Self-influence and influence pseudotensors of d-dimensional spheres

(1)

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Self-influence and influence pseudotensors of d-dimensional spheres

Sébastien Brisard, Luc Dormieux, Karam Sab

To cite this version:

Sébastien Brisard, Luc Dormieux, Karam Sab. Self-influence and influence pseudotensors of d-

dimensional spheres. 2013, pp.53. �hal-00876028v2�

(2)

Self-influence and influence pseudotensors of d-dimensional spheres

S. Brisard ^∗ L. Dormieux ^∗ K. Sab ^∗ December 30, 2013

This document is a companion to Ref. [1]. It gathers analytical expressions for the self-influence and influence pseudotensors of d-dimensional spheres (d = 2, 3).

1 How to read this document? 1

2 Self-influence pseudotensors of d-dimensional spheres 2

2.1 Self-influence pseudotensors of disks (d = 2) . . . . 2 2.2 Self-influence pseudotensors of spheres (d = 3) . . . . 3

3 Elementary influence pseudotensors of d-dimensional spheres 9

3.1 Elementary influence pseudotensors of disks (d = 2) . . . . 9 3.2 Elementary influence pseudotensors of spheres (d = 3) . . . . 16

1 How to read this document?

Fourth-rank pseudotensors T

i jkl

are represented in tabular form as follows

Two-dimensional space (d = 2)



 



T 1111 T 1122 T 1112

T ₂₂₁₁ T ₂₂₂₂ T ₂₂₁₂ T ₁₂₁₁ T ₁₂₂₂ T ₁₂₁₂



 



Three-dimensional space (d = 3)



 

 

T ₁₁₁₁ T ₁₁₂₂ T ₁₁₃₃ T ₁₁₂₃ T ₁₁₃₁ T ₁₁₁₂ T ₂₂₁₁ T ₂₂₂₂ T ₂₂₃₃ T ₂₂₂₃ T ₂₂₃₁ T ₂₂₁₂ T 3311 T 3322 T 3333 T 3323 T 3331 T 3312

T 2311 T 2322 T 2333 T 2323 T 2331 T 2312

T 3111 T 3122 T 3133 T 3123 T 3131 T 3112

T 1211 T 1222 T 1233 T 1223 T 1231 T 1212



 

 

It should be noted that the above representations should be understood as arrays, not matrices. Indeed, none of the components have been premultiplied by 2 (resp.

√

2), as required by the Voigt (resp. Mandel) notation, which precludes performing linear algebra operations (such as matrix-vector products) on these tables.

This document follows closely the notations of Ref. [1]. In addition, a α and a β denote the radiuses of the spherical inclusions α and β, respectively. It is recalled that r is the center-to-center distance of the two inclusions. Also, µ 0 (resp.

∗

Universit´e Paris-Est, Laboratoire Navier (UMR 8205), CNRS, ENPC, IFSTTAR, F-77455 Marne-la-Vall´ee, France

(3)

ν 0 ) denotes the shear modulus (resp. Poisson ratio) of the reference material. For d = 2, plane strain elasticity is assumed;

plane stress elasticity is retrieved with the classical substitution ν 0 ! ν 0 (1 + ν 0 )

⁻¹

(leaving µ 0 unchanged).

Expressions of the self-influence pseudotensors S

^k

_α

^•^l^•

and influence pseudotensors T

^k

_αβ

^•^l^•

(r) are given for k

₊

= k 1 + · · · + k

d

≤ p and l

₊

= l 1 + · · · + l

d

≤ p, with p = 3 (d = 2) and p = 2 (d = 3); pseudotensors which are omitted below are identically null. In addition, it is assumed that r = re

d

for the influence pseudotensors; in other words, the influence pseudotensors are expressed in a local basis attached to the two inclusions under consideration. Such pseudotensors are called elementary. In the general case of an arbitrarily oriented center-to-center vector, a change of basis must be performed according to Ref. [1] to retrieve the influence pseudotensors from the elementary influence pseudotensors provided in the present document.

2 Self-influence pseudotensors of d-dimensional spheres

2.1 Self-influence pseudotensors of disks (d = 2)

S ⁰⁰⁰⁰ _α = π a ² _α 16 µ ₀ (1 − ν ₀ )



 



5 − 8 ν 0 −1 0

−1 5 − 8 ν 0 0

0 0 3 − 4 ν 0



 



S ⁰⁰⁰² _α = π a ⁴ _α 64 µ ₀ (1 − ν ₀ )



 



5 − 8 ν 0 −1 0

−1 5 − 8 ν 0 0

0 0 3 − 4 ν 0



 



S ⁰⁰²⁰ _α = π a ⁴ _α 64 µ 0 (1 − ν 0 )



 



5 − 8 ν 0 − 1 0

− 1 5 − 8 ν ₀ 0

0 0 3 − 4 ν ₀



 



S ⁰¹⁰¹ _α = π a ⁴ _α 64 µ 0 (1 − ν 0 )



 



3 − 4 ν ₀ −1 0

−1 7 − 12 ν ₀ 0

0 0 3 − 4 ν ₀



 



S ⁰¹¹⁰ _α = π a ⁴ _α (1 − 2 ν 0 ) 64 µ 0 (1 − ν 0 )



 



0 0 1 0 0 1 1 1 0



 



S ⁰²⁰⁰ _α = π a ⁴ _α 64 µ ₀ (1 − ν ₀ )



 



5 − 8 ν 0 −1 0

−1 5 − 8 ν 0 0

0 0 3 − 4 ν 0



 



S ⁰²⁰² _α = π a ⁶ _α 768 µ 0 (1 − ν 0 )



 



21 − 32 ν 0 −5 0

−5 37 − 64 ν 0 0

0 0 19 − 24 ν ₀



 



S ⁰²¹¹ _α = π a ⁶ _α 768 µ 0 (1 − ν 0 )



 



0 0 3 − 4 ν ₀

0 0 1 − 4 ν ₀

3 − 4 ν ₀ 1 − 4 ν ₀ 0



 



S ⁰²²⁰ _α = π a ⁶ _α 768 µ 0 (1 − ν 0 )



 



11 − 16 ν ₀ −3 0

−3 11 − 16 ν 0 0

0 0 5 − 8 ν 0



 



S ¹⁰⁰¹ _α = π a ⁴ _α (1 − 2 ν 0 ) 64 µ ₀ (1 − ν ₀ )



 



0 0 1 0 0 1 1 1 0



 



S ¹⁰¹⁰ _α = π a ⁴ _α 64 µ ₀ (1 − ν ₀ )



 



7 − 12 ν 0 −1 0

−1 3 − 4 ν 0 0

0 0 3 − 4 ν 0



 



S ¹¹⁰² _α = π a ⁶ _α 768 µ 0 (1 − ν 0 )



 



0 0 3 − 4 ν 0

0 0 1 − 4 ν ₀

3 − 4 ν ₀ 1 − 4 ν ₀ 0



 



(4)

S ¹¹¹¹ _α = π a ⁶ _α 768 µ 0 (1 − ν 0 )



 



11 − 16 ν 0 −3 0

− 3 11 − 16 ν ₀ 0

0 0 5 − 8 ν ₀



 



S ¹¹²⁰ _α = π a ⁶ _α 768 µ 0 (1 − ν 0 )



 



0 0 1 − 4 ν ₀

0 0 3 − 4 ν ₀

1 − 4 ν ₀ 3 − 4 ν ₀ 0



 



S ²⁰⁰⁰ _α = π a ⁴ _α 64 µ 0 (1 − ν 0 )



 



5 − 8 ν 0 −1 0

−1 5 − 8 ν 0 0

0 0 3 − 4 ν 0



 



S ²⁰⁰² _α = π a ⁶ _α 768 µ ₀ (1 − ν ₀ )



 



11 − 16 ν 0 −3 0

−3 11 − 16 ν 0 0

0 0 5 − 8 ν 0



 



S ²⁰¹¹ _α = π a ⁶ _α 768 µ 0 (1 − ν 0 )



 



0 0 1 − 4 ν 0

0 0 3 − 4 ν 0

1 − 4 ν 0 3 − 4 ν 0 0



 



S ²⁰²⁰ _α = π a ⁶ _α 768 µ 0 (1 − ν 0 )



 



37 − 64 ν ₀ − 5 0

−5 21 − 32 ν ₀ 0

0 0 19 − 24 ν ₀



 



2.2 Self-influence pseudotensors of spheres (d = 3)

S ⁰⁰⁰⁰⁰⁰ _α = 2 π a ³ _α 45 µ 0 (1 − ν 0 )



 

 

7 − 10 ν 0 −1 −1 0 0 0

−1 7 − 10 ν 0 −1 0 0 0

−1 −1 7 − 10 ν 0 0 0 0

0 0 0 4 − 5 ν 0 0 0

0 0 0 0 4 − 5 ν ₀ 0

0 0 0 0 0 4 − 5 ν ₀



 

 

S ⁰⁰⁰⁰⁰² _α = 2 π a ⁵ _α 225 µ 0 (1 − ν 0 )



 

 

7 − 10 ν ₀ −1 −1 0 0 0

−1 7 − 10 ν ₀ −1 0 0 0

−1 −1 7 − 10 ν ₀ 0 0 0

0 0 0 4 − 5 ν 0 0 0

0 0 0 0 4 − 5 ν 0 0

0 0 0 0 0 4 − 5 ν 0



 

 

S ⁰⁰⁰⁰²⁰ _α = 2 π a ⁵ _α 225 µ ₀ (1 − ν ₀ )



 

 

7 − 10 ν 0 −1 −1 0 0 0

−1 7 − 10 ν 0 −1 0 0 0

−1 −1 7 − 10 ν 0 0 0 0

0 0 0 4 − 5 ν 0 0 0

0 0 0 0 4 − 5 ν 0 0

0 0 0 0 0 4 − 5 ν 0



 

 

S ⁰⁰⁰²⁰⁰ _α = 2 π a ⁵ _α 225 µ 0 (1 − ν 0 )



 

 

7 − 10 ν ₀ − 1 − 1 0 0 0

−1 7 − 10 ν ₀ −1 0 0 0

−1 −1 7 − 10 ν ₀ 0 0 0

0 0 0 4 − 5 ν ₀ 0 0

0 0 0 0 4 − 5 ν ₀ 0

0 0 0 0 0 4 − 5 ν 0



 

 

S ⁰⁰¹⁰⁰¹ _α = 2 π a ⁵ _α 525 µ ₀ (1 − ν ₀ )



 

 

11 − 14 ν 0 −1 −3 0 0 0

−1 11 − 14 ν 0 −3 0 0 0

−3 −3 27 − 42 ν 0 0 0 0

0 0 0 11 − 14 ν 0 0 0

0 0 0 0 11 − 14 ν 0 0

0 0 0 0 0 6 − 7 ν 0



 

 

(5)

S ⁰⁰¹⁰¹⁰ _α = π a ⁵ _α 525 µ 0 (1 − ν 0 )



 

 

0 0 0 −2 0 0

0 0 0 8 − 14 ν ₀ 0 0

−2 8 − 14 ν ₀ 8 − 14 ν ₀ 0 0 0

0 0 0 0 0 5 − 7 ν ₀

0 0 0 0 5 − 7 ν ₀ 0



 

 

S ⁰⁰¹¹⁰⁰ _α = π a ⁵ _α 525 µ ₀ (1 − ν ₀ )



 

 

0 0 0 0 8 − 14 ν 0 0

0 0 0 0 −2 0

0 0 0 0 8 − 14 ν 0 0

0 0 0 0 0 5 − 7 ν 0

8 − 14 ν 0 −2 8 − 14 ν 0 0 0 0

0 0 0 5 − 7 ν 0 0 0



 

 

S ⁰⁰²⁰⁰⁰ _α = 2 π a ⁵ _α 225 µ 0 (1 − ν 0 )



 

 

7 − 10 ν 0 −1 −1 0 0 0

−1 7 − 10 ν 0 −1 0 0 0

−1 −1 7 − 10 ν 0 0 0 0

0 0 0 4 − 5 ν ₀ 0 0

0 0 0 0 4 − 5 ν ₀ 0

0 0 0 0 0 4 − 5 ν ₀



 

 

S ⁰⁰²⁰⁰² _α = 2 π a ⁷ _α 11025 µ 0 (1 − ν 0 )



 

 

107 − 146 ν ₀ −13 −21 0 0 0

−13 107 − 146 ν ₀ −21 0 0 0

−21 −21 211 − 338 ν ₀ 0 0 0

0 0 0 100 − 121 ν 0 0 0

0 0 0 0 100 − 121 ν 0 0

0 0 0 0 0 60 − 73 ν 0



 

 

S ⁰⁰²⁰¹¹ _α = 4 π a ⁷ _α 11025 µ ₀ (1 − ν ₀ )



 

 

0 0 0 −1 0 0

0 0 0 9 − 12 ν 0 0 0

0 0 0 4 − 12 ν 0 0 0

−1 9 − 12 ν 0 4 − 12 ν 0 0 0 0

0 0 0 0 0 5 − 6 ν 0

0 0 0 0 5 − 6 ν 0 0



 

 

S ⁰⁰²⁰²⁰ _α = 2 π a ⁷ _α 11025 µ 0 (1 − ν 0 )



 

 

57 − 86 ν ₀ − 7 − 7 0 0 0

−7 49 − 62 ν ₀ −11 0 0 0

−7 −11 49 − 62 ν ₀ 0 0 0

0 0 0 20 − 31 ν ₀ 0 0

0 0 0 0 30 − 37 ν ₀ 0

0 0 0 0 0 30 − 37 ν 0



 

 

S ⁰⁰²¹⁰¹ _α = 4 π a ⁷ _α 11025 µ ₀ (1 − ν ₀ )



 

 

0 0 0 0 9 − 12 ν 0 0

0 0 0 0 −1 0

0 0 0 0 4 − 12 ν 0 0

0 0 0 0 0 5 − 6 ν 0

9 − 12 ν 0 −1 4 − 12 ν 0 0 0 0

0 0 0 5 − 6 ν 0 0 0



 

 

S ⁰⁰²¹¹⁰ _α = 2 π a ⁷ _α 11025 µ 0 (1 − ν 0 )



 

 

0 0 0 0 0 6 ν 0 − 2

0 0 0 0 0 6 ν ₀ − 2

0 0 0 0 0 − 2

0 0 0 0 3 ν ₀ − 5 0

0 0 0 3 ν ₀ − 5 0 0

6 ν ₀ − 2 6 ν ₀ − 2 −2 0 0 0



 

 

S ⁰⁰²²⁰⁰ _α = 2 π a ⁷ _α 11025 µ ₀ (1 − ν ₀ )



 

 

49 − 62 ν ₀ −7 −11 0 0 0

−7 57 − 86 ν 0 −7 0 0 0

−11 −7 49 − 62 ν 0 0 0 0

0 0 0 30 − 37 ν 0 0 0

0 0 0 0 20 − 31 ν 0 0

0 0 0 0 0 30 − 37 ν 0



 

 

(6)

S ⁰¹⁰⁰⁰¹ _α = π a ⁵ _α 525 µ 0 (1 − ν 0 )



 

 

0 0 0 −2 0 0

0 0 0 8 − 14 ν ₀ 0 0

−2 8 − 14 ν ₀ 8 − 14 ν ₀ 0 0 0

0 0 0 0 0 5 − 7 ν ₀

0 0 0 0 5 − 7 ν ₀ 0



 

 

S ⁰¹⁰⁰¹⁰ _α = 2 π a ⁵ _α 525 µ ₀ (1 − ν ₀ )



 

 

11 − 14 ν 0 −3 −1 0 0 0

−3 27 − 42 ν 0 −3 0 0 0

−1 −3 11 − 14 ν 0 0 0 0

0 0 0 11 − 14 ν 0 0 0

0 0 0 0 6 − 7 ν 0 0

0 0 0 0 0 11 − 14 ν 0



 

 

S ⁰¹⁰¹⁰⁰ _α = π a ⁵ _α 525 µ 0 (1 − ν 0 )



 

 

0 0 0 0 0 8 − 14 ν 0

0 0 0 0 0 −2

0 0 0 0 5 − 7 ν ₀ 0

0 0 0 5 − 7 ν ₀ 0 0

8 − 14 ν ₀ 8 − 14 ν ₀ −2 0 0 0



 

 

S ⁰¹¹⁰⁰² _α = 4 π a ⁷ _α 11025 µ 0 (1 − ν 0 )



 

 

0 0 0 −1 0 0

0 0 0 9 − 12 ν ₀ 0 0

0 0 0 4 − 12 ν ₀ 0 0

−1 9 − 12 ν 0 4 − 12 ν 0 0 0 0

0 0 0 0 0 5 − 6 ν 0

0 0 0 0 5 − 6 ν 0 0



 

 

S ⁰¹¹⁰¹¹ _α = 2 π a ⁷ _α 2205 µ ₀ (1 − ν ₀ )



 

 

5 − 6 ν 0 −1 −1 0 0 0

−1 13 − 18 ν 0 −3 0 0 0

−1 −3 13 − 18 ν 0 0 0 0

0 0 0 6 − 9 ν 0 0 0

0 0 0 0 5 − 6 ν 0 0

0 0 0 0 0 5 − 6 ν 0



 

 

S ⁰¹¹⁰²⁰ _α = 4 π a ⁷ _α 11025 µ 0 (1 − ν 0 )



 

 

0 0 0 − 1 0 0

0 0 0 4 − 12 ν ₀ 0 0

0 0 0 9 − 12 ν ₀ 0 0

−1 4 − 12 ν ₀ 9 − 12 ν ₀ 0 0 0

0 0 0 0 0 5 − 6 ν ₀

0 0 0 0 5 − 6 ν 0 0



 

 

S ⁰¹¹¹⁰¹ _α = π a ⁷ _α 2205 µ ₀ (1 − ν ₀ )



 

 

0 0 0 0 0 4 − 6 ν 0

0 0 0 0 0 −2

0 0 0 0 1 − 3 ν 0 0

0 0 0 1 − 3 ν 0 0 0

4 − 6 ν 0 4 − 6 ν 0 −2 0 0 0



 

 

S ⁰¹¹¹¹⁰ _α = π a ⁷ _α 2205 µ 0 (1 − ν 0 )



 

 

0 0 0 0 4 − 6 ν 0 0

0 0 0 0 − 2 0

0 0 0 0 4 − 6 ν ₀ 0

0 0 0 0 0 1 − 3 ν ₀

4 − 6 ν ₀ −2 4 − 6 ν ₀ 0 0 0

0 0 0 1 − 3 ν ₀ 0 0



 

 

S ⁰¹¹²⁰⁰ _α = 2 π a ⁷ _α 11025 µ ₀ (1 − ν ₀ )



 

 

0 0 0 −2 0 0

0 0 0 6 ν 0 − 2 0 0

−2 6 ν 0 − 2 6 ν 0 − 2 0 0 0

0 0 0 0 0 3 ν 0 − 5

0 0 0 0 3 ν 0 − 5 0



 

 

(7)

S ⁰²⁰⁰⁰⁰ _α = 2 π a ⁵ _α 225 µ 0 (1 − ν 0 )



 

 

7 − 10 ν 0 −1 −1 0 0 0

− 1 7 − 10 ν ₀ − 1 0 0 0

− 1 − 1 7 − 10 ν ₀ 0 0 0

0 0 0 4 − 5 ν ₀ 0 0

0 0 0 0 4 − 5 ν ₀ 0

0 0 0 0 0 4 − 5 ν ₀



 

 

S ⁰²⁰⁰⁰² _α = 2 π a ⁷ _α 11025 µ ₀ (1 − ν ₀ )



 

 

57 − 86 ν 0 −7 −7 0 0 0

−7 49 − 62 ν 0 −11 0 0 0

−7 −11 49 − 62 ν 0 0 0 0

0 0 0 20 − 31 ν 0 0 0

0 0 0 0 30 − 37 ν 0 0

0 0 0 0 0 30 − 37 ν 0



 

 

S ⁰²⁰⁰¹¹ _α = 4 π a ⁷ _α 11025 µ 0 (1 − ν 0 )



 

 

0 0 0 −1 0 0

0 0 0 4 − 12 ν 0 0 0

0 0 0 9 − 12 ν 0 0 0

− 1 4 − 12 ν ₀ 9 − 12 ν ₀ 0 0 0

0 0 0 0 0 5 − 6 ν ₀

0 0 0 0 5 − 6 ν ₀ 0



 

 

S ⁰²⁰⁰²⁰ _α = 2 π a ⁷ _α 11025 µ 0 (1 − ν 0 )



 

 

107 − 146 ν ₀ −21 −13 0 0 0

−21 211 − 338 ν ₀ −21 0 0 0

−13 −21 107 − 146 ν ₀ 0 0 0

0 0 0 100 − 121 ν 0 0 0

0 0 0 0 60 − 73 ν 0 0

0 0 0 0 0 100 − 121 ν 0



 

 

S ⁰²⁰¹⁰¹ _α = 2 π a ⁷ _α 11025 µ ₀ (1 − ν ₀ )



 

 

0 0 0 0 6 ν 0 − 2 0

0 0 0 0 −2 0

0 0 0 0 6 ν 0 − 2 0

0 0 0 0 0 3 ν 0 − 5

6 ν 0 − 2 −2 6 ν 0 − 2 0 0 0

0 0 0 3 ν 0 − 5 0 0



 

 

S ⁰²⁰¹¹⁰ _α = 4 π a ⁷ _α 11025 µ 0 (1 − ν 0 )



 

 

0 0 0 0 0 9 − 12 ν ₀

0 0 0 0 0 4 − 12 ν ₀

0 0 0 0 0 −1

0 0 0 0 5 − 6 ν ₀ 0

0 0 0 5 − 6 ν ₀ 0 0

9 − 12 ν 0 4 − 12 ν 0 −1 0 0 0



 

 

S ⁰²⁰²⁰⁰ _α = 2 π a ⁷ _α 11025 µ ₀ (1 − ν ₀ )



 

 

49 − 62 ν 0 −11 −7 0 0 0

−11 49 − 62 ν 0 −7 0 0 0

−7 −7 57 − 86 ν 0 0 0 0

0 0 0 30 − 37 ν 0 0 0

0 0 0 0 30 − 37 ν 0 0

0 0 0 0 0 20 − 31 ν 0



 

 

S ¹⁰⁰⁰⁰¹ _α = π a ⁵ _α 525 µ 0 (1 − ν 0 )



 

 

0 0 0 0 8 − 14 ν 0 0

0 0 0 0 − 2 0

0 0 0 0 8 − 14 ν ₀ 0

0 0 0 0 0 5 − 7 ν ₀

8 − 14 ν ₀ −2 8 − 14 ν ₀ 0 0 0

0 0 0 5 − 7 ν ₀ 0 0



 

 

S ¹⁰⁰⁰¹⁰ _α = π a ⁵ _α 525 µ ₀ (1 − ν ₀ )



 

 

0 0 0 0 0 8 − 14 ν ₀

0 0 0 0 0 8 − 14 ν 0

0 0 0 0 0 −2

0 0 0 0 5 − 7 ν 0 0

0 0 0 5 − 7 ν 0 0 0

8 − 14 ν 0 8 − 14 ν 0 −2 0 0 0



 

 

(8)

S ¹⁰⁰¹⁰⁰ _α = 2 π a ⁵ _α 525 µ 0 (1 − ν 0 )



 

 

27 − 42 ν 0 −3 −3 0 0 0

− 3 11 − 14 ν ₀ − 1 0 0 0

− 3 − 1 11 − 14 ν ₀ 0 0 0

0 0 0 6 − 7 ν ₀ 0 0

0 0 0 0 11 − 14 ν ₀ 0

0 0 0 0 0 11 − 14 ν ₀



 

 

S ¹⁰¹⁰⁰² _α = 4 π a ⁷ _α 11025 µ ₀ (1 − ν ₀ )



 

 

0 0 0 0 9 − 12 ν 0 0

0 0 0 0 −1 0

0 0 0 0 4 − 12 ν 0 0

0 0 0 0 0 5 − 6 ν 0

9 − 12 ν 0 −1 4 − 12 ν 0 0 0 0

0 0 0 5 − 6 ν 0 0 0



 

 

S ¹⁰¹⁰¹¹ _α = π a ⁷ _α 2205 µ 0 (1 − ν 0 )



 

 

0 0 0 0 0 4 − 6 ν 0

0 0 0 0 0 −2

0 0 0 0 1 − 3 ν ₀ 0

0 0 0 1 − 3 ν ₀ 0 0

4 − 6 ν ₀ 4 − 6 ν ₀ −2 0 0 0



 

 

S ¹⁰¹⁰²⁰ _α = 2 π a ⁷ _α 11025 µ 0 (1 − ν 0 )



 

 

0 0 0 0 6 ν ₀ − 2 0

0 0 0 0 −2 0

0 0 0 0 6 ν ₀ − 2 0

0 0 0 0 0 3 ν 0 − 5

6 ν 0 − 2 −2 6 ν 0 − 2 0 0 0

0 0 0 3 ν 0 − 5 0 0



 

 

S ¹⁰¹¹⁰¹ _α = 2 π a ⁷ _α 2205 µ ₀ (1 − ν ₀ )



 

 

13 − 18 ν 0 −1 −3 0 0 0

−1 5 − 6 ν 0 −1 0 0 0

−3 −1 13 − 18 ν 0 0 0 0

0 0 0 5 − 6 ν 0 0 0

0 0 0 0 6 − 9 ν 0 0

0 0 0 0 0 5 − 6 ν 0



 

 

S ¹⁰¹¹¹⁰ _α = π a ⁷ _α 2205 µ 0 (1 − ν 0 )



 

 

0 0 0 − 2 0 0

0 0 0 4 − 6 ν ₀ 0 0

−2 4 − 6 ν ₀ 4 − 6 ν ₀ 0 0 0

0 0 0 0 0 1 − 3 ν ₀

0 0 0 0 1 − 3 ν 0 0



 

 

S ¹⁰¹²⁰⁰ _α = 4 π a ⁷ _α 11025 µ ₀ (1 − ν ₀ )



 

 

0 0 0 0 4 − 12 ν 0 0

0 0 0 0 −1 0

0 0 0 0 9 − 12 ν 0 0

0 0 0 0 0 5 − 6 ν 0

4 − 12 ν 0 −1 9 − 12 ν 0 0 0 0

0 0 0 5 − 6 ν 0 0 0



 

 

S ¹¹⁰⁰⁰² _α = 2 π a ⁷ _α 11025 µ 0 (1 − ν 0 )



 

 

0 0 0 0 0 6 ν 0 − 2

0 0 0 0 0 6 ν ₀ − 2

0 0 0 0 0 − 2

0 0 0 0 3 ν ₀ − 5 0

0 0 0 3 ν ₀ − 5 0 0

6 ν ₀ − 2 6 ν ₀ − 2 −2 0 0 0



 

 

S ¹¹⁰⁰¹¹ _α = π a ⁷ _α 2205 µ ₀ (1 − ν ₀ )



 

 

0 0 0 0 4 − 6 ν ₀ 0

0 0 0 0 −2 0

0 0 0 0 4 − 6 ν 0 0

0 0 0 0 0 1 − 3 ν 0

4 − 6 ν 0 −2 4 − 6 ν 0 0 0 0

0 0 0 1 − 3 ν 0 0 0



 

 

(9)

S ¹¹⁰⁰²⁰ _α = 4 π a ⁷ _α 11025 µ 0 (1 − ν 0 )



 

 

0 0 0 0 0 9 − 12 ν 0

0 0 0 0 0 4 − 12 ν ₀

0 0 0 0 0 − 1

0 0 0 0 5 − 6 ν ₀ 0

0 0 0 5 − 6 ν ₀ 0 0

9 − 12 ν ₀ 4 − 12 ν ₀ −1 0 0 0



 

 

S ¹¹⁰¹⁰¹ _α = π a ⁷ _α 2205 µ ₀ (1 − ν ₀ )



 

 

0 0 0 −2 0 0

0 0 0 4 − 6 ν 0 0 0

−2 4 − 6 ν 0 4 − 6 ν 0 0 0 0

0 0 0 0 0 1 − 3 ν 0

0 0 0 0 1 − 3 ν 0 0



 

 

S ¹¹⁰¹¹⁰ _α = 2 π a ⁷ _α 2205 µ 0 (1 − ν 0 )



 

 

13 − 18 ν 0 −3 −1 0 0 0

−3 13 − 18 ν 0 −1 0 0 0

−1 −1 5 − 6 ν 0 0 0 0

0 0 0 5 − 6 ν ₀ 0 0

0 0 0 0 5 − 6 ν ₀ 0

0 0 0 0 0 6 − 9 ν ₀



 

 

S ¹¹⁰²⁰⁰ _α = 4 π a ⁷ _α 11025 µ 0 (1 − ν 0 )



 

 

0 0 0 0 0 4 − 12 ν ₀

0 0 0 0 0 9 − 12 ν ₀

0 0 0 0 0 −1

0 0 0 0 5 − 6 ν 0 0

0 0 0 5 − 6 ν 0 0 0

4 − 12 ν 0 9 − 12 ν 0 −1 0 0 0



 

 

S ²⁰⁰⁰⁰⁰ _α = 2 π a ⁵ _α 225 µ ₀ (1 − ν ₀ )



 

 

7 − 10 ν 0 −1 −1 0 0 0

−1 7 − 10 ν 0 −1 0 0 0

−1 −1 7 − 10 ν 0 0 0 0

0 0 0 4 − 5 ν 0 0 0

0 0 0 0 4 − 5 ν 0 0

0 0 0 0 0 4 − 5 ν 0



 

 

S ²⁰⁰⁰⁰² _α = 2 π a ⁷ _α 11025 µ 0 (1 − ν 0 )



 

 

49 − 62 ν ₀ − 7 − 11 0 0 0

−7 57 − 86 ν ₀ −7 0 0 0

−11 −7 49 − 62 ν ₀ 0 0 0

0 0 0 30 − 37 ν ₀ 0 0

0 0 0 0 20 − 31 ν ₀ 0

0 0 0 0 0 30 − 37 ν 0



 

 

S ²⁰⁰⁰¹¹ _α = 2 π a ⁷ _α 11025 µ ₀ (1 − ν ₀ )



 

 

0 0 0 −2 0 0

0 0 0 6 ν 0 − 2 0 0

−2 6 ν 0 − 2 6 ν 0 − 2 0 0 0

0 0 0 0 0 3 ν 0 − 5

0 0 0 0 3 ν 0 − 5 0



 

 

S ²⁰⁰⁰²⁰ _α = 2 π a ⁷ _α 11025 µ 0 (1 − ν 0 )



 

 

49 − 62 ν 0 −11 −7 0 0 0

− 11 49 − 62 ν ₀ − 7 0 0 0

− 7 − 7 57 − 86 ν ₀ 0 0 0

0 0 0 30 − 37 ν ₀ 0 0

0 0 0 0 30 − 37 ν ₀ 0

0 0 0 0 0 20 − 31 ν ₀



 

 

S ²⁰⁰¹⁰¹ _α = 4 π a ⁷ _α 11025 µ ₀ (1 − ν ₀ )



 

 

0 0 0 0 4 − 12 ν ₀ 0

0 0 0 0 −1 0

0 0 0 0 9 − 12 ν 0 0

0 0 0 0 0 5 − 6 ν 0

4 − 12 ν 0 −1 9 − 12 ν 0 0 0 0

0 0 0 5 − 6 ν 0 0 0



 

 

(10)

S ²⁰⁰¹¹⁰ _α = 4 π a ⁷ _α 11025 µ 0 (1 − ν 0 )



 

 

0 0 0 0 0 4 − 12 ν 0

0 0 0 0 0 9 − 12 ν ₀

0 0 0 0 0 − 1

0 0 0 0 5 − 6 ν ₀ 0

0 0 0 5 − 6 ν ₀ 0 0

4 − 12 ν ₀ 9 − 12 ν ₀ −1 0 0 0



 

 

S ²⁰⁰²⁰⁰ _α = 2 π a ⁷ _α 11025 µ ₀ (1 − ν ₀ )



 

 

211 − 338 ν 0 −21 −21 0 0 0

−21 107 − 146 ν 0 −13 0 0 0

−21 −13 107 − 146 ν 0 0 0 0

0 0 0 60 − 73 ν 0 0 0

0 0 0 0 100 − 121 ν 0 0

0 0 0 0 0 100 − 121 ν 0



 

 

3 Elementary influence pseudotensors of d-dimensional spheres

3.1 Elementary influence pseudotensors of disks (d = 2)

T ⁰⁰⁰⁰ _αβ = π a ² _α a ² _β 8 µ 0 (1 − ν 0 ) r ²



 



1 − 4 ν ₀ 1 0 1 4 ν ₀ − 3 0

0 0 1



 



− 3 π a ² _α a ² _β a ² _β + a ² _α 16 µ 0 (1 − ν 0 ) r ⁴



 



−1 1 0 1 −1 0

0 0 1



 



T ⁰⁰⁰¹ _αβ = π a ² _α a ⁴ _β 16 µ ₀ (1 − ν ₀ ) r ³



 



4 ν 0 − 1 −1 0

−1 3 − 4 ν 0 0

0 0 −1



 



+ π a ² _α a ⁴ _β

2 a ² _β + 3 a ² _α 16 µ ₀ (1 − ν ₀ ) r ⁵



 



−1 1 0 1 − 1 0

0 0 1



 



T ⁰⁰⁰² _αβ = π a ² _α a ⁴ _β 32 µ 0 (1 − ν 0 ) r ²



 



1 − 4 ν ₀ 1 0 1 4 ν ₀ − 3 0

0 0 1



 



− π a ² _α a ⁶ _β (1 − 2 ν 0 ) 16 µ 0 (1 − ν 0 ) r ⁴



 



−1 0 0

0 1 0

0 0 0



 



+ 3 π a ⁴ _α a ⁴ _β 64 µ 0 (1 − ν 0 ) r ⁴



 



1 −1 0

−1 1 0

0 0 −1



 



−

5 π a ² _α a ⁶ _β

a ² _β + 2 a ² _α 64 µ ₀ (1 − ν ₀ ) r ⁶



 



−1 1 0 1 −1 0

0 0 1



 



T ⁰⁰¹⁰ _αβ = π a ² _α a ⁴ _β 16 µ 0 (1 − ν 0 ) r ³



 



0 0 2 ν ₀ + 1

0 0 2 ν ₀ − 3

2 ν ₀ + 1 2 ν ₀ − 3 0



 



+ π a ² _α a ⁴ _β

2 a ² _β + 3 a ² _α 16 µ 0 (1 − ν 0 ) r ⁵



 



0 0 −1

0 0 1

−1 1 0



 



T ⁰⁰¹¹ _αβ = π a ² _α a ⁶ _β 32 µ 0 (1 − ν 0 ) r ⁴



 



0 0 −2 ν 0 − 1

0 0 3 − 2 ν 0

−2 ν 0 − 1 3 − 2 ν 0 0



 



Self-influence and influence pseudotensors of d-dimensional spheres

HAL Id: hal-00876028

https://hal-enpc.archives-ouvertes.fr/hal-00876028v2

Submitted on 30 Dec 2013

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