Anisotropic Structures - Theory and Design Strutture anisotrope: - PowerPoint PPT Presentation

International Doctorate in Civil and Environmental Engineering Anisotropic Structures - Theory and Design Strutture anisotrope: teoria e progetto Paolo VANNUCCI Lesson 2 - April 9, 2019 - DICEA - Universit´ a di Firenze 1 / 100

Topics of the second lesson • Anisotropic elasticity - Part 2 2 / 100

Reduction of the E ijkl by elastic symmetries We consider now the effects of elastic symmetries on tensor E ; we will see that, depending upon the symmetry, some components E ijkl vanish while some other can become functions of other components. In the end, elastic symmetries reduce the number of the independent Cartesian components of E . It is worth to work on the C ij rather than on the E ijkl because simpler. Before going on, we recall the equations that are needed in the following: • invariance of the strain energy { ε } ⊤ [ C ] { ε } = ([ R ] { ε } ) ⊤ [ C ][ R ] { ε } ∀{ ε } (1) 3 / 100

• orthogonal tensor describing a symmetry with respect to a plane whose normal is n = ( n 1 , n 2 , n 3 ):   1 − 2 n 2 − 2 n 1 n 2 − 2 n 1 n 3 1 U = I − 2 n ⊗ n = 1 − 2 n 2 (2) − 2 n 2 n 3   2   1 − 2 n 2 sym 3 • rotation matrix corresponding, in the Kelvin’s notation, to U :   U 2 U 2 U 2 √ √ √ 2 U 12 U 13 2 U 13 U 11 2 U 11 U 12 11 12 13 U 2 U 2 U 2 √ √ √ 2 U 22 U 23 2 U 23 U 21 2 U 21 U 22 21 22 23   U 2 U 2 U 2 √ √ √ 2 U 32 U 33 2 U 33 U 31 2 U 31 U 32 [ R ] =  31 32 33  √ √ √   2 U 21 U 31 2 U 22 U 32 2 U 23 U 33 U 23 U 32 + U 22 U 33 U 33 U 21 + U 31 U 23 U 31 U 22 + U 32 U 21   √ √ √ 2 U 31 U 11 2 U 32 U 12 2 U 33 U 13 U 32 U 13 + U 33 U 12 U 31 U 13 + U 33 U 11 U 31 U 12 + U 32 U 11 √ √ √ 2 U 11 U 21 2 U 12 U 22 2 U 13 U 23 U 12 U 23 + U 13 U 22 U 11 U 23 + U 13 U 21 U 11 U 22 + U 12 U 21 (3) 4 / 100

Triclinic bodies A triclinic body has no material symmetries, so eq. (1) cannot be written → it is not possible to reduce the number of independent elastic components, that remains fixed to 21:   C 11 C 12 C 13 C 14 C 15 C 16   C 22 C 23 C 24 C 25 C 26     C 33 C 34 C 35 C 36   [ C ] = . (4)     C 44 C 45 C 46     sym C 55 C 56     C 66 5 / 100

Monoclinic bodies The only symmetry of a monoclinic body is a reflection in a plane. Without loss in generality, we can suppose to be x 3 = 0 the symmetry plane ⇒ n = (0 , 0 , 1). In such a case it is, see eqs. (2) and (3),   1 0 0 0 0 0   0 1 0 0 0 0     1 0 0   0 0 1 0 0 0   U =  ⇒ [ R ] = , (5)  0 1 0       0 0 0 − 1 0 0   0 0 − 1   0 0 0 0 − 1 0     0 0 0 0 0 1 that applied to eq. (1) gives the condition 6 / 100

C 14 ε 1 ε 4 + C 24 ε 2 ε 4 + C 34 ε 3 ε 4 + C 15 ε 1 ε 5 + ∀ ε ⇐ ⇒ (6) C 25 ε 2 ε 5 + C 35 ε 3 ε 5 + C 46 ε 4 ε 6 + C 56 ε 5 ε 6 = 0 , C 14 = C 24 = C 34 = C 15 = C 25 = C 35 = C 46 = C 56 = 0 . (7) Hence, a monoclinic body depends upon only 13 distinct elastic moduli:   C 11 C 12 C 13 0 0 C 16   C 22 C 23 0 0 C 26     C 33 0 0 C 36   [ C ] = . (8)     C 44 C 45 0     sym C 55 0     C 66 7 / 100

Orthotropic bodies Let us now add another plane of symmetry orthogonal to the previous one, say the plane x 2 = 0 ⇒ n = (0 , 1 , 0). With the same procedure, we get successively:   1 0 0 0 0 0   0 1 0 0 0 0     1 0 0   0 0 1 0 0 0   U =  ⇒ [ R ] = (9)  0 − 1 0       0 0 0 − 1 0 0   0 0 1   0 0 0 0 1 0     0 0 0 0 0 − 1 8 / 100

( C 14 ε 1 + C 24 ε 2 + C 34 ε 3 + C 45 ε 5 ) ε 4 + (10) ( C 16 ε 1 + C 26 ε 2 + C 36 ε 3 + C 56 ε 5 ) ε 6 = 0 ∀ ε ⇐ ⇒ C 14 = C 24 = C 34 = C 45 = C 16 = C 26 = C 36 = C 56 = 0 . So, the existence of the second plane of symmetry has added the four supplementary conditions C 16 = C 26 = C 36 = C 45 = 0 (11) to the previous eight ones, reducing hence to only 9 the number of distinct elastic moduli. Let us now suppose the existence of a third plane of symmetry, orthogonal to the previous ones, the plane x 1 = 0 ⇒ n = (1 , 0 , 0). With the same procedure, we get: 9 / 100

  1 0 0 0 0 0   0 1 0 0 0 0     − 1 0 0   0 0 1 0 0 0   U =  ⇒ [ R ] = , (12) 0 1 0        0 0 0 1 0 0   0 0 1   0 0 0 0 − 1 0     0 0 0 0 0 − 1 ( C 15 ε 1 + C 25 ε 2 + C 35 ε 3 + C 45 ε 4 ) ε 5 + (13) ( C 16 ε 1 + C 26 ε 2 + C 36 ε 3 + C 46 ε 4 ) ε 6 = 0 ∀ ε ⇐ ⇒ C 15 = C 25 = C 35 = C 45 = C 16 = C 26 = C 36 = C 46 = 0 . Rather surprisingly, this new symmetry condition does not give any supplementary condition to those in (7) and (11). ⇒ the existence of 2 orthogonal planes of elastic symmetry is physically impossible: only the presence of 1 or 3 mutually orthogonal planes of symmetry is admissible. 10 / 100

The class of orthotropic materials is very important, because a lot of materials or structures belong to it. An orthotropic material depends hence upon 9 distinct elastic moduli and its matrix [C] looks like   C 11 C 12 C 13 0 0 0   C 22 C 23 0 0 0     C 33 0 0 0   [ C ] = . (14)     C 44 0 0     sym C 55 0     C 66 11 / 100

Axially symmetric bodies There are only 4 possible cases of axial symmetries for crystals: the 2-, 3-, 4- and 6-fold axis of symmetry (say x 3 ). Let us begin with a 2-fold axis of symmetry; the covering operation corresponds hence to a rotation of π about x 3 →   1 0 0 0 0 0   0 1 0 0 0 0     − 1 0 0   0 0 1 0 0 0   U =  ⇒ [ R ] = , (15) 0 − 1 0        0 0 0 − 1 0 0   0 0 1   0 0 0 0 − 1 0     0 0 0 0 0 1 and we can observe that [ R ] is the same of the monoclinic case → a 2-fold axis of symmetry coincides with a plane of symmetry. 12 / 100

For a 3-fold axis of symmetry, the covering operation corresponds to a rotation of 2 π/ 3 about x 3 →   � 1 3 3 0 0 0 − 4 4 8 √ �   3 1 3 0 0 0  − 1 3  0   4 4 8 2 2   √ 0 0 1 0 0 0   3 − 1 U =  ⇒ [ R ] =  − 0    2 2 √  − 1 3 0 0 0 0  −  2 2 0 0 1   √ 3 − 1  0 0 0 0  2 2   � � 3 3 − 1 − 0 0 0 8 8 2 (16) and condition (1) gives 14 conditions on the components of [ C ]: C 16 = C 26 = C 34 = C 35 = C 36 = C 45 = 0 , C 22 = C 11 , C 55 = C 44 , C 23 = C 13 , C 24 = − C 14 , (17) √ √ C 25 = − C 15 , C 56 = 2 C 14 , C 46 = 2 C 15 , C 66 = C 11 − C 12 13 / 100

So, there are only 7 distinct elastic moduli:   C 11 C 12 C 13 C 14 C 15 0   C 11 C 13 − C 14 − C 15 0     C 33 0 0 0   [ C ] = . (18) √     C 44 0 − 2 C 15   √   sym C 44 2 C 14     C 11 − C 12 14 / 100

For a 4-fold axis of symmetry, the covering operation corresponds to a rotation of π/ 2 about x 3 →   0 1 0 0 0 0   1 0 0 0 0 0     0 1 0   0 0 1 0 0 0   U =  ⇒ [ R ] = . (19) − 1 0 0        0 0 0 0 − 1 0   0 0 1   0 0 0 1 0 0     0 0 0 0 0 − 1 The result are 14 conditions different from the (17): C 14 = C 24 = C 34 = C 15 = C 25 = C 35 = (20) C 45 = C 36 = C 46 = C 56 = 0 , C 22 = C 11 , C 55 = C 44 , C 23 = C 13 , C 26 = − C 16 15 / 100

This gives an elastic matrix [ C ] still depending upon only 7 distinct moduli, but different from the previous case:   C 11 C 12 C 13 0 0 C 16   C 11 C 13 0 0 − C 16     C 33 0 0 0   [ C ] = . (21)     C 44 0 0     sym C 44 0     C 66 16 / 100

The last case of 6-fold axis of symmetry has as covering operation a rotation of π/ 3 about x 3 →  �  1 3 3 0 0 0 4 4 8 √ �   3 1 3 0 0 0 −  1 3  0  4 4 8  2 2   √ 0 0 1 0 0 0   U = 3 1  ⇒ [ R ] =  − 0    √ 2 2  1 3 0 0 0 − 0   2 2 0 0 1   √ 3 1  0 0 0 0  2 2   � � 3 3 − 1 0 0 0 − 8 8 2 (22) The result are 16 conditions: C 14 = C 24 = C 34 = C 15 = C 25 = C 35 = (23) C 45 = C 16 = C 26 = C 36 = C 46 = C 56 = 0 , C 22 = C 11 , C 55 = C 44 , C 23 = C 13 , C 66 = C 11 − C 12 17 / 100

Finally, the elastic matrix [ C ] depends upon only 5 moduli:   C 11 C 12 C 13 0 0 0   C 11 C 13 0 0 0     C 33 0 0 0   [ C ] = . (24)     C 44 0 0     sym C 44 0     C 11 − C 12 18 / 100