Department of Polymer Engineering, The University of Akron,. Akron, Ohio ... nematic âN-operatorsâ build up on the additive group of traceless second rank 3D.
Algebraic theory of linear viscoelastic nematodynamics Part 1: Algebra of nematic operators Arkady I. Leonov Department of Polymer Engineering, The University of Akron, Akron, Ohio 44325-0301, USA. Abstract This first part of the paper develops algebraic theory of linear anisotropic, six-parametric nematic “N-operators” build up on the additive group of traceless second rank 3D tensors. These operators have been implicitly used in continual theories of nematic liquid crystals and weakly elastic nematic elastomers. It is shown that there exists a noncommutative, multiplicative group N6 of N-operators build up on a manifold in 6D space of parameters. Positive N-operators, which in physical applications holds thermodynamic stability constraints, form a subgroup of group N6 on a more complicated manifold in parametric space. A three-parametric, commutative transversal-isotropic subgroup S3 ⊂ N6 of positive symmetric nematic operators is also briefly discussed. The special case of singular, non-negative symmetric N-operators reveals the algebraic structure of nematic soft deformation modes.
1
Introduction One of the central problems in theoretical descriptions of liquid crystalline polymers (LCP) is a lack of general continual (or ”field”) theory, which could consistently describe their specific viscoelastic nematic properties. It seems that this situation is caused by using an awkward common tensor/matrix form of operations in nematic theories (e.g. see [1-3]), which does not allow displaying their simple algebraic structure. It makes difficult (if possible) elaborating a general theory of linear or weakly nonlinear dynamic behavior of LCP’s. This paper reveals the algebraic structure of the continual nematic theories and presents it in a simple form. This will allow us to demonstrate in the second part of the paper some remarkable features of linear nematic viscoealsticity of LCP. 1.1. Definitions and general properties Consider a set X of traceless 3D second rank Cartesian tensors x = {xij } ∈ X : tr x = 0. The set X is forms an additive group defined on the field of real numbers. The group X is naturally decomposed in the sum, X = X s + X a , ( X s ∩ X a = 0 ) of two additive
subgroups of symmetric X s and asymmetric X a matrices, so that ∀ x ∈ X : x = x s + x a , where x s ∈ X s and x a ∈ X a . We introduce on X linear, axially symmetric operations transforming X into itself. The axial symmetry is characterized by a given unit vector n (director), disposed arbitrarily relative to axes of a chosen Cartesian coordinate system. A linear operation, invariant relative to transformation n → − n , is called nematic operation (or simply Noperator). The implicit definition of N-operation in the common tensor presentation is: y = r0 x s + r1[ nn ⋅ x s + x s ⋅ nn − 2nn( x s : nn)] + r2 ( nn − δ / 3)( x s : nn) + r3 (nn ⋅ x a − x a ⋅ nn) s
y = r4 ( nn ⋅ x s − x s ⋅ nn) − r5 ( nn ⋅ x a + x a ⋅ nn).
(1.1)
a
Here (nn)ij = ni n j , the symbol “ ⋅ ” means the tensor (matrix) multiplication, “:” the trace operation, δ is the unit tensor, and rk are the six independent ordered basis parameters, characterizing the operation in (1.1), denoted as: r = (r0 , r1 ,..., r5 ) . It is seen that
2
y ∈ X s , y ∈ X a , i.e. the operation (1.1) transforms X → X . Relations (1.1) have been s
a
used as constitutive equations for viscous and weakly elastic nematic cases [1-5]. The a
possible term ~ x was excluded from (1.1) due to physical arguments (e.g. see [3-5]), We now denote the N-operator as N r (n) , and symbolically present (1.1) as
∀ x, y ∈ X : y = N r (n)i x . Although the parameters rk are generally independent, there is still one physically significant, “Onsager” case of their relation, r4 = − r3 . So we define the Onsager N-operator (or ON-operator) as: N or (n) ≡ N r (n)
r4 =− r3
.
For any N-operator N r (n) we can introduce on X the quadratic form, a scalar P , defined as: P ≡ xiN r (n)i x ≡ r0 x
s 2
+ 2r1 nn : x s + (r2 − 2r1 )(nn : x s ) 2 − 4r3* nn : ( x s ⋅ x a ) − 2r5 nn : x a . (1.2) 2
2
r3* ≡ (r3 − r4 ) / 2 .
In case of ON operator N or (n) , when P → P o , the quadratic form P o has potential properties: 2 y = ∂P o / ∂ x s , 2 y = ∂P o / ∂ x a . s
a
Operator N r (n) is called positive if ∀ x ∈ X : P ≡ xiN r (n)i x > 0 . The same holds
for ON operator N or (n). Theorem 1. (i) The N-operator N r (n) is positive ( P > 0 ) iif: r ∈ R6+ : r0 > 0;
r0 + r1 > 0; 3 / 2r0 + r2 > 0; ( r0 + r1 ) r5 > ( r3 − r4 ) 2 / 4 ≡ r3*2 . (1.31,2,3,4)
(ii) Any positive N-operator N r (n) has inverse, N −r 1 ( n). Proof Using an orthogonal transformation, we choose the coordinate system, whose axis 1 is directed along the director. In this coordinate system (1.1,2), written in the component form, are reduced to: s s y11s = (r0 + 2 / 3r2 ) x11 ; y22 =r0 x22 − x11r2 / 3; y33s =r0 x33 − x11r2 / 3; y23 = r0 x23
⎧ y1sk = (r0 + r1 ) x1sk + r3 x1ak ⎨ a s a ⎩ y1k = − r4 x1k + r5 x1k
(k = 2,3)
(1.11a) (1.11b)
3
P = (3 / 2r0 + r2 ) x112 + 2r0 [( x22 + x11 / 2) 2 + xs223 ]+2 ∑ [(r0 + r1 ) xs21k + 2r3* xa1k xs1k + r5 xa21k ] (1.21) k = 2,3
Here the traceless condition, x11 + x22 + x33 = 0 , has been used to exclude x33 from (1.21). Demanding P > 0 and using independence of terms in (1.2) yields inequalities (1.3). Note that the inequality (1.34) yields the inequality: ( r0 + r1 ) r5 > − r3 r4 .
(1.3.5)
When (1.31,2,3,5) holds for equations (1.11), there is a unique linear dependence of xij on yij , which means the existence of a unique inverse operation N −r 1 ( n).
Remark 1.1 Theorem 1 holds for ON-operators N or (n) when r3* = r3 ( P → P o ) . To make inverse resolution of equations (1.1) it is not necessary to use in the parametric space, r ∈ R6 , the manifold R6+ with inequalities (1.31-4). The necessary and sufficient conditions for this resolution are: r0 ≠ 0, 3 / 2r0 + r2 ≠ 0, ( r0 + r1 ) r5 + r3 r4 ≠ 0 . Nevertheless, it is more convenient to use in the following the positive conditions of the resolution: r ∈ R6+ : r0 > 0;
3 / 2r0 + r2 > 0; ( r0 + r1 ) r5 + r3 r4 > 0 .
(1.31,3,5)
N-operator is called N+-operator if its basis parameters satisfy less restrictive inequalities (1.31,3,5) than (1.31-4), valid for positive N-operators. Evidently, R6+ ⊆ R6+ , i.e. N+operators are not necessarily positive though any positive N-operator is N+-operator. 1.2. Basis N-operators and their multiplicative properties The tensor/matrix presentations [6] of basis N-operators a k (n) (k=0,1,...,5) in (1) is explicitly defined via fourth rank numerical tensors (or simply 4-tensors) {a k (n)}ijαβ as:
{a0}ijαβ = aij(0) αβ = 1/ 2(δ iα δ j β + δ i β δ jα − 2 / 3δ ijδ αβ )
(1.41)
{a1 (n)}ijαβ = aij(1)αβ = 1/ 2(δ iα⊥ n j nβ + δ j⊥α ni nβ + δ i⊥β n j nα + δ j⊥β ni nα ) δ ij⊥ = δ ij − ni n j
(1.42)
{a 2 (n)}ijαβ = aij(2) αβ = ( ni n j − 1/ 3δ ij )( nα nβ − 1/ 3δαβ )
(143)
{a3 (n)}ijαβ = aij(3)αβ = 1/ 2(δ iα n j nβ + δ jα ni nβ − δ iβ n j nα − δ jβ ni nα )
(1.44)
4
{a 4 (n)}ijαβ = aij(4) αβ ( n) = 1/ 2(δ iα n j nβ + δ iβ n j nα − δ jα ni nβ − δ j β ni nα ) = {a3 ( n)}αβ ij
(1.45)
{a5 (n)}ijαβ = aij(5)αβ (n) = 1/ 2(δ iβ n j nα − δ iα n j nβ + δ jα ni nβ − δ jβ ni nα )
(1.46)
The basis 4-tensors in (41-6), are traceless with respect of the first and the second pairs of indices, i.e. {a k ( n)}iiαβ = {a k ( n)}ijαα = 0 (k = 0,..,5). The following symmetry properties hold for the basis 4-tensors: {a k ( n)}ijαβ = {a k ( n)} jiαβ = {a k ( n)}ijβα = {a k ( n)}αβ ij
(k = 0,1,2)
(1.51)
{a3 (n)}ijαβ = {a3 (n)} jiαβ = −{a3 (n)}ij βα
(1.52)
{a4 (n)}ijαβ = −{a3 (n)} jiαβ = {a3 (n)}ij βα
(1.53)
{a5 ( n)}ijαβ = −{a5 ( n)} jiαβ = −{a5 ( n)}ij βα = {a5 ( n)}αβ ij .
(1.54)
Formulae (1.41-3) and (1.51) show that the 4-tensors {a k ( n)}ijαβ (k = 0,1,2) are symmetric relative to transposition of the first and second indices, the third and fourth indices, as well as the first and second pairs of indices; the tensor {a3 ( n)}ijαβ is symmetric relative to transposition of the first and second indices, and skew symmetric when transposing the third and fourth indices; the tensor {a 4 ( n)}ijαβ is asymmetric relative to transposition of the first and second indices, and symmetric when transposing the third and fourth indices; and the tensor {a5 ( n)}ijαβ is skew symmetric relative to transposition of the first and second, as well as of the third and fourth indices, and symmetric when transposing the first and second pairs of indices. The symmetry properties of the above 4-tensors a k (n) (k = 0,..,5) show that they represent irreducible (and therefore linearly independent) set of traceless 4-th rank tensors. Therefore thy are called basis tensors. The products of the tensors of different ranks are defined as: s) (r ) r) a s (n)iar (n) ⇒ aij( αβ aβανγ , ar (n)i x ⇒ aij( αβ xβα .
(1.6)
The definitions in (6) disclose the sense of operation “ i ” symbolically used in Section 2. The products a s (n)iar (n) , established directly are presented in Table 1. Table 1. Products of basis tensors ai ia j
5
0
1
2
3
4
5
0
a0
a1
a3
0
0
1
a1
0
a2 0
a3 0 0
0
2 3
a1 0 0
a2 0
0
0
4
a4 0
a4 0
−a1 0
a3 0
a4
a5
j i
5
(2/3) a 2 0
−a 5 0
0 0
It is seen that except i,j = 0,1,2, the multiplication of basis tensors is non-commutative, for example, a0 ia3 = a3 ≠ a3 ia 0 = 0 . 1.3. Multiplicative group of N-operators Using basis operators a k (n) , equation (1.1) can be rewritten in the operator form: y = N r ( n )i x,
5
N r ( n) ≡ ∑ rk a k ( n),
(1.71)
k =0
or equivalently as: 2
y = ∑ rk a k (n)i x s + r3a3 ( n)i x a , s
y a = r4a 4 ( n)i x s + r5a5 ( n)i x a .
k =0
(1.72)
The product of two N-operators is defined in the common way:
N p ( n) ≡ N q ( n)i N r ( n) =
5
∑qra
k ,m=0
k m k
6
(n)ia m (n) = ∑ pk a k (n).
(1.8)
k =0
With the use of multiplicative Table 1, the basic scalars pk for resulting operation are found from the fundamental equation: p0 = q0 r0 , p1 = q0 r1 + q1r0 + q1r1 − q3 r4 , p2 = q2 r0 + q0 r2 + 2 / 3q2 r2 p3 = (q0 + q1 )r3 + q3 r5 , p4 = q4 (r0 + r1 ) + q5 r4 , p5 = − q4 r3 + q5 r5
(1.9)
Even in the Onsager case, when q4 = −q3 and r4 = −r3 , generally p4 ≠ − p3 . It means that N p (n) ≠ Nop (n) i.e. that the product of two ON-operators is not an ON-operator. Theorem 2
6
The set of N+-operators N r (n) , whose basis parameters r ∈ R6+ satisfy inequalities
(1.31,3,5), constitute a non-commutative, multiplicative, six-parametric group
N6, which
has the fundamental group equations (1.9). Proof
(i) The definition and basic properties of the unit N-operator I(n) due to the Table 1 are: I (n) = a 0 + a 5 (n), N r ( n)iI ( n) = I ( n)i N r ( n) = N r ( n)
(1.10)
Because of (1.2),(1.21) the unit N-operator is positive, and therefore it is a N+operator. (ii) If N r (n) is N+-operator, its inverse, N −r 1 (n) ≡ N rˆ (n) , existing due to theorem 1, should satisfy the common condition, N rˆ (n)iN r (n) = N r (n)iN rˆ (n) = I(n) which for parameters in (9) yields: p0 = 1, p1 = p2 = p3 = p4 = 0, p5 = 1 .
(1.11)
The basis parameters rˆk of inverse N-operator N rˆ (n) are found using (1.11) and (1.9) as: rˆ0 =
1 , r0
rˆ1 =
−(r3 r4 + r1r5 ) / r0 , r5 (r0 + r1 ) + r3 r4
rˆ2 =
−r2 / r0 r0 + 2 / 3r2
(1.12)
− r3 −r4 r0 + r1 rˆ3 = , rˆ4 = , rˆ5 = r5 (r0 + r1 ) + r3 r4 r5 (r0 + r1 ) + r3 r4 r5 (r0 + r1 ) + r3r4
With the use of inequalities (1.31,2,3,4) and (1.12) it is checked directly that N −r 1 (n) is a N+-operator. If N r (n) is positive, N −r 1 (n) is positive too. (iii) The product N p (n) = N r (n)iN q (n) of two N+ operators is N+ operator, because ∀r,q ∈ R6+ : p ∈ R6+ . This follows from the direct calculations with the use of (1.9):
p0 = q0 r0 > 0, 3 / 2 p0 + p2 = 3 / 2(q0 + 2 / 3q2 )(r0 + 2 / 3r2 ) > 0 p5 ( p0 + p1 ) + p4 p3 = [q5 (q0 + q1 ) + q4 q3 ][r5 (r0 + r1 ) + r4 r3 ] > 0
.
(1.13)
The properties of N+-operators established in (i)-(iii) prove theorem 2. Remark 2.1
Due to (1.9), p0 + p1 = (q0 + q1 )(r0 + r1 ) − q3 r4 , p5 = q5 r5 − q4 r3 , therefore the product of two positive N+-operators is positive only under additional constraints: q3 r4 , q4 r3 < 0 .
7
Remark 2.2 As a consequence of Remark 2.1, the product of two positive ON-operators is generally N+-operator, which is positive only under additional constraint, q3 r3 > 0 . Remark 2.3 The multiplicative group
N6 of N+-operators also constitutes the additive group relative
to the addition of its elements, where N q (n) + N r (n) = N q + r (n) . Therefore the set N+operators constitutes the associative ring Ń6 relative to both, addition and multiplication operations. In the following we are interested only in the multiplicative properties of Noperators. Example: Dual N-operations Two linear nematic transformations, z = N q ( n)i y = N r ( n)i x where N r (n) and N q (n) are N-operators, are called dual. When both the N r (n) and N q (n) are of N+ type, there always are the unique dependences, y = N p ( n)i x and x = N pˆ ( n)i y where
N p (n) = N q−1 (n)iN r (n), N pˆ (n) = N −p1 (n) = N −r 1 (n)iN q (n).
(1.14)
Formulae (1.9) and (1.12) express the basis scalars p and pˆ for dual N+-operators via given basis scalars r and q . In case N p (n) = N oq−1 (n)iN or (n), where N or (n) and N oq (n) are
positive ON-operators, the parameters p are: p0 =
r0 , q0
p1 =
(r0 + r1 )q5 − r3 q3 r0 − , q5 (q0 + q1 ) − q32 q0
p2 =
r2 q0 − r0 q2 , q0 (q0 + 2 / 3q2 )
r (q + q ) − q3 (r0 + r1 ) r5 q3 − r3 q5 r (q + q ) − r3 q3 , p4 = , p5 = 5 0 1 p3 = 3 0 1 2 2 q5 (q0 + q1 ) − q3 q5 (q0 + q1 ) − q3 q5 (q0 + q1 ) − q32
(1.15)
Due to (1.14) respective formulae for the basis parameters pˆ of inverse dual N+ operation are obtained from (1.15) by substitution r ↔ q . Applying (1.13) to the terms in (1.15) for positive ON-operators N or (n) and N oq (n) , shows that in this case the sufficient condition for N p (n) to be positive is: r3 q3 < 0 . 1.4. Spectral properties of N-operators
8
This Section considers the case of non-degenerating (NG) N r (n) operators whose basis
parameters r do not vanish: rk ≠ 0 (k = 0,1, 2,3, 4,5) .
(1.16)
The spectral problem for a NG operator N r (n) is formulated in the standard way:
N r (n)i x = ν x , or [N r (n) −ν I(n)]i x = 0 .
(1.171)
5
Here N r (n) = ∑ rk a k (n) , I (n) = a 0 + a5 (n) , ν being a generally complex eigenvalue, and k =0
x(ν ) ∈ X is a respective “eigentensor”. Theorem 3 (i) The spectral points (egenvalues) of problem (1.171) for any NG operator N r (n) , are:
ν 1 = r0 , ν 2 = r0 + 2 / 3r2 , ν 3 = 1/ 2(r0 + r1 + r5 + d ) , ν 4 = 1/ 2(r0 + r1 + r5 − d ) d 2 = (r0 + r1 + r5 )2 − 4[r5 (r0 + r1 ) + r3 r4 ] ≡ (r0 + r1 − r5 ) 2 − 4r3r4 .
(1.18)
(ii) The corresponding eigentensors x(ν k ) are found as x(ν k ) = Q(ν k , n)i x 0 where x 0 is a given tensor, and the “eigenoperators” Q(ν k , n) are: Q(ν 1 , n) = c1 (a0 − a1 − 3 / 2a 2 ), Q(ν 2 , n) = c2a 2 , Q(ν 3 , n) = c3 (a1 − λ1a3 ) + c4 (a4 + λ1a5 ) Q(ν 4 , n) = c5 (a1 − λ2a3 ) + c6 (a4 + λ2a5 )
{λ1 = (r0 + r1 −ν 3 ) / r4 , λ2 = (r0 + r1 −ν 4 ) / r4 } (1.19)
(iii) In case of the dual operator N p (n) consisting of two ON positive operators, with the parameters pk defined in (1.15), all eigenvalues in (1.18) are real positive. Proof (i) A common, analytical continuation ν → νˆ is used to find eigenvalues. Substituting r0 → r0 −νˆ and r5 → r5 −νˆ , and using (1.12) results in formal finding the basis parameters rˆk (νˆ, rk ) of operator N −r 1 (νˆ, n) . The eigenvalues (1.18) are then found as singular points νˆ = ν for the basis parameters rˆk (νˆ, rk ) of the inverse operator N −r 1 (νˆ, n) . 5
(ii) Instead of x(ν ) , the N-“eigenoperator” Q(ν , n) = ∑ qk (ν )a k with known eigenvalues k =0
ν k , is now searched from the equation: N r (n)iQ(ν , n) −ν Q(ν , n) ≡ N r (ν , n)iQ(ν , n) = 0 ; N r (ν , n) = N r (n) −ν I (n) .
(1.172) 9
Evidently, the tensor x = Q(ν , n)i x 0 with x 0 being any given tensor is an eigentensor,
because it identically satisfies (1.232). To find the solution (1.232) one can use (1.9) with r0 → r0 −ν , r5 → r5 −ν . Demanding then due to (1.172) p = 0 , yields:
(r0 +ν )q0 = 0, (r0 + r1 −ν )q1 + r1q0 − r4 q3 = 0, (r0 + 2 / 3r2 −ν )q2 + r2 q0 = 0 (q0 + q1 )r3 + q3 (r5 −ν ) = 0, (r0 + r1 −ν )q4 + r4 q5 = 0, − r3q4 + (r5 −ν )q5 = 0 For each value of ν from (1.18), the parameters q (ν ) of eigenoperator Q(ν , n) are found
from the above equations as: q(ν 1 ) = c1 (1, −1, −3 / 2, 0, 0, 0), q(ν 2 ) = c2 (0, 0,1, 0, 0, 0), q(ν 3 ) = (0, c3 , 0, λ1c3 , c4 , −λ1c4 ) q(ν 4 ) = (0, c5 , 0, λ2 c5 , c6 , −λ2 c6 )
{λ1 = (r0 + r1 −ν 3 ) / r5 , λ2 = (r0 + r1 −ν 4 ) / r5 }
(1.20)
Here c1 , c2 ,..., c6 are arbitrary constants. Additional solutions that use specific relations between parameters rk have been rejected because they are not robust. (iii) In case of the dual operator N p (n) consistent of two ON positive operators, one should use in (24) the substitution rk → pk (ri , q j ) with pk presented in (1.15). The proof is given in the following steps: 1) ν 1 = p0 = r0 / q0 > 0 ; 2) ν 2 = p0 + 2 / 3 p2 = (r0 + 2 / 3r2 ) /(q0 + 2 / 3q2 ) > 0 ; 3) p0 + p1 + p5 =
(r0 + r1 )q5 + (q0 + q1 )r5 − 2r3 q3 > 0 , because q5 (q0 + q1 ) − q32
(r0 + r1 )q5 + (q0 + q1 )r5 − 2q3 r3 ≥ 2[(r0 + r1 )r5 (q0 + q1 )q5 ]1/ 2 − 2q3 r3 > 2( q3r3 − q3r3 ) ; [(r0 + r1 )q5 − (q0 + q1 )r5 ]2 + 4Φ 4) d = , Φ = r5 (r0 + r1 )r32 g ( x), 2 2 [q5 (q0 + q1 ) − q3 ] 2
g ( x) = ( x − α )( x − β ) ( x ≡ q3 / r3 ) , α = (q0 + q1 ) /(r0 + r1 ),
β = q5 / r5 .
5) min g ( x) = g ( xm ) = g{1/ 2(α + β )} = −1/ 4(α − β ) 2 . 6) d 2 ≥
[(r0 + r1 )q5 − (q0 + q1 )r5 ]2 + 4 min Φ [(r0 + r1 )q5 − (q0 + q1 )r5 ]2 [q5 (q0 + q1 ) − q32 ] = > 0. [q5 (q0 + q1 ) − q32 ]2 r5 (r0 + r1 )[q5 (q0 + q1 ) − q32 ]2
While the above dual N p (n) operator is generally not positive, its eigenvalues are positive. Remark 3.1
10
Along with the part (iii) of Theorem 3.1, easy analysis reveals various cases of behavior of eigenvalues ν k in (1.18). (i) In general case of NG operators N r (n) , the egenvalues ν 1 ,ν 2 are real but generally
have arbitrary signs, the egenvalues ν 3 ,ν 4 being generally complex and conjugated. (ii) In case of N+ operators when N r (n) ∈ N6, the egenvalues ν 1 ,ν 2 are positive, the egenvalues ν 3 ,ν 4 being generally complex and conjugated. (iii) In case of positive N+ operators, the egenvalues ν 1 ,ν 2 are positive, the egenvalues
ν 3 ,ν 4 being generally complex and conjugated, with Re(ν 3 ,ν 4 ) > 0 . All eigenvalues ν k in (1.24) are real positive if r3r4 < 0 , as in particular case of positive ON operators where r4 = − r3 . Remark 3.2
Arbitrary parameters ck in (1.15) can be established from various physical conditions. One of them is: 4
∑ Q(ν , n) = I(n) = a i =1
i
0
+ a 5 ( n) .
(1.21)
Using (1.20) and (1.21) yields: c1 = 1, c2 = 3 / 2, c3 = (r0 + r1 −ν 4 ) / d , c4 = − r4 / d , c5 = (ν 3 − r0 − r1 ) / d , c6 = r4 / d (1.22)
Here parameters rk ,ν 3 ,ν 4 , and d have been defined in (1.18). 1.5. Symmetric N-operators: Transversal isotropy (TI)
1.5.1. General properties
This Section briefly describes the nematic operations on the subgroup X s ⊂ X of traceless second rank symmetric tensors x s ∈ X s : tr x s = 0. Therefore the lower index “s” is omitted here. The linear symmetric N-operator S r (n) on X s is defined as: 2
2
k =0
k =0
y = S r (n)i x = ∑ rk a k ( n)i x, or S r ( n) = ∑ rk a k (n)
(1.23)
11
Here a k (n) are the basis tensors defined in (1.4), and rk are the real-valued basic scalar ordered parameters {r}, characterizing operation. The common tensor presentation of symmetric operation (1.1), and corresponding quadratic form Ps are: y = r0 x + r1[nn ⋅ x + x ⋅ nn − 2nn( x : nn)] + r2 (nn − δ / 3)( x : nn) .
Ps ≡ xiS r (n)i x ≡ r0 x
s 2
+ 2r1 nn : x s + (r2 − 2r1 )(nn : x s ) 2 . 2
(1.24)
(1.25)
Equations (1.23) and (1.24) could also be obtained from (1.1) when ya ≡ 0 , using the normalizing procedure [4,5].
In this case the second equation in (1.1) is used for
expressing xa via xs and n . Substituting this dependence xa = xa ( xs , n) into (1) results in the equations (1.23), (1.24). As seen, this procedure does not violate the non-degrading conditions (1.16). Relation (1.30) shows that symmetric N-operators are transversally
isotropic. Therefore they are called TI-operators.
1.5.2. Multiplicative group
Due to the Table1, the products of TI-operators are commutative: S p ( n ) = S r ( n ) iS q ( n ) = S q ( n ) i S r ( n ) .
(1.26)
Here the basis parameters pk are found from the fundamental equation:
p0 = r0 q0 , p1 = r0 q1 + r1q0 + r1q1 , p2 = r0 q2 + r2 q0 + 2 / 3r2 q2 .
(1.27)
A TI-operator is called positive if ∀ x ∈ X the quadratic form Ps = xiS r (n)i x > 0 .
Theorem 4 TI operator S r (n) is positive iif
r0 > 0, r0 + r1 > 0, r0 + 2 / 3r2 > 0 .
(1.28)
Proof is the same as for the Theorem 1. Theorem 5
The set of positive TI operators constitutes commutative three parametric TI group S3. Proof
(i) Direct calculations show that the product (1.26) of two positive TI-operators is positive. (ii) Direct calculations show that the unit TI operation is I = a0 , so 12
∀S r (n) : I iS r (n) = S r (n)iI = S r (n) S r (n) . (iii) The basis scalar parameters rˆk of inverse positive TI-operator S −r 1 (n) ≡ S rˆ (n) are found from the relation S −r 1 ( n)iS r ( n) = S rˆ ( n)iS r ( n) = a 0 ( n) as: rˆ0 =
r /r 1 , rˆ1 = − 1 0 , r0 r0 + r1
rˆ2 = −
r2 / r0 . r0 + 2 / 3r2
(1.29)
Due to (1.28), (1.29), any positive TI-operator S r (n) has inverse positive. The dual linear transformations, z = S q ( n )i y = S r ( n )i x with positive TI operators S q ( n ) and S r ( n ) define the direct y = S p (n)i x , and inverse, x = S pˆ (n)i y , linear relations,
where S p ( n ) = S q−1 ( n )iS r ( n ) , S pˆ ( n ) = S −p1 ( n ) = S −r 1 ( n )iS q ( n ) ,
(1.30)
and parameters p are: p0 =
r0 , q0
p1 =
r1q0 − q1r0 , q0 + q1
p2 =
r2 q0 − q2 r0 . q0 + 2 / 3q2
(1.311)
Parameters pˆ of inverse operation found by substitution q ↔ r are: pˆ 0 =
q0 , r0
pˆ1 =
q1r0 − r1q0 , r0 + r1
pˆ 2 =
q2 r0 − r2 q0 . r0 + 2 / 3r2
(1.312)
1.5.3. Eigenvalue problem
The formulation of the eigenvalue problem, similar to (1.171) is:
[S r (n) −ν I]i x = 0 , or S r (n)iQ(ν , n) −ν Q(ν , n) ≡ S r (ν , n)iQ(ν , n) = 0 . 2
2
k =0
k =0
(1.32)
Here S r (n) = ∑ rk a k (n) ∈ S3, I = a0 , Q(ν , n) = ∑ qk (ν )a k (n) and x = x(ν ) ∈ X s . Theorem 6
∀N r ( n) ∈
S3, the spectral points (egenvalues) of problem (1.32) are: ν 1 = r0 , ν 2 = r0 + r1 , ν 3 = r0 + 2 / 3r2 .
(1.33)
The corresponding eigentensors x(ν k ) are found as x(ν k ) = Q(ν k , n)i x 0 where x 0 is a given tensor, and the “eigenoperators” Q(ν k , n) are given by:
13
Q(ν 1 , n) = c1 (a0 − a1 − 3 / 2a 2 ) , Q(ν 2 , n) = c2a1 , Q(ν 3 , n) = c3a 2 .
(1.34)
Proof employs the same technique as used in proof of Theorem 3. Remark 6.1 Due to (1,28), ν k > 0 . Remark 6.2 If the arbitrary parameters ck in (1.34) are found from the physical condition, 3
∑ Q(ν , n) = I(n) = a i =1
i
0
,
(1.35)
their values are: c1 = c2 = 1, c3 = 3 / 2 .
(1.36)
1.5.4. Singular TI operators Consider now the limiting marginal situation, when some inequalities in (1.34) turn out to be equalities. If once again non-degeneration conditions, rk ≠ 0 (k = 0,1, 2) , with r0 > 0 , are used, there might be only two independent marginal conditions: r0 + r1 = 0, r0 + 2 / 3r2 = 0
(1.371)
When one of these conditions is satisfied, TI operator is called partially soft. When both of them are satisfied, TI operator is called completely soft. In both the partially or complete soft cases, the quadratic form Ps is positively semi-definite (non-negative). The nearly marginal situation are defined as those that reduce (1.431) to: r0 + r1 = r0δ , r0 + 2 / 3r2 = 2 / 3r0κ (0 < δ , κ 0) with the parameters r , represented as: α (n) = a0 (n) − a1 (n) − 3 / 2a 2 (n) , r = r0 (1, −1, −3 / 2) .
(1.38)
It is seen that the marginal TI operator α(n) is singular, i.e. α −1 (n) does not exist. The operator α(n) has the property:
α(n)iα(n) = α (n).
(1.39)
14
Consider now a pair S r (n) = Sδ1 ,κ1 (n) and S q (n) = Sδ 2 ,κ 2 (n) of positive, nearly marginal TI operators, with parameters: r = (1, δ1 − 1, κ1 − 3 / 2) and q = (1, δ 2 − 1, κ 2 − 3 / 2) , where 0 < δ1 , κ1
