II.1.021),

where the dependent variable  , 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 to express H  as a function of H  . Hence, admitting that the 2H G determinant in this G-space does not vanish, we can invert (II.1.021) and write H

det( 2H G)  0 and



H H  .





G 1

H H  .

(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 this space). Else the law exist in a space of dimension one unit less (a subspace if G3H) for each vanishing 2H G eigenvalue; and in this subspace the 2H G will be seen complete. The polyadic 2H G characterizes the medium completely for the phenomenon governed by the law (II.1.01). Thus we 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 hand we must observe that the polyadic coordinates for the specified phenomenon are different for different observers (because each one chooses his own vector base). A Postulate, Specific Magnitudes and

G Symmetry.

In view of physical usefulness we might admit the following Postulate: There exists a continuous variable function of scalar value 2W with several continuous derivatives, defining a new physical magnitude by the law

2W  H 

H H  .

H 

H .

  H

H 2H G H. H  .

(II.1.031).

To simplify the mathematical handling we will introduce the new variables H 

 , |H  | H

2W=|H  | 2W0 ,

and

2w=

2W0 2W  , |H  | | H  |

(II.1.04),

so that - besides the unchanged law (II.1.01) - we have,

  2 H G H. H   H  H. 2w  H  H. H   H  H. H   H

2W0  ˆ H

H H  .

  H

 2H

GH , H  H 2H  . G

H .

(II.1.02), H H  .

(II.1.03).

ˆ ,

(II.1.031).

We could name magnitudes H  and 2W0 "specific magnitudes", or "magnitudes by unit of (modulus)" since | H  | represents a quantity of the magnitude H  . From (II.1.02) and (II.1.03) we deduce  H ˆ : 2H

G

 H

G 

ˆ

H .



( 2H G  2H G H )

 , that is, 2H

G

H H ˆ = 0 , .

H

intensity

which is possible if and only if

 2H

GH ,

(II.1.05).

Hence, the acceptance of the postulate (II.1.03) carries the 2H G 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 are independent because its norm is equal to one. Taking the point O of this euclidean G-space as origin of polyadics, H  can modulus. 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.).

be seen as the H-adic position of a point on a hiperspherical surface (or, simply, a spherical surface) of unit radius centered at O; it defines a hiperdirection in this space (or, simply, a direction). In elasticity, for H=1 and G=3 for example, H  is a vector p representing the stress vector on a plane with normal unit vector H   n ; and 2w is the normal stress, , on this plane. Still in elasticity3, for H=2 and G=6, H  is the stress dyadic  on a subspace of dimension six4 at the point O (of the 6-space spanned by six independent H  ) with normal unit dyadic H  ; and 2W0 represents twice the specific density energy (the stored strain energy by unit of volume at this point), although the value 2W is more commonly used. In view of the isomorphism with vector spaces we call H  ( H  ) the H-adic (specific) projection of 2H G in the direction H  . Similarly, 2W (2w) is the scalar (specific) projection of H  ( H  ) in the direction H  ; it is also called the radial (specific) value of 2H G relative to the direction H  . For two different directions H   and H   we write (in accordance with polyadic algebra)

2w   H ˆ 

H 2H G H. H ˆ   H ˆ  H. 2HG H. H ˆ  , .

(II.1.061),

2w   H ˆ 

H H   .

 H ˆ 

H H  .

(II.1.06).

Hence the proposition: The 2HG projection H   relative to a direction H   projected on a second direction H   , is equal 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 to note that in the theory of elasticity equality (II.1.06) translates into Betti's law. For H=1 - in which case H  is a force vector, H ˆ is 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 a system of forces f2, f2', ...) is equal to the work that should be produced by this latter, in virtue of the application of the first". For H=2 - in which case

 is a stress dyadic and

ˆ a 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 to the 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.1.02 1) and (II.1.031) when the independent and continuous variable H  assumes all possible finite values of a certain defined domain that will be not discussed here from physical 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 {H 1 , H  2 , ..., H  G } constitute an orthogonal base, then {H ˆ 1 , H ˆ 2 , ..., H ˆ G } - the set of the unit dyadics of the former - 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 called orthonormal bases; for these bases we can write H

ˆ 1 : H ˆ j   ij

(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 are 3 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.

dyads formed with vectors of a orthonormal vector base {i j k } , that is, ˆ 1  îî, ˆ 2  ˆjˆj, ˆ 3  kˆ kˆ , ˆ 4  ˆjkˆ , ....,ˆ 9  ˆjî , 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 assumed to have unit norms: ˆ 1 , ˆ 2 , ...,ˆ G ; this means,

H1  H 2  ... H G



ˆ i : ˆ j   ij

(i,j=1,2, ..., G),

(II.2.02 1).

The metric matrix associated to this set is [ˆ i :ˆ j ] , 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 2H

G  G i ˆ i ˆ i

G in the form:

(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 and Cowin, 1994; Helbig, 1994). Let us suppose now that 2HG has a double eigenvalue, say G G 1  G G . 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 cross product of this G-1 eigendyadics,  ˆ 1ˆ 2 ...ˆ G 1  , 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 eigenHadics 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 we can 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 makes w an extreme then dw=0 in this direction. By differentiating (II.1.03) we get: 2dw  2 Hˆ H. 2HG H. dHˆ  0 . From (II.1.01), we deduce



H .

d H   0 ; hence, we conclude that

 and



orthogonal to the same plane (hyperplane) tangent to the spherical surface H  and



H .

H H  .

H H  1 . .

are orthogonal to d H  , that is, This means that the H-adics



 must be parallels.

H

By (II.1.031) we write5: 2w=|H | cos( H  , H  ) , whence we deduce that the 2w extreme value is |H| if the H-adics H  and H  are parallels (a maximum corresponding to the null angle and a minimum to 180). The parallel condition may be expressed in the form H   2 H G H. H   X H  , where X and H  are a scalar and a H-adic to be determined, which, as we 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 directions H   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: 2H

G  Gu

u

H

u

H

(sum for u=1,2, ..., G),

(II.3.01),

since 2H

H H  1  G1 H  1 , .

H H 2 .

G2

 2 , ...,

H

(II.3.011),

and H

 1

H 2H G H. H  2 .

 0 = H  1

H 2H G H. H  3 .

 .....=

 2

H 2H G H. H  3 = .

5 - For multiple dot multiplication of polyadics essentially same concepts hold as for scalar multiplication of vectors.

... ,

(II.3.012).

The 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; its eigenH-adics are its principal directions and constitute the principal (orthonormal) H-adic base in the point O. Referred to this principal H-adic base, the 2HG associated matrix is a (GxG) diagonal matrix, its diagonal elements being the 2HG eigenvalues; 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=( H 

H H  u )2 .

Gu   (  H.

 1 ) 2 G 1  ( H 

H

H H  2 )2 .

G 2  ...,

(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 the corresponding 2HG eigenH-adic direction. If we denote by Eu and Su the projections (coordinates) of

 and

 on the eigenH-adic of base H  u , that is, if

we put H



H H u .

 E u and



H .

 u S u ,

H

(II.3.05),

the law (II.1.02) is then equivalent to the system

S1  G 1 E 1 S 2  G 2 E 2 , ... S  G E G 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 the symmetric polyadic 2H G, the ratio of the H-order magnitudes with the same subscript is equal to the corresponding 2H-adic eigenvalue.

II.4 - The Projection Norm and Octahedral Directions. For an arbitrary direction

 in the vicinity of the point O of the G-space we can write, with respect to the principal base

{ ˆ 1 , ˆ 2 , ..., ˆ G } : H

  ( H 

H H  u ) H  u .

(u=1,2, ..., G),

(II.4.01),

being G

(

ˆ

H H ˆ u ) 2 .

 1,

(II.4.011),

because



H H  1 . .

The numbers



H H u .

are the G principal director cosines of the direction. In general they are

all different, but for a particular direction they can be all equal. For a given and ordered set of G squares, whose sum is equal to one, there are 2G directions (that is, all the arrangements with repetition of the signs + and – taken G by G with the G ) whose director cosines have the same modulus. modulus of the director cosines, (AR) G 2 2 We shall call octahedral directions, or octahedral H-adics of 2HG, the unit H-adics equally inclined to its principal directions. Denoting a octahedral direction by H  oct we can write from (II.4.01), H

 oct  ( H  oct

H H  u ) u .

and from (II.4.011), since the cosines (cos oct) are all equal:

(u=1,2, ..., G),

(II.4.01 2),

for (u=1,2, ..., G)

ˆ oct

H H ˆ u .

 cos  oct  



G , G

(II.4.01 3).

So, for G=2,  oct  45  , for G=3,  oct  54  44' , for G=4,  oct  60  , for G=9, oct  7031'44" etc.. For any octahedral direction to which w=woct corresponds we have

2w oct = H ˆ oct

H 2H G H. H ˆ oct .

 ( H ˆ oct

H H ˆ u ) 2 G u .



1 G

G

(II.4.014),

that is: 2H

In any octahedral direction, the eigenvalues.

G radial value 2woct is equal to the invariant average of the

For an arbitrary direction any in the space we can write the norm of the correspondent 2HG projection as

||H  || H ˆ

H 2H G H. 2H G H. H ˆ .

= H ˆ

H 2H 2 H H G . ˆ , .

with

 H 2H G = 2H G 2 .

(II.4.02).



The polyadic 2H G 2 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 write 2H



G2 = (G u ) 2

 u H  u ,

H

(u=1,2, ..., G)

(II.4.02 1);

whence we deduce 2H



G 2E 

 (G

(II.4.022).

Hence:

| H |  ( H 

H H  u ) 2 (G u ) 2 , .

(II.4.03),

or, writing in full: ||  || ( H ˆ

H H ˆ 1 ) 2 (G1 ) 2 .

 ( H ˆ

H H ˆ 2 ) 2 (G 2 ) 2 .

 ... ( H ˆ

H H ˆ G ) 2 (G G ) 2 .

(II.4.031).

The Gu are invariant, that is, they don't depend on the H  . Hence, ||H|| varies with the square of the 3H direction cosines ( H  H H  ) 2 whose sum, in conformity with (II.4.01 1), is equal to one. Thus, when these are all equal - for .

octahedral directions - representing by H  oct the 2HG projection correspondent to any one of the octahedral directions, we have:

||H  oct ||

1 G

 (G 1



1 G



G 2E ,

(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 eigenvalue (or equal to the G-th part of the 2HG dot square scalar). Let us calculate now the difference between the eigenvalues, that is, 1 G



G 2E  (

1 G

G square eigenvalues average and the square of the average

GE )2 

G u 2 1 2 ) .  (G u )  ( G G

Developing the squares in the second member, grouping pieces conveniently and noting the presence of new perfect squares we have

G u 2 1 1 2 )  2 [(G 1  G 2 ) 2  (G 1  G 3 ) 2  ...  (G 1  G G ) 2   (G u )  ( G G G (G 2  G 3 ) 2  (G 2  G 4 ) 2  ...  (G 2  G G ) 2  ...  (G G 1  G G ) 2 ] . Since we have C 2G squares of differences inside the brackets, we write:

 G u 2 G 1  G) 2 1 2 )  ,  (G u )  ( G G 2G C 2

(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 1)/ 2G of the average of its squares difference two by two. The 3H principal 2HG invariants are the coefficients of its characteristic equation XG 

G 1E X G1 

G 2E X G2  ... 



G (G  2 ) E X 2 +

G (G 1 ) E X  det( 2H G)  0 ,

(II.4.05),

where: for i=1,2, ..., G-1, 2H G i E 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 2H

~ 2 E



 1 2H [( G E ) 2 ( 2H G 2 ) E ] , 2

(II.4.06).

Hence: The sum of the pairwise products of eigenvalues of 2HG is equal to one half the difference between the square of their sum and the sum of their squares.

II.5 - The Transversal Value of the Proportionality Polyadic. From 2w=|H | cos( H  , H  ) we see that  H  : | H | (2w) 2 . Hence, there always exists a positive number, say t2, which complement (2w)2 to ||H||. We can write:

 H  : |H |2 =(2w) 2  t 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 t2 is the square of the tangential stress vector  on a plane defined by a normal unit 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. For H=2, t2 is an always existing positive number which has to be summed to the square of the specific energy density 2w to obtain the norm of the specific stress dyadic. We write (II.4.07) as t 2+(2w)2=||2=||||. Applying (II.4.04) to the case of octahedral direction, considering (II.4.03 2) and (II.4.014) and simplifying we get

t oct 

1 G) 2 , G

(II.5.011),

and state: The 2HG transversal value t oct relative to octahedral directions is equal to the G-th part of the square 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 eigenvalues are all equal, to be, 2HG is a scalar polyadic. We write:

t oct  0



G G

 with G=G1  G 2  ... and | H  oct | G 2 ,

(II.5.012).

If at least two of 2HG eigenvalues are different, then toct0. 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 a multivariable function we must extreme the function

F  t 2  L H  (with the conditional equation

H H H  .  1 )

H H  .

(II.5.02),

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):

F  t2  ||H  ||  w   H ˆ=   2 L    2L H ˆ  H  ,  H ˆ  H ˆ  H ˆ  H ˆ

(II.5.021),

where H0 is the null H-adic. Calculating the derivatives and simplifying we write (II.5.02 1) in compact polyadic notation:  F  2[ 2 H G 2  2(2w)  H 

2H G -L 2H  ] H H   = H , .

(II.5.022);

or, with respect to the 2HG eigenH-adic base:

2[(G u ) 2  2(2w)G u  L]( H 

H H  u ) H  u .

H  ,

(II.5.023).

The linear combination (II.5.023) implies the nullity of all eigenH-adic (H  u ) factors, since these H-adics form a base). Hence

[(G u ) 2  2(2w)G u  L]( H 

H H  u ) 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:

 [( G 1 ) 2  2(2w )G 1  L]( H ˆ   [( G 2 ) 2  2(2w )G 2  L]( H ˆ  ...   2 H [( G G )  2(2w )G G  L]( ˆ

H H ˆ 1 ) .

0

H H ˆ 2 ) .

0

H H ˆ G ) .

 0,

(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 eigenHadics 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-adics H  1 and H  2 . Let us suppose that H  - the unit H-adic that makes the 2HG squared transversal value t2 stationary – is not orthogonal to any 2HG eigenH-adic. Then we get from system (II.5.04)

(G1 ) 2  2(2w)G1  (G 2 ) 2  2(2w)G 2  ...  (G G ) 2  2(2w)G G  L ,

(II.5.041).

As G1 and G2 are single eigenvalues, G1  G 2  G 3 , G 4 , ..., G G . The first two members of (II.5.041) give G1 +G 2 4w ; the first and the third give G1 +G 3 4w , and so on. But this is a contradiction because we might accept that G 2=G3=G4= ...= G G . Hence H  must be orthogonal to at least one 2HG eigenH-adic. If H  was perpendicular to H  1 , the system (II.5.041) would be reduced to at most G-1 equations and we could write (G 2 ) 2  2(2w)G 2  ...  (G G ) 2  2(2w)G G  L , whence we could deduce G 2 +G 3 4w , G 2 +G 4 4w , ..., to be G2=G4=...= G G . But this is also a contradiction (G2 is a

single eigenvalue). By the same reason H  cannot be perpendicular to H  2 . Theorem 2: In the 2-space defined by a pair of orthogonal eigenH-adics of a symmetric 2H-adic there exist two other H-adics, unitary and orthogonal to each other, which bisect the supplementary angles of the two 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 (between which couldn't exist H  1 nor H  2 ) we still should find contradictions because the corresponding conditions would imply the equality of G1 and G2. But if

 should be perpendicular to

 3 , H  4 , ..., H  G - in which case

 1 and  2 and still would make the square of the the system (II.5.041) would be reduced to the following two equations the 2-principal space defined by

 would belong to

G transversal value stationary -

(G1 ) 2  2(2w)G1  (G 2 ) 2  2(2w)G 2  L ; we would deduce G1 +G 2 4w , or, taking into account the correspondent expression (II.3.05) of the 2HG radial value 2w: G +G  2[( H  H H  ) 2 G  ( H  H H  ) 2 G ] . Noting that, in the 2-space, H   ( H  H H  ) H   ( H  H H  ) H  1

2 H

with ( 

. 1 1 H H 2 H  H H 2 1)  (  .  2 ) .



H H 1  .

to H 

 1 , we have: G1 +G 2  2{(  H

this, remembering that G1–G20: H

(

2 / 2 we get the solution

 H. H  1 ) 2  1/ 2 . H   H  (+) under

. 1 1 H H 2 H  H H 2  1 ) G1 [1 (  .  1 ) ]G 2 } ; . H

Hence: which



 ˆ

we can simplify

ˆ

H H ˆ 1 .

(+)

makes the angle of 45 with

H H ˆ 2 .

  2 / 2 . Related to H

 1 ; related

H H 1   .

2 / 2 , the solution H  (-) makes the angle of 135 with H  1 . We arrive to an analogous conclusion with respect to H  2 . Hence H  (+) and H  (-) bisect the supplementary principal directions defined by H  1 and H  2 ; evidently they are perpendicular to each other. Corollary 1: If the symmetric proportionality 2HG-adic has N single eigenvalues (1N G), there exists C 2N (combinations of N taken two by two) H-adics that make the square of its transversal value |t| stationary, each H-adic belonging 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 the corresponding eigenH-adics; and if N is the number of single eigenvalues, these exist in number of C 2N . Theorem 3: The radial value of the symmetric proportionality 2H-adic, 2w, relative to any bisector direction is equal to one half the sums of the eigenvalues related to the corresponding bisected principal directions. Indeed, for the bisector corresponding

  ( H  1  H  2 ) 2 / 2 in the 2-space (H  1 ,

G radial value, 2w12: we have 4w 12  ( H ˆ 1  H ˆ 2 ) H

 u and

 2 ), equation (II.1.03) gives the

( H ˆ 1  H ˆ 2 ) . Substituting 2HG for  (G1  G 2 ) / 2 . In general, then, we write for

H 2H G H. .

(II.3.01), expanding and considering (II.3.01 1) and (II.3.012), we get 2w 12 the principal directions

 v : 1 2w uv  (G u  G v ) , 2

(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 equal to one half the modulus of the difference of the eigenvalues related to the correspondent bisected principal directions. We can calculate t2 for the particular case of the (orthogonal) bisector directions considered in the demonstration of the Theorem 3, that is, H   ( H  1  H  2 ) 2 / 2 . Noting that, from (II.4.03),

| H |  ( H 

H H  u ) 2 (G u ) 2 .

1 [ ( H  1  H  2 ) 2

H .

 u ]2 (G u ) 2 ,

we have, developing the sum in u: ||H  || [(G1 ) 2  (G 2 ) 2 ] / 2 . Hence, using (II.4.07) and considering (II.4.11) we write:

t 2  [(G1 ) 2  (G 2 ) 2 ] / 2  (G1  G 2 ) 2 / 4  (G1  G 2 ) 2 / 4 , that is, | t || G1  G 2 | / 2 . We should obtain the same result for the bisector direction H   ( H   H  ) 2 / 2 . 1

In general, then, we write for the principal directions

|t uv |

 u and

 v :

|G u  G 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 all positive or all negative, this value is not the |t|max because this maximum is |G6|/2 and occurs in the 9-space (the zero eigenvalue must now be considered).

II.6 - Cauchy's and Lamé's quadric. For det2HG0 and (II.1.03), considering (II.1.01), we can deduce, in the G-space: H



H 2 H 2 G .

H H  1 , .

(II.6.01),

and

Q  H

H 2H G H. H   .

2w/ | 2w | 1 ,

(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., H

  H ˆ / | 2w | ,

(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. These are quadric surfaces centered at the point O. The first – representing the variation of  – is the Lamé's ellipsoid. The second – representing the variations of 2w and called Cauchy's quadric or indicator quadric – is either an ellipsoid or a hyperboloid (of one or two sheets) depending on the coordinates of the 2HG-adic at the point. Correspondingly, 2HG is called 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 H  can be easy determined. To calculate 2w it is enough to determine the point Y where H  intercepts the indicator Q, since according 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 can conclude: 1) If Q+ is a real ellipsoid, Q- wills no real graphical representation, because it is a imaginary ellipsoid. In this case 2w>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 asymptotic cone whose equation is C  H  H. 2H G H. H  = 0 . The point Y, intersection of H  with Q, could be on Q+, on Q-, or even might not exist (if H  should be parallel to any generator of the cone). In the first case, the angle between H and H  is always acute and 2w>0; in the second case the angle 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 regions where 2w>0 and 2w<0. The orthogonal projections of these hyper curves on the coordinate planes are ellipses or hyperbolic arches.

Reduced 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 space is referred to the (principal) base (at the point) defined by the 3 H orthonormal eigenH-adics, H  u of 2HG. In this case, (II.6.01) and (II.6.02) are written in the respective forms (H 

H H ˆ u .

/ Gu )2  1 and ( H 

H H 2 u) .

G u  1 , where the

left-hand-sides are sums on u. If Su and Yu are the coordinates of  and  in the H-adic principal base we can write: H

and (II.6.04),  ( | G u | Yu )2  1 , (S u / G u ) 2  1 where the sign of each term in the left-hand-side of the second equation is the sign of Gu. In reduced form, the indicator quadric 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 equations

2W = H ˆ H. 2H G  2 2 H (2W)  T  ˆ H H H 1  ˆ . ˆ 

H H ˆ . H 2H 2 H H G . ˆ , .

(II.7.01),

enclosing the 2HG radial value 2w and the 2HG transversal value |t| corresponding to the variable H-adic H  . With respect to the orthonormal base of the 2HG eigenH-adics, we can - in view of (II.3.04) and (II.4.02 1) - transform the right-hand-side of these equations and put H  H. H  u  E u for all u=1,2, ..., G to obtain

2w = (E 1 ) 2 G 1  (E 2 ) 2 G 2  ...  (E G ) 2 G G  2 2 2 2 2 2 2 2 (2w )  t  (E 1 ) (G 1 )  (E 2 ) (G 2 )  ...  (E G ) (H G ) , 2 2 2 1  (E 1 )  (E 2 )  ...  (E G ) 

(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 G1  G 2  G 3 ...  G 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

 1 ,

 2 and

H

 3 , in which case E4=E5= ...

H

= E G =0 and the system (II.7.02) is reduced to

2w=(E 1 ) 2 G 1  (E 2 ) 2 G 2  (E 3 ) 2 G 3  2 2 2 2 2 2 2 2 (2 w)  t  (E 1 ) (G 1 )  (E 2 ) (G 2 )  (E 3 ) (G 3 ) 1 (E ) 2  (E ) 2  (E ) 2 1 2 3 

(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 this point 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)2 and (E3)2. Solving (II.7.03) we get, remembering that G 1G2 G3:

 (2 w - G 2 )( 2 w - G 3 )  t 2 2 ( E )   1 (G 3 - G 1 )(G 2 - G 1 )   (2 w - G 3 )( 2w - G 1 )  t 2  2 ( E )   2 (G 2 - G 3 )(G 2 - G 1 )   2 (E 3 ) 2  (2 w - G 1 )( 2 w - G 2 )  t  (G 3 - G 1 )(G 3 - G 2 ) whence

(II.7.04),

(2 w-G 2 )(2 w-G 3 )  t 2  0  2 (2 w-G 3 )(2 w-G 1 )  t  0 (2 w-G )(2 w-G )  t 2  0 1 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

(2w-G 2 )(2 w-G 3 )  t 2  (

G 3 -G 2 2 G 3 -G 2 2 ) ( ) , 2 2

or, transforming the left-hand-side:

t 2  (2 w 

G 3 +G 2 2 G 3 -G 2 2 ) ( ) , 2 2

(II.7.06).

The inequality (II.7.06) represents points not interior to the semicircle centered in C 23  ((G 2  G 3 ) / 2,0) with radius R 23  (G 3  G 2 ) / 2 . Similar interpretation hold for the other two inequalities in (II.7.05), the second representing points not exterior to the semicircle centered at C13  ((G 1  G 3 ) / 2,0) with radius R 13  (G 3  G 1 ) / 2 and the third, points not interior to the semicircle centered at C12  ((G 2  G 1 ) / 2,0) with radius R 12  (G 2  G 1 ) / 2 . 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 for the particular case of all G>0).

This graphical representation on the variations of the radial and transversal values of 2HG is called Mohr's representation because of its analog in the theory of elasticity; the bounding circles, Mohr's 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 without 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 the locus of points equally inclined against spherical surface



H H  1 , .

 3 ; in other words, we ask: if N describes the parallel of co-latitude 3 of the

what curve does N describe in the Mohr plane?

The equation of this curve is obtained by elimination of E 1 and E2 in the system (II.7.03); we have:

t 2  (2 w 

G1  G 2 2 G 2  G1 2 ) ( )  cos2 3 (G 3  G 1 )(G 3  G 2 ) , 2 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 us draw in the Mohr plane, as shown in Figure .2, the line r 3 from (G3 ,0) that makes an angle 3 with the |t| axis. This line intersects the circles (C13,R13) and (C23,R23) at A2 e A1 , respectively. Since G2A1 e G1A2 are both perpendicular to r3, they are 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):

C12 A 12 (

 C12 G 3 cos 2  3  (G 3 

G1  G 2 2 ) cos 2  3 , 2

A 1A 2 2 G G G  G1 2 )  ( 1 2 ) 2 sen 2  3  ( 2 ) (1  cos 2  3 ), 2 2 2 2

C12 A 1  C12 A 12  (

A 1A 2 2 G  G1 2 ) ( 2 )  (G 1  G 3 )(G 2  G 3 ) cos 2  3 , 2 2

(II.7.08).

Hence the radius of the circle (II.7.06) is C12A1 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 C12A1 can vary:

C12 A 1  (

for 3  0 ,

for 3   / 2 ,

G 2  G1 2 G  G2 )  (G 1  G 3 )(G 2  G 3 )  G 3  1  C12 G 3 ; 2 2

C12 A 1 

G 2  G1 G  G2  G2  1  C12 G 2 . 2 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 H  1 , but now we must have

1   / 2  3 in order to satisfy the third equation in (II.7.03). The locus of the points in the Mohr plane that correspond to 2w for directions with the same inclination against H  1 is the circle 2

t 2  [2w  (G 2  G 3 ) / 2) 2  [(G 3  G 2 ) / 2] 2  cos 2  (G 3  G1 )(G 2  G1 )  C 23B 2 ,

(II.7.09),

We grade the semicircle (C12,R12) in 1 in the same way as before. Now it is possible to locate immediately the point N whose coordinates are 2w and |t| (Figure 3). It is evident that the points of the semicircle centered at C 13 with radius R13 are related to directions with 2   / 2 , i.e., with direction perpendicular to the plane defined by directions H  1 and H  3 . These are precisely the directions with respect to which 2w assumes extreme values. It is evident, also, that the extreme value of |t| is R 13, i.e.,

| t | max  R 13  (G 3  G1 ) / 2 ,

(II.7.10).

There is also a third circle passing through the point N (not shown in the figures) with center at C 13 with the equation

t 2  [2w  (G 3  G1 ) / 2] 2  [(G 3  G1 ) / 2] 2  cos 2  2 (G 3  G 2 )(G1  G 2 ) ,

(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.

Note: In the theory of elasticity with H=1, the radial value 2w of the stress dyadic represents normal stress and its tangential value |t| tangential stress; with H=2, i.e., for Green's tetradic, the radial value 2W for a certain unit strain dyadic  , represents stored energy. The tangential value |t| is a new variable which we might call complementary energy, since 2t|2w|max . Hence, in connection with Green's tetradic we can generalize the concept of Mohr's 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 valid area, 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

 1 , H  2 , H  3 and H  4 , in which case

E5= ... = E G =0 and the system (II.7.02) is reduced to

2w=(E 1 ) 2 G 1  (E 2 ) 2 G 2  (E 3 ) 2 G 3  (E 4 ) 2 G 4  2 2 2 2 2 2 2 2 2 2 (2 w )  t  (E 1 ) (G 1 )  (E 2 ) (G 2 )  (E 3 ) (G 3 )  (E 4 ) (G 4 ) 1 (E ) 2  (E ) 2  (E ) 2  (E ) 2 1 2 3 4 

(II.7.11).

Substituting (E4)2 from the last equation into the first two and rearrange we obtain an equivalent system in the unknown E12, E22 and E32:

2w - G 4 = (E 1 ) 2 (G 1 - G 4 )  (E 2 ) 2 (G 2 - G 4 )  (E 3 ) 2 (G 3 - G 4 )   2 2 2 2 2 2 2 2 2 2 2 2 (2w )  (G 4 )  t  (E 1 ) [(G 1 )  (G 4 ) ]  (E 2 ) [ (G 2 )  (G 4 ) ]  (E 3 ) [(G 3 )  (G 4 ) ]  1  (E 4 ) 2  (E 1 ) 2  (E 2 ) 2  (E 3 ) 2 ,   The determinant of this system is (G 2  G1 )(G 3  G1 )(G 3  G 2 )  0 ; the solution as a function of E42 is:

(G 3 - G 1 )(G 2 - G 1 )( E 1 ) 2  (2w - G 4 )[ 2w - (G 2 + G 3 - G 4 )]  t 2  (G 4 - G 2 )( G 4 - G 3 )[1 - (E 4 ) 2 ]   2 2 2 (G 3 - G 2 )(G 2 - G 1 )( E 2 )  (2w - G 4 )[ 2w - (G 1 + G 3 - G 4 )]  t  (G 4 - G 1 )( G 4 - G 3 )[1 - (E 4 ) ] ,  (G - G 1 )(G 3 - G 2 )( E 3 ) 2  (2w - G 4 )[ 2w - (G 1 + G 2 - G 4 )]  t 2  (G 4 - G 1 )( G 4 - G 2 )[1 - (E 4 ) 2 ]   3 i.e.,

(2 w-G 4 )[2 w-(G 2 +G 3 -G 4 )]  t 2  (G 4 -G 2 )(G 4 -G 3 )[1-(E 4 ) 2 ] 0  2 2 (2 w-G 4 )[2 w-(G 1 +G 3 -G 4 )]  t  (G 4 -G 1 )(G 4 -G 3 )[1-(E 4 ) ] 0 (2 w-G )[2 w-(G +G -G )]  t 2  (G -G )(G -G )[1-(E ) 2 ] 0 4 1 2 4 4 1 4 2 4 

(II.7.12),

If we add [(G 2  G 3 ) / 2  G 4 ] 2 to both sides of the first inequality in (II.7.12) we find after some manipulation:

t 2  (2 w 

G2 G3 2 G3 G2 2 ) ( )  (G 4  G 2 )(G 4  G 3 )( E 4 ) 2 . 2 2

As 0(E4)21 we can write

(

G3 G2 2 2 G G3 2 G G3 2 )  t  (2 w  2 )  (G 4  2 ) , 2 2 2

(II.7.121).

From the second and the third inequalities we obtain also two other inequalities; a second, viz.:

( and

G 2  G1 2 2 G G2 2 G G2 2 )  t  (2 w  1 )  (G 4  1 ) , 2 2 2

(II.7.122),

t 2  (2 w 

G1  G 3 2 G G3 2 )  (G 4  1 ) , 2 2

(II.7.123).

The meaning of these inequalities is obvious: (II.7.121), for example, represent the set of points not interior to the circle centered at ( ((G 2  G 3 ) / 2,0) ) with radius (G 3  G 2 ) / 2 and not exterior to the circle with the same center and radius G 4  (G 2  G 3 ) / 2 . Hence, in this 4-space the set of points in the 2wx|t| plane that satisfies the system (II.7.12) contains all points not exterior to the circle centered at ( ((G 2  G 3 ) / 2,0) ) with radius G 4  (G 2  G 3 ) / 2 and not interior to the following two circles: a first centered at ((G 1  G 2 ) / 2,0) with radius (G 2  G 1 ) / 2 and a second centered at ((G 2  G 3 ) / 2,0) with radius (G 3  G 2 ) / 2 . As the 3-spaces defined by ( H  1 , H  2 , H  3 ), ( H  2 , H  3 , H  4 ) etc. are subspaces of the considered 4-space, the corresponding areas could be valid areas; hence we must exclude the points not interior to the semicircle centered at ((G 3  G 4 ) / 2,0) with radius (G 4  G 3 ) / 2 . If we had taken (E1)2 from the last equation (II.7.11) in place of (E 4)2 we 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 with direction cosines cos1, cos2 etc. It is convenient to observe that the sum of two arbitrary direction angles that satisfy the third equation (II.7.11) can not be less than 90 . Indeed, for any two the sum of the squares of the respective cosines must be less than one, say cos2 1  cos2  2  1 , i.e.,

cos2 1  sen 2  2  cos2 (90   2 ) , or 1  90   2 . Let us consider initially the directions inclined of the given angle  i against the reference H-adic against

 m . When N describes the spherical surface



H H  1 .

 i and  m

(with fixed  i and  m ) what curve does the

corresponding N describe in the Mohr plane ?. As in the previous case, the equation of this curve can be obtained by eliminating the cosines E j and Ek in the system (II.7.11). In this way we find

t 2  (2w 

G j G k 2 2 )  R jk 2

(II.7.13),

with 2

Rjk  (

Gk  G j 2

) 2  (G k  G i )(G j  G i )cos2  i  (G k  G m )(G j  G m )cos2  m ,

(II.7.131).

Suppose now that four angles satisfy the third equation of the system (II.7.11). For elimination we can select six pairs of angles. This is the same as saying that from (II.7.13), with j and k running from 1 to 4 (with jk), we can obtain the equations of six circles, each centered at the "Mohr's centers" with radius R *jk. But this six circles will have necessarily a 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 Mohr's Circles. Some particular properties of these circles can be listed. For example: the direction perpendicular to the base H-adic  4 (  4  90 ) belongs to the 3-space of base H  1 , H  2 , H  3 . With the choice j=1, k=2 and i=3 in (II.7.13) and (II.7.13 1) 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 the geometrical meaning are the same as in section II.7.1. For  i   m  90 the considered direction belongs to the 2-space of base H  j , H  k . Hence the correspondent H

Mohr 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 2

C14 F4  C14 G 2 G 2 F4 ,

and

C14 F1  C14 G 3 G 2 F1

G1  G 4 2 ) , 2

and

C14 G 3  (G 3 

(II.7.14).

But 2

C14 G 2  (G 2 

G1  G 4 2 ) 2

(II.7.15).

Then, since the perpendicular from a point of a circle over a diameter is the geometric average between the segments determined by its foot, 2

G 2 F4  G 1G 2  G 2 G 3  (G 2  G 1 )(G 3  G 2 ) , 2

G 3 F1  G 3G 4  G 2 G 3  (G 4  G 3 )(G 3  G 2 ) ,

(II.7.16), (II.7.161).

Substituting (II.7.15), (II.7.16) and (II.7.16 1) into the equalities (II.7.14) we conclude: 2

C14 F4  C14 F1  (

G1  G 4 2 )  G 3G 2  G 1G 3  G 2 G 4 , 2

(II.7.17).

For octahedral directions we have: cos 2 1  cos 2 2  cos 2 3  cos 2 4  1/ 4 . Denoting the radii of the correspondent circles by Rjkoct we write from (II.7.131):

4 R jkoct2  4(

Gk G j 2

) 2  (G k  G i )(G j  G i )  (G m  G k )(G m  G j ) .

Developing the left-hand-side, adding pieces conveniently and representing by  R 2 the sum of the squares of the radii of the Mohr circles, i.e.,

R

(

G 4  G1 2 G 4  G 2 2 G 4  G3 2 G3  G 2 2 G3  G1 2 G 2  G1 2 ) ( ) ( ) ( ) ( ) ( ) , 2 2 2 2 2 2

(II.7.18),

we obtain,

4 R jkoct2   R 2  (

G k G j 2

)2 (

G m Gi 2 ) , 2

(II.7.19).

Summing up the six expressions obtained from (II.7.19) giving to the subscripts 1, 2, 3 and 4 we deduce that

1 R 12oct 2  R 34oct 2   R 2  R 13oct 2  R 24oct 2  R 14oct 2  R 23oct 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 radial values are positive; to each point of the valid area correspond one and only one point of the corresponding indicator ellipsoid. 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 the sphere. We could say also that to the points belonging to that segment correspond directions parallels to the cone's generators. 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 the law of proportionality. In the previous paragraphs we have shown that to each pair ( H, H) corresponds a point P in the valid area of the Mohr's plane with coordinates (2w,t), see Figure 4.

A necessary and sufficient condition for the point P to be not exterior to the valid area – that is, a condition for the existence of the proportionality established by the law (II.1.02 1) - is that its distance d to center of the greatest radius circle be less than this radius, i.e., less than the greatest t, the invariant t max which is a characteristic of the medium. Hence, if the eigenvalues are all positive, t max=|Gmax|/2, if they are all negative, tmax=|Gmin|/2 and if Gmin<0 and Gmax>0, tmax=|Gmax-Gmin|/2. In any case, we see that the double of tmax is the modulus of the difference between the greatest and the smalest of the 2HG eigenvalues (zero included). On the other hand, we can write

d 2  (| 2w | t max ) 2  t 2  (t max ) 2 . Hence,

||H  || |H  |2   | 2w | , G max  G min 2 t max

(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 the eigenvalues Gi and Gi+1 (diameter points of a certain circle) the inequality (2w-Gi)(Gi+1-2w)  t2 must also be satisfied. Thus,

||H  || G i 1Gi  | 2w | , Gi 1  Gi

(II.8.02).

Hence, for any i,

||H  || ||H  || Gi 1Gi  2w  , 2 tmax Gi 1  Gi

(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 be proportional to the correspondent strain dyadic through the Green's tetradic of this solid is that, for any strain dyadic: 1) - the quotient of the stress dyadic norm to the double of the modulus of the difference between the extreme tetradic eigenvalues do not surpass the double the respective specific density energy stored in the vicinity of that point; 2) - the quotient of the sum of the stress dyadic norm with the product of two any consecutive eigenvalues to the modulus of the sum of these eigenvalues is not less than twice the respective specific energy density stored in the vicinity 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 Hooke's tetradic or Green's isotropic tetradic). Hence, from (II.8.01), it follows that ||  || 2w(3  2) ; and (II.8.02) generates || || 2w  2 and

||  || (3  22  2w(3  4) . These inequalities must be physically interpreted. Analogous inequalities may be derived for materials with other symmetries.

REFERENCES 1 2 3 4 5 6 7 8

I.1

Drew, T. B., Handbook of vector and Polyadic Analysis, Reinhold Pub. Corp., 1961. Gibbs, J. W. and Wilson, E. B., Vector Analysis, Yale University Press, New Haven, 1901. Helbig, K., Foundations of Anisotropy for Exploration Seismic, Handbook of Geophysical Exploration, vol. 22, Pergamon, 1994. Jaeger, J. C., Elasticity, Fracture and Flow, Methuen, 1969. 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. Mehrabadi M. M. and Cowin, S. C., Eigentensors of Linear Anisotropic Materials, Quarterly Jounal of Mechanics and Applied Mathematics.,vol. 43, Pt. 1, 1990. Ruggeri, E. R. F., Introdução à Teoria do Campo, Imprensa da UFOP (Universidade Federal de Ouro Preto), 1984. Ruggeri, E. R. F., Fundamentals of Polyadic Calculus, under way, registered in the Biblioteca Nacional do Rio de Janeiro under number 173298, book 291, sheet 445 on may/1999; to be printed.

Physical Magnitudes, Polyadics and Euclidean Space.

I.2 - Physical laws, Linear Transformations and Polyadic Geometry.

I.3 - Linear Transformations, Experimental Measurements and Statistical Polyadics.

I.4 - The Physical Phenomenon is Equivalent to a System of Linear Polyadic Equations.

I.5 - Eigenvalues and Eigenpolyadics.

II.1 - A particular situation, largely useful in Physics. II.2 – The

G Characteristic Elements or Eigensystems. 9

II.3 – The stationary proportionality polyadic specific radial value (2w). II.4 - The Projection Norm and Octahedral Directions.

II.5 - The Transversal Value of the Proportionality Polyadic. II.6 - Cauchy's and Lamé's quadric.

II.7 - Graphical Representations. 17 II.7.2 - H  varies in a 4-space, 5-space .... II.8 - A Criterion of Proportionality. 22