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# Polyadic formulation of linear physical laws

Published on: Mar 4, 2016
Published in: Technology

#### Transcripts - Polyadic formulation of linear physical laws

• 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 333133: ,or  31321VVVA 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 3333,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 . .
• 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 haveG1uu2uHHoctHoctHH2HHoctHoct 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 deduceG12u2E2H)(GG , (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 Hoct 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 G1 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: